3.7.45 \(\int \frac {1}{x^3 (a+b \sqrt {c+d x})^2} \, dx\) [645]

Optimal. Leaf size=306 \[ \frac {a b^2 \left (a^2+11 b^2 c\right ) d^2}{2 c \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )}-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2 \left (a+b \sqrt {c+d x}\right )}-\frac {b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )^2 x \left (a+b \sqrt {c+d x}\right )}-\frac {a b \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2} \left (a^2-b^2 c\right )^4}+\frac {b^4 \left (5 a^2+b^2 c\right ) d^2 \log (x)}{\left (a^2-b^2 c\right )^4}-\frac {2 b^4 \left (5 a^2+b^2 c\right ) d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^4} \]

[Out]

-1/2*a*b*(-15*b^4*c^2-10*a^2*b^2*c+a^4)*d^2*arctanh((d*x+c)^(1/2)/c^(1/2))/c^(3/2)/(-b^2*c+a^2)^4+b^4*(b^2*c+5
*a^2)*d^2*ln(x)/(-b^2*c+a^2)^4-2*b^4*(b^2*c+5*a^2)*d^2*ln(a+b*(d*x+c)^(1/2))/(-b^2*c+a^2)^4+1/2*a*b^2*(11*b^2*
c+a^2)*d^2/c/(-b^2*c+a^2)^3/(a+b*(d*x+c)^(1/2))+1/2*(-a+b*(d*x+c)^(1/2))/(-b^2*c+a^2)/x^2/(a+b*(d*x+c)^(1/2))-
1/2*b*d*(3*a*b*c-(2*b^2*c+a^2)*(d*x+c)^(1/2))/c/(-b^2*c+a^2)^2/x/(a+b*(d*x+c)^(1/2))

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Rubi [A]
time = 0.29, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {378, 1412, 837, 815, 649, 212, 266} \begin {gather*} \frac {a b^2 d^2 \left (a^2+11 b^2 c\right )}{2 c \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )}-\frac {a-b \sqrt {c+d x}}{2 x^2 \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )}-\frac {b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt {c+d x}\right )}{2 c x \left (a^2-b^2 c\right )^2 \left (a+b \sqrt {c+d x}\right )}+\frac {b^4 d^2 \log (x) \left (5 a^2+b^2 c\right )}{\left (a^2-b^2 c\right )^4}-\frac {2 b^4 d^2 \left (5 a^2+b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^4}-\frac {a b d^2 \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2} \left (a^2-b^2 c\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*Sqrt[c + d*x])^2),x]

[Out]

(a*b^2*(a^2 + 11*b^2*c)*d^2)/(2*c*(a^2 - b^2*c)^3*(a + b*Sqrt[c + d*x])) - (a - b*Sqrt[c + d*x])/(2*(a^2 - b^2
*c)*x^2*(a + b*Sqrt[c + d*x])) - (b*d*(3*a*b*c - (a^2 + 2*b^2*c)*Sqrt[c + d*x]))/(2*c*(a^2 - b^2*c)^2*x*(a + b
*Sqrt[c + d*x])) - (a*b*(a^4 - 10*a^2*b^2*c - 15*b^4*c^2)*d^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*c^(3/2)*(a^2
- b^2*c)^4) + (b^4*(5*a^2 + b^2*c)*d^2*Log[x])/(a^2 - b^2*c)^4 - (2*b^4*(5*a^2 + b^2*c)*d^2*Log[a + b*Sqrt[c +
d*x]])/(a^2 - b^2*c)^4

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 378

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 1412

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \left (a+b \sqrt {c+d x}\right )^2} \, dx &=d^2 \text {Subst}\left (\int \frac {1}{\left (a+b \sqrt {x}\right )^2 (-c+x)^3} \, dx,x,c+d x\right )\\ &=\left (2 d^2\right ) \text {Subst}\left (\int \frac {x}{(a+b x)^2 \left (-c+x^2\right )^3} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2 \left (a+b \sqrt {c+d x}\right )}+\frac {d^2 \text {Subst}\left (\int \frac {-2 a b c+4 b^2 c x}{(a+b x)^2 \left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )}\\ &=-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2 \left (a+b \sqrt {c+d x}\right )}-\frac {b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )^2 x \left (a+b \sqrt {c+d x}\right )}+\frac {d^2 \text {Subst}\left (\int \frac {2 a b c \left (a^2-7 b^2 c\right )+4 b^2 c \left (a^2+2 b^2 c\right ) x}{(a+b x)^2 \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )}{4 c^2 \left (a^2-b^2 c\right )^2}\\ &=-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2 \left (a+b \sqrt {c+d x}\right )}-\frac {b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )^2 x \left (a+b \sqrt {c+d x}\right )}+\frac {d^2 \text {Subst}\left (\int \left (-\frac {2 a b^3 c \left (a^2+11 b^2 c\right )}{\left (a^2-b^2 c\right ) (a+b x)^2}-\frac {8 b^5 c^2 \left (5 a^2+b^2 c\right )}{\left (-a^2+b^2 c\right )^2 (a+b x)}+\frac {2 b c \left (-a \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right )-4 b^3 c \left (5 a^2+b^2 c\right ) x\right )}{\left (a^2-b^2 c\right )^2 \left (c-x^2\right )}\right ) \, dx,x,\sqrt {c+d x}\right )}{4 c^2 \left (a^2-b^2 c\right )^2}\\ &=\frac {a b^2 \left (a^2+11 b^2 c\right ) d^2}{2 c \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )}-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2 \left (a+b \sqrt {c+d x}\right )}-\frac {b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )^2 x \left (a+b \sqrt {c+d x}\right )}-\frac {2 b^4 \left (5 a^2+b^2 c\right ) d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^4}+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {-a \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right )-4 b^3 c \left (5 a^2+b^2 c\right ) x}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )^4}\\ &=\frac {a b^2 \left (a^2+11 b^2 c\right ) d^2}{2 c \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )}-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2 \left (a+b \sqrt {c+d x}\right )}-\frac {b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )^2 x \left (a+b \sqrt {c+d x}\right )}-\frac {2 b^4 \left (5 a^2+b^2 c\right ) d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^4}-\frac {\left (2 b^4 \left (5 a^2+b^2 c\right ) d^2\right ) \text {Subst}\left (\int \frac {x}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^4}-\frac {\left (a b \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )^4}\\ &=\frac {a b^2 \left (a^2+11 b^2 c\right ) d^2}{2 c \left (a^2-b^2 c\right )^3 \left (a+b \sqrt {c+d x}\right )}-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2 \left (a+b \sqrt {c+d x}\right )}-\frac {b d \left (3 a b c-\left (a^2+2 b^2 c\right ) \sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )^2 x \left (a+b \sqrt {c+d x}\right )}-\frac {a b \left (a^4-10 a^2 b^2 c-15 b^4 c^2\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2} \left (a^2-b^2 c\right )^4}+\frac {b^4 \left (5 a^2+b^2 c\right ) d^2 \log (x)}{\left (a^2-b^2 c\right )^4}-\frac {2 b^4 \left (5 a^2+b^2 c\right ) d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^4}\\ \end {align*}

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Mathematica [A]
time = 0.60, size = 301, normalized size = 0.98 \begin {gather*} \frac {1}{2} \left (\frac {a^5 c-b^5 c^2 (c-2 d x) \sqrt {c+d x}+a^2 b^3 c (2 c-d x) \sqrt {c+d x}-a^4 b (c+d x)^{3/2}+a b^4 c \left (c^2-3 c d x-11 d^2 x^2\right )-a^3 b^2 \left (2 c^2-3 c d x+d^2 x^2\right )}{c \left (-a^2+b^2 c\right )^3 x^2 \left (a+b \sqrt {c+d x}\right )}+\frac {\left (-a^5 b+10 a^3 b^3 c+15 a b^5 c^2\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2} \left (a^2-b^2 c\right )^4}+\frac {2 b^4 \left (5 a^2+b^2 c\right ) d^2 \log (-d x)}{\left (a^2-b^2 c\right )^4}-\frac {4 b^4 \left (5 a^2+b^2 c\right ) d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*Sqrt[c + d*x])^2),x]

[Out]

((a^5*c - b^5*c^2*(c - 2*d*x)*Sqrt[c + d*x] + a^2*b^3*c*(2*c - d*x)*Sqrt[c + d*x] - a^4*b*(c + d*x)^(3/2) + a*
b^4*c*(c^2 - 3*c*d*x - 11*d^2*x^2) - a^3*b^2*(2*c^2 - 3*c*d*x + d^2*x^2))/(c*(-a^2 + b^2*c)^3*x^2*(a + b*Sqrt[
c + d*x])) + ((-(a^5*b) + 10*a^3*b^3*c + 15*a*b^5*c^2)*d^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(c^(3/2)*(a^2 - b^2
*c)^4) + (2*b^4*(5*a^2 + b^2*c)*d^2*Log[-(d*x)])/(a^2 - b^2*c)^4 - (4*b^4*(5*a^2 + b^2*c)*d^2*Log[a + b*Sqrt[c
+ d*x]])/(a^2 - b^2*c)^4)/2

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Maple [A]
time = 0.07, size = 303, normalized size = 0.99

method	result	size
derivativedivides	\(2 d^{2} \left (-\frac {\frac {-\frac {a b \left (-7 b^{4} c^{2}+6 a^{2} b^{2} c +a^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 c}+\left (-\frac {1}{2} c^{2} b^{6}-a^{2} b^{4} c +\frac {3}{2} a^{4} b^{2}\right ) \left (d x +c \right )+\left (-\frac {9}{4} a \,b^{5} c^{2}+\frac {5}{2} a^{3} b^{3} c -\frac {1}{4} a^{5} b \right ) \sqrt {d x +c}+\frac {3 c^{3} b^{6}}{4}+\frac {3 a^{2} b^{4} c^{2}}{4}-\frac {7 a^{4} b^{2} c}{4}+\frac {a^{6}}{4}}{d^{2} x^{2}}+\frac {b \left (-\frac {\left (4 c^{2} b^{5}+20 a^{2} b^{3} c \right ) \ln \left (-d x \right )}{2}+\frac {\left (-15 a \,b^{4} c^{2}-10 a^{3} b^{2} c +a^{5}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{4 c}}{\left (-b^{2} c +a^{2}\right )^{4}}+\frac {b^{4} a}{\left (-b^{2} c +a^{2}\right )^{3} \left (a +b \sqrt {d x +c}\right )}-\frac {b^{4} \left (b^{2} c +5 a^{2}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{\left (-b^{2} c +a^{2}\right )^{4}}\right )\)	\(303\)
default	\(2 d^{2} \left (-\frac {\frac {-\frac {a b \left (-7 b^{4} c^{2}+6 a^{2} b^{2} c +a^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{4 c}+\left (-\frac {1}{2} c^{2} b^{6}-a^{2} b^{4} c +\frac {3}{2} a^{4} b^{2}\right ) \left (d x +c \right )+\left (-\frac {9}{4} a \,b^{5} c^{2}+\frac {5}{2} a^{3} b^{3} c -\frac {1}{4} a^{5} b \right ) \sqrt {d x +c}+\frac {3 c^{3} b^{6}}{4}+\frac {3 a^{2} b^{4} c^{2}}{4}-\frac {7 a^{4} b^{2} c}{4}+\frac {a^{6}}{4}}{d^{2} x^{2}}+\frac {b \left (-\frac {\left (4 c^{2} b^{5}+20 a^{2} b^{3} c \right ) \ln \left (-d x \right )}{2}+\frac {\left (-15 a \,b^{4} c^{2}-10 a^{3} b^{2} c +a^{5}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{4 c}}{\left (-b^{2} c +a^{2}\right )^{4}}+\frac {b^{4} a}{\left (-b^{2} c +a^{2}\right )^{3} \left (a +b \sqrt {d x +c}\right )}-\frac {b^{4} \left (b^{2} c +5 a^{2}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{\left (-b^{2} c +a^{2}\right )^{4}}\right )\)	\(303\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a+b*(d*x+c)^(1/2))^2,x,method=_RETURNVERBOSE)

[Out]

2*d^2*(-1/(-b^2*c+a^2)^4*((-1/4*a*b*(-7*b^4*c^2+6*a^2*b^2*c+a^4)/c*(d*x+c)^(3/2)+(-1/2*c^2*b^6-a^2*b^4*c+3/2*a
^4*b^2)*(d*x+c)+(-9/4*a*b^5*c^2+5/2*a^3*b^3*c-1/4*a^5*b)*(d*x+c)^(1/2)+3/4*c^3*b^6+3/4*a^2*b^4*c^2-7/4*a^4*b^2
*c+1/4*a^6)/d^2/x^2+1/4*b/c*(-1/2*(4*b^5*c^2+20*a^2*b^3*c)*ln(-d*x)+(-15*a*b^4*c^2-10*a^3*b^2*c+a^5)/c^(1/2)*a
rctanh((d*x+c)^(1/2)/c^(1/2))))+b^4/(-b^2*c+a^2)^3*a/(a+b*(d*x+c)^(1/2))-b^4*(b^2*c+5*a^2)/(-b^2*c+a^2)^4*ln(a
+b*(d*x+c)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (289) = 578\).
time = 0.57, size = 659, normalized size = 2.15 \begin {gather*} \frac {1}{4} \, d^{2} {\left (\frac {4 \, {\left (b^{6} c + 5 \, a^{2} b^{4}\right )} \log \left (d x\right )}{b^{8} c^{4} - 4 \, a^{2} b^{6} c^{3} + 6 \, a^{4} b^{4} c^{2} - 4 \, a^{6} b^{2} c + a^{8}} - \frac {8 \, {\left (b^{6} c + 5 \, a^{2} b^{4}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{8} c^{4} - 4 \, a^{2} b^{6} c^{3} + 6 \, a^{4} b^{4} c^{2} - 4 \, a^{6} b^{2} c + a^{8}} - \frac {{\left (15 \, a b^{5} c^{2} + 10 \, a^{3} b^{3} c - a^{5} b\right )} \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{{\left (b^{8} c^{5} - 4 \, a^{2} b^{6} c^{4} + 6 \, a^{4} b^{4} c^{3} - 4 \, a^{6} b^{2} c^{2} + a^{8} c\right )} \sqrt {c}} - \frac {2 \, {\left (7 \, a b^{4} c^{3} + 6 \, a^{3} b^{2} c^{2} - a^{5} c + {\left (11 \, a b^{4} c + a^{3} b^{2}\right )} {\left (d x + c\right )}^{2} - {\left (2 \, b^{5} c^{2} - a^{2} b^{3} c - a^{4} b\right )} {\left (d x + c\right )}^{\frac {3}{2}} - {\left (19 \, a b^{4} c^{2} + 5 \, a^{3} b^{2} c\right )} {\left (d x + c\right )} + 3 \, {\left (b^{5} c^{3} - a^{2} b^{3} c^{2}\right )} \sqrt {d x + c}\right )}}{a b^{6} c^{6} - 3 \, a^{3} b^{4} c^{5} + 3 \, a^{5} b^{2} c^{4} - a^{7} c^{3} + {\left (b^{7} c^{4} - 3 \, a^{2} b^{5} c^{3} + 3 \, a^{4} b^{3} c^{2} - a^{6} b c\right )} {\left (d x + c\right )}^{\frac {5}{2}} + {\left (a b^{6} c^{4} - 3 \, a^{3} b^{4} c^{3} + 3 \, a^{5} b^{2} c^{2} - a^{7} c\right )} {\left (d x + c\right )}^{2} - 2 \, {\left (b^{7} c^{5} - 3 \, a^{2} b^{5} c^{4} + 3 \, a^{4} b^{3} c^{3} - a^{6} b c^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 2 \, {\left (a b^{6} c^{5} - 3 \, a^{3} b^{4} c^{4} + 3 \, a^{5} b^{2} c^{3} - a^{7} c^{2}\right )} {\left (d x + c\right )} + {\left (b^{7} c^{6} - 3 \, a^{2} b^{5} c^{5} + 3 \, a^{4} b^{3} c^{4} - a^{6} b c^{3}\right )} \sqrt {d x + c}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*(d*x+c)^(1/2))^2,x, algorithm="maxima")

[Out]

1/4*d^2*(4*(b^6*c + 5*a^2*b^4)*log(d*x)/(b^8*c^4 - 4*a^2*b^6*c^3 + 6*a^4*b^4*c^2 - 4*a^6*b^2*c + a^8) - 8*(b^6
*c + 5*a^2*b^4)*log(sqrt(d*x + c)*b + a)/(b^8*c^4 - 4*a^2*b^6*c^3 + 6*a^4*b^4*c^2 - 4*a^6*b^2*c + a^8) - (15*a
*b^5*c^2 + 10*a^3*b^3*c - a^5*b)*log((sqrt(d*x + c) - sqrt(c))/(sqrt(d*x + c) + sqrt(c)))/((b^8*c^5 - 4*a^2*b^
6*c^4 + 6*a^4*b^4*c^3 - 4*a^6*b^2*c^2 + a^8*c)*sqrt(c)) - 2*(7*a*b^4*c^3 + 6*a^3*b^2*c^2 - a^5*c + (11*a*b^4*c
+ a^3*b^2)*(d*x + c)^2 - (2*b^5*c^2 - a^2*b^3*c - a^4*b)*(d*x + c)^(3/2) - (19*a*b^4*c^2 + 5*a^3*b^2*c)*(d*x
+ c) + 3*(b^5*c^3 - a^2*b^3*c^2)*sqrt(d*x + c))/(a*b^6*c^6 - 3*a^3*b^4*c^5 + 3*a^5*b^2*c^4 - a^7*c^3 + (b^7*c^
4 - 3*a^2*b^5*c^3 + 3*a^4*b^3*c^2 - a^6*b*c)*(d*x + c)^(5/2) + (a*b^6*c^4 - 3*a^3*b^4*c^3 + 3*a^5*b^2*c^2 - a^
7*c)*(d*x + c)^2 - 2*(b^7*c^5 - 3*a^2*b^5*c^4 + 3*a^4*b^3*c^3 - a^6*b*c^2)*(d*x + c)^(3/2) - 2*(a*b^6*c^5 - 3*
a^3*b^4*c^4 + 3*a^5*b^2*c^3 - a^7*c^2)*(d*x + c) + (b^7*c^6 - 3*a^2*b^5*c^5 + 3*a^4*b^3*c^4 - a^6*b*c^3)*sqrt(
d*x + c)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (289) = 578\).
time = 1.18, size = 1252, normalized size = 4.09 \begin {gather*} \left [-\frac {2 \, b^{8} c^{6} - 4 \, a^{2} b^{6} c^{5} + 4 \, a^{6} b^{2} c^{3} - 2 \, a^{8} c^{2} - 4 \, {\left (b^{8} c^{4} + 4 \, a^{2} b^{6} c^{3} - 5 \, a^{4} b^{4} c^{2}\right )} d^{2} x^{2} - 2 \, {\left (b^{8} c^{5} + 3 \, a^{2} b^{6} c^{4} - 9 \, a^{4} b^{4} c^{3} + 5 \, a^{6} b^{2} c^{2}\right )} d x - {\left ({\left (15 \, a b^{7} c^{2} + 10 \, a^{3} b^{5} c - a^{5} b^{3}\right )} d^{3} x^{3} + {\left (15 \, a b^{7} c^{3} - 5 \, a^{3} b^{5} c^{2} - 11 \, a^{5} b^{3} c + a^{7} b\right )} d^{2} x^{2}\right )} \sqrt {c} \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 8 \, {\left ({\left (b^{8} c^{3} + 5 \, a^{2} b^{6} c^{2}\right )} d^{3} x^{3} + {\left (b^{8} c^{4} + 4 \, a^{2} b^{6} c^{3} - 5 \, a^{4} b^{4} c^{2}\right )} d^{2} x^{2}\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left ({\left (b^{8} c^{3} + 5 \, a^{2} b^{6} c^{2}\right )} d^{3} x^{3} + {\left (b^{8} c^{4} + 4 \, a^{2} b^{6} c^{3} - 5 \, a^{4} b^{4} c^{2}\right )} d^{2} x^{2}\right )} \log \left (x\right ) - 2 \, {\left (2 \, a b^{7} c^{5} - 6 \, a^{3} b^{5} c^{4} + 6 \, a^{5} b^{3} c^{3} - 2 \, a^{7} b c^{2} - {\left (11 \, a b^{7} c^{3} - 10 \, a^{3} b^{5} c^{2} - a^{5} b^{3} c\right )} d^{2} x^{2} - {\left (5 \, a b^{7} c^{4} - 9 \, a^{3} b^{5} c^{3} + 3 \, a^{5} b^{3} c^{2} + a^{7} b c\right )} d x\right )} \sqrt {d x + c}}{4 \, {\left ({\left (b^{10} c^{6} - 4 \, a^{2} b^{8} c^{5} + 6 \, a^{4} b^{6} c^{4} - 4 \, a^{6} b^{4} c^{3} + a^{8} b^{2} c^{2}\right )} d x^{3} + {\left (b^{10} c^{7} - 5 \, a^{2} b^{8} c^{6} + 10 \, a^{4} b^{6} c^{5} - 10 \, a^{6} b^{4} c^{4} + 5 \, a^{8} b^{2} c^{3} - a^{10} c^{2}\right )} x^{2}\right )}}, -\frac {b^{8} c^{6} - 2 \, a^{2} b^{6} c^{5} + 2 \, a^{6} b^{2} c^{3} - a^{8} c^{2} - 2 \, {\left (b^{8} c^{4} + 4 \, a^{2} b^{6} c^{3} - 5 \, a^{4} b^{4} c^{2}\right )} d^{2} x^{2} - {\left (b^{8} c^{5} + 3 \, a^{2} b^{6} c^{4} - 9 \, a^{4} b^{4} c^{3} + 5 \, a^{6} b^{2} c^{2}\right )} d x + {\left ({\left (15 \, a b^{7} c^{2} + 10 \, a^{3} b^{5} c - a^{5} b^{3}\right )} d^{3} x^{3} + {\left (15 \, a b^{7} c^{3} - 5 \, a^{3} b^{5} c^{2} - 11 \, a^{5} b^{3} c + a^{7} b\right )} d^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + 4 \, {\left ({\left (b^{8} c^{3} + 5 \, a^{2} b^{6} c^{2}\right )} d^{3} x^{3} + {\left (b^{8} c^{4} + 4 \, a^{2} b^{6} c^{3} - 5 \, a^{4} b^{4} c^{2}\right )} d^{2} x^{2}\right )} \log \left (\sqrt {d x + c} b + a\right ) - 2 \, {\left ({\left (b^{8} c^{3} + 5 \, a^{2} b^{6} c^{2}\right )} d^{3} x^{3} + {\left (b^{8} c^{4} + 4 \, a^{2} b^{6} c^{3} - 5 \, a^{4} b^{4} c^{2}\right )} d^{2} x^{2}\right )} \log \left (x\right ) - {\left (2 \, a b^{7} c^{5} - 6 \, a^{3} b^{5} c^{4} + 6 \, a^{5} b^{3} c^{3} - 2 \, a^{7} b c^{2} - {\left (11 \, a b^{7} c^{3} - 10 \, a^{3} b^{5} c^{2} - a^{5} b^{3} c\right )} d^{2} x^{2} - {\left (5 \, a b^{7} c^{4} - 9 \, a^{3} b^{5} c^{3} + 3 \, a^{5} b^{3} c^{2} + a^{7} b c\right )} d x\right )} \sqrt {d x + c}}{2 \, {\left ({\left (b^{10} c^{6} - 4 \, a^{2} b^{8} c^{5} + 6 \, a^{4} b^{6} c^{4} - 4 \, a^{6} b^{4} c^{3} + a^{8} b^{2} c^{2}\right )} d x^{3} + {\left (b^{10} c^{7} - 5 \, a^{2} b^{8} c^{6} + 10 \, a^{4} b^{6} c^{5} - 10 \, a^{6} b^{4} c^{4} + 5 \, a^{8} b^{2} c^{3} - a^{10} c^{2}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*(d*x+c)^(1/2))^2,x, algorithm="fricas")

[Out]

[-1/4*(2*b^8*c^6 - 4*a^2*b^6*c^5 + 4*a^6*b^2*c^3 - 2*a^8*c^2 - 4*(b^8*c^4 + 4*a^2*b^6*c^3 - 5*a^4*b^4*c^2)*d^2
*x^2 - 2*(b^8*c^5 + 3*a^2*b^6*c^4 - 9*a^4*b^4*c^3 + 5*a^6*b^2*c^2)*d*x - ((15*a*b^7*c^2 + 10*a^3*b^5*c - a^5*b
^3)*d^3*x^3 + (15*a*b^7*c^3 - 5*a^3*b^5*c^2 - 11*a^5*b^3*c + a^7*b)*d^2*x^2)*sqrt(c)*log((d*x + 2*sqrt(d*x + c
)*sqrt(c) + 2*c)/x) + 8*((b^8*c^3 + 5*a^2*b^6*c^2)*d^3*x^3 + (b^8*c^4 + 4*a^2*b^6*c^3 - 5*a^4*b^4*c^2)*d^2*x^2
)*log(sqrt(d*x + c)*b + a) - 4*((b^8*c^3 + 5*a^2*b^6*c^2)*d^3*x^3 + (b^8*c^4 + 4*a^2*b^6*c^3 - 5*a^4*b^4*c^2)*
d^2*x^2)*log(x) - 2*(2*a*b^7*c^5 - 6*a^3*b^5*c^4 + 6*a^5*b^3*c^3 - 2*a^7*b*c^2 - (11*a*b^7*c^3 - 10*a^3*b^5*c^
2 - a^5*b^3*c)*d^2*x^2 - (5*a*b^7*c^4 - 9*a^3*b^5*c^3 + 3*a^5*b^3*c^2 + a^7*b*c)*d*x)*sqrt(d*x + c))/((b^10*c^
6 - 4*a^2*b^8*c^5 + 6*a^4*b^6*c^4 - 4*a^6*b^4*c^3 + a^8*b^2*c^2)*d*x^3 + (b^10*c^7 - 5*a^2*b^8*c^6 + 10*a^4*b^
6*c^5 - 10*a^6*b^4*c^4 + 5*a^8*b^2*c^3 - a^10*c^2)*x^2), -1/2*(b^8*c^6 - 2*a^2*b^6*c^5 + 2*a^6*b^2*c^3 - a^8*c
^2 - 2*(b^8*c^4 + 4*a^2*b^6*c^3 - 5*a^4*b^4*c^2)*d^2*x^2 - (b^8*c^5 + 3*a^2*b^6*c^4 - 9*a^4*b^4*c^3 + 5*a^6*b^
2*c^2)*d*x + ((15*a*b^7*c^2 + 10*a^3*b^5*c - a^5*b^3)*d^3*x^3 + (15*a*b^7*c^3 - 5*a^3*b^5*c^2 - 11*a^5*b^3*c +
a^7*b)*d^2*x^2)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) + 4*((b^8*c^3 + 5*a^2*b^6*c^2)*d^3*x^3 + (b^8*c^4 +
4*a^2*b^6*c^3 - 5*a^4*b^4*c^2)*d^2*x^2)*log(sqrt(d*x + c)*b + a) - 2*((b^8*c^3 + 5*a^2*b^6*c^2)*d^3*x^3 + (b^
8*c^4 + 4*a^2*b^6*c^3 - 5*a^4*b^4*c^2)*d^2*x^2)*log(x) - (2*a*b^7*c^5 - 6*a^3*b^5*c^4 + 6*a^5*b^3*c^3 - 2*a^7*
b*c^2 - (11*a*b^7*c^3 - 10*a^3*b^5*c^2 - a^5*b^3*c)*d^2*x^2 - (5*a*b^7*c^4 - 9*a^3*b^5*c^3 + 3*a^5*b^3*c^2 + a
^7*b*c)*d*x)*sqrt(d*x + c))/((b^10*c^6 - 4*a^2*b^8*c^5 + 6*a^4*b^6*c^4 - 4*a^6*b^4*c^3 + a^8*b^2*c^2)*d*x^3 +
(b^10*c^7 - 5*a^2*b^8*c^6 + 10*a^4*b^6*c^5 - 10*a^6*b^4*c^4 + 5*a^8*b^2*c^3 - a^10*c^2)*x^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (a + b \sqrt {c + d x}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a+b*(d*x+c)**(1/2))**2,x)

[Out]

Integral(1/(x**3*(a + b*sqrt(c + d*x))**2), x)

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Giac [A]
time = 3.24, size = 521, normalized size = 1.70 \begin {gather*} \frac {{\left (b^{6} c d^{2} + 5 \, a^{2} b^{4} d^{2}\right )} \log \left (-d x\right )}{b^{8} c^{4} - 4 \, a^{2} b^{6} c^{3} + 6 \, a^{4} b^{4} c^{2} - 4 \, a^{6} b^{2} c + a^{8}} - \frac {2 \, {\left (b^{7} c d^{2} + 5 \, a^{2} b^{5} d^{2}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{9} c^{4} - 4 \, a^{2} b^{7} c^{3} + 6 \, a^{4} b^{5} c^{2} - 4 \, a^{6} b^{3} c + a^{8} b} - \frac {{\left (15 \, a b^{5} c^{2} d^{2} + 10 \, a^{3} b^{3} c d^{2} - a^{5} b d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{2 \, {\left (b^{8} c^{5} - 4 \, a^{2} b^{6} c^{4} + 6 \, a^{4} b^{4} c^{3} - 4 \, a^{6} b^{2} c^{2} + a^{8} c\right )} \sqrt {-c}} - \frac {7 \, a b^{6} c^{4} d^{2} - a^{3} b^{4} c^{3} d^{2} - 7 \, a^{5} b^{2} c^{2} d^{2} + a^{7} c d^{2} + {\left (11 \, a b^{6} c^{2} d^{2} - 10 \, a^{3} b^{4} c d^{2} - a^{5} b^{2} d^{2}\right )} {\left (d x + c\right )}^{2} - {\left (2 \, b^{7} c^{3} d^{2} - 3 \, a^{2} b^{5} c^{2} d^{2} + a^{6} b d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - {\left (19 \, a b^{6} c^{3} d^{2} - 14 \, a^{3} b^{4} c^{2} d^{2} - 5 \, a^{5} b^{2} c d^{2}\right )} {\left (d x + c\right )} + 3 \, {\left (b^{7} c^{4} d^{2} - 2 \, a^{2} b^{5} c^{3} d^{2} + a^{4} b^{3} c^{2} d^{2}\right )} \sqrt {d x + c}}{2 \, {\left (b^{2} c - a^{2}\right )}^{4} {\left (\sqrt {d x + c} b + a\right )} c d^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*(d*x+c)^(1/2))^2,x, algorithm="giac")

[Out]

(b^6*c*d^2 + 5*a^2*b^4*d^2)*log(-d*x)/(b^8*c^4 - 4*a^2*b^6*c^3 + 6*a^4*b^4*c^2 - 4*a^6*b^2*c + a^8) - 2*(b^7*c
*d^2 + 5*a^2*b^5*d^2)*log(abs(sqrt(d*x + c)*b + a))/(b^9*c^4 - 4*a^2*b^7*c^3 + 6*a^4*b^5*c^2 - 4*a^6*b^3*c + a
^8*b) - 1/2*(15*a*b^5*c^2*d^2 + 10*a^3*b^3*c*d^2 - a^5*b*d^2)*arctan(sqrt(d*x + c)/sqrt(-c))/((b^8*c^5 - 4*a^2
*b^6*c^4 + 6*a^4*b^4*c^3 - 4*a^6*b^2*c^2 + a^8*c)*sqrt(-c)) - 1/2*(7*a*b^6*c^4*d^2 - a^3*b^4*c^3*d^2 - 7*a^5*b
^2*c^2*d^2 + a^7*c*d^2 + (11*a*b^6*c^2*d^2 - 10*a^3*b^4*c*d^2 - a^5*b^2*d^2)*(d*x + c)^2 - (2*b^7*c^3*d^2 - 3*
a^2*b^5*c^2*d^2 + a^6*b*d^2)*(d*x + c)^(3/2) - (19*a*b^6*c^3*d^2 - 14*a^3*b^4*c^2*d^2 - 5*a^5*b^2*c*d^2)*(d*x
+ c) + 3*(b^7*c^4*d^2 - 2*a^2*b^5*c^3*d^2 + a^4*b^3*c^2*d^2)*sqrt(d*x + c))/((b^2*c - a^2)^4*(sqrt(d*x + c)*b
+ a)*c*d^2*x^2)

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Mupad [B]
time = 5.96, size = 1441, normalized size = 4.71 \begin {gather*} \frac {\frac {\left (5\,a^3\,b^2\,d^2+19\,c\,a\,b^4\,d^2\right )\,\left (c+d\,x\right )}{2\,\left (b^2\,c-a^2\right )\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}+\frac {\left (a^3\,b^2\,d^2+11\,c\,a\,b^4\,d^2\right )\,{\left (c+d\,x\right )}^2}{2\,c\,\left (a^6-3\,a^4\,b^2\,c+3\,a^2\,b^4\,c^2-b^6\,c^3\right )}-\frac {a\,\left (-a^4\,d^2+6\,a^2\,b^2\,c\,d^2+7\,b^4\,c^2\,d^2\right )}{2\,\left (b^2\,c-a^2\right )\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}+\frac {b\,\left (a^2\,d^2+2\,c\,b^2\,d^2\right )\,{\left (c+d\,x\right )}^{3/2}}{2\,c\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}-\frac {3\,b^3\,c\,d^2\,\sqrt {c+d\,x}}{2\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}}{a\,{\left (c+d\,x\right )}^2+b\,{\left (c+d\,x\right )}^{5/2}+a\,c^2-2\,a\,c\,\left (c+d\,x\right )-2\,b\,c\,{\left (c+d\,x\right )}^{3/2}+b\,c^2\,\sqrt {c+d\,x}}+\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (\frac {10\,b^4\,d^2}{{\left (b^2\,c-a^2\right )}^3}-\frac {12\,b^6\,c\,d^2}{{\left (b^2\,c-a^2\right )}^4}\right )+\frac {\ln \left (\frac {a\,b^4\,d^4\,\left (a^6+2\,a^4\,b^2\,c-103\,a^2\,b^4\,c^2-44\,b^6\,c^3\right )}{4\,c^2\,{\left (b^2\,c-a^2\right )}^6}-\frac {b\,d^2\,\left (\frac {b^2\,d^2\,\left (a^2\,\sqrt {c+d\,x}+4\,a\,b\,c+3\,b^2\,c\,\sqrt {c+d\,x}\right )\,\left (a^5\,\sqrt {c^3}+4\,b^5\,c^4+20\,a^2\,b^3\,c^3-10\,a^3\,b^2\,c\,\sqrt {c^3}-15\,a\,b^4\,c^2\,\sqrt {c^3}\right )}{2\,c^3\,{\left (b^2\,c-a^2\right )}^4}-\frac {b^3\,d^2\,\sqrt {c+d\,x}\,\left (-a^4+19\,a^2\,b^2\,c+6\,b^4\,c^2\right )}{c\,{\left (b^2\,c-a^2\right )}^3}+\frac {a\,b^2\,d^2\,\left (7\,b^2\,c-a^2\right )}{2\,c\,{\left (b^2\,c-a^2\right )}^2}\right )\,\left (a^5\,\sqrt {c^3}+4\,b^5\,c^4+20\,a^2\,b^3\,c^3-10\,a^3\,b^2\,c\,\sqrt {c^3}-15\,a\,b^4\,c^2\,\sqrt {c^3}\right )}{4\,c^3\,{\left (b^2\,c-a^2\right )}^4}+\frac {a^2\,b^5\,d^4\,{\left (a^2+11\,c\,b^2\right )}^2\,\sqrt {c+d\,x}}{4\,c^2\,{\left (b^2\,c-a^2\right )}^6}\right )\,\left (4\,b^6\,c^4\,d^2+20\,a^2\,b^4\,c^3\,d^2+a^5\,b\,d^2\,\sqrt {c^3}-10\,a^3\,b^3\,c\,d^2\,\sqrt {c^3}-15\,a\,b^5\,c^2\,d^2\,\sqrt {c^3}\right )}{4\,\left (a^8\,c^3-4\,a^6\,b^2\,c^4+6\,a^4\,b^4\,c^5-4\,a^2\,b^6\,c^6+b^8\,c^7\right )}+\frac {\ln \left (\frac {a\,b^4\,d^4\,\left (a^6+2\,a^4\,b^2\,c-103\,a^2\,b^4\,c^2-44\,b^6\,c^3\right )}{4\,c^2\,{\left (b^2\,c-a^2\right )}^6}-\frac {b\,d^2\,\left (\frac {b^2\,d^2\,\left (a^2\,\sqrt {c+d\,x}+4\,a\,b\,c+3\,b^2\,c\,\sqrt {c+d\,x}\right )\,\left (4\,b^5\,c^4-a^5\,\sqrt {c^3}+20\,a^2\,b^3\,c^3+10\,a^3\,b^2\,c\,\sqrt {c^3}+15\,a\,b^4\,c^2\,\sqrt {c^3}\right )}{2\,c^3\,{\left (b^2\,c-a^2\right )}^4}-\frac {b^3\,d^2\,\sqrt {c+d\,x}\,\left (-a^4+19\,a^2\,b^2\,c+6\,b^4\,c^2\right )}{c\,{\left (b^2\,c-a^2\right )}^3}+\frac {a\,b^2\,d^2\,\left (7\,b^2\,c-a^2\right )}{2\,c\,{\left (b^2\,c-a^2\right )}^2}\right )\,\left (4\,b^5\,c^4-a^5\,\sqrt {c^3}+20\,a^2\,b^3\,c^3+10\,a^3\,b^2\,c\,\sqrt {c^3}+15\,a\,b^4\,c^2\,\sqrt {c^3}\right )}{4\,c^3\,{\left (b^2\,c-a^2\right )}^4}+\frac {a^2\,b^5\,d^4\,{\left (a^2+11\,c\,b^2\right )}^2\,\sqrt {c+d\,x}}{4\,c^2\,{\left (b^2\,c-a^2\right )}^6}\right )\,\left (4\,b^6\,c^4\,d^2+20\,a^2\,b^4\,c^3\,d^2-a^5\,b\,d^2\,\sqrt {c^3}+10\,a^3\,b^3\,c\,d^2\,\sqrt {c^3}+15\,a\,b^5\,c^2\,d^2\,\sqrt {c^3}\right )}{4\,\left (a^8\,c^3-4\,a^6\,b^2\,c^4+6\,a^4\,b^4\,c^5-4\,a^2\,b^6\,c^6+b^8\,c^7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*(c + d*x)^(1/2))^2),x)

[Out]

(((5*a^3*b^2*d^2 + 19*a*b^4*c*d^2)*(c + d*x))/(2*(b^2*c - a^2)*(a^4 + b^4*c^2 - 2*a^2*b^2*c)) + ((a^3*b^2*d^2
+ 11*a*b^4*c*d^2)*(c + d*x)^2)/(2*c*(a^6 - b^6*c^3 - 3*a^4*b^2*c + 3*a^2*b^4*c^2)) - (a*(7*b^4*c^2*d^2 - a^4*d
^2 + 6*a^2*b^2*c*d^2))/(2*(b^2*c - a^2)*(a^4 + b^4*c^2 - 2*a^2*b^2*c)) + (b*(a^2*d^2 + 2*b^2*c*d^2)*(c + d*x)^
(3/2))/(2*c*(a^4 + b^4*c^2 - 2*a^2*b^2*c)) - (3*b^3*c*d^2*(c + d*x)^(1/2))/(2*(a^4 + b^4*c^2 - 2*a^2*b^2*c)))/
(a*(c + d*x)^2 + b*(c + d*x)^(5/2) + a*c^2 - 2*a*c*(c + d*x) - 2*b*c*(c + d*x)^(3/2) + b*c^2*(c + d*x)^(1/2))
+ log(a + b*(c + d*x)^(1/2))*((10*b^4*d^2)/(b^2*c - a^2)^3 - (12*b^6*c*d^2)/(b^2*c - a^2)^4) + (log((a*b^4*d^4
*(a^6 - 44*b^6*c^3 + 2*a^4*b^2*c - 103*a^2*b^4*c^2))/(4*c^2*(b^2*c - a^2)^6) - (b*d^2*((b^2*d^2*(a^2*(c + d*x)
^(1/2) + 4*a*b*c + 3*b^2*c*(c + d*x)^(1/2))*(a^5*(c^3)^(1/2) + 4*b^5*c^4 + 20*a^2*b^3*c^3 - 10*a^3*b^2*c*(c^3)
^(1/2) - 15*a*b^4*c^2*(c^3)^(1/2)))/(2*c^3*(b^2*c - a^2)^4) - (b^3*d^2*(c + d*x)^(1/2)*(6*b^4*c^2 - a^4 + 19*a
^2*b^2*c))/(c*(b^2*c - a^2)^3) + (a*b^2*d^2*(7*b^2*c - a^2))/(2*c*(b^2*c - a^2)^2))*(a^5*(c^3)^(1/2) + 4*b^5*c
^4 + 20*a^2*b^3*c^3 - 10*a^3*b^2*c*(c^3)^(1/2) - 15*a*b^4*c^2*(c^3)^(1/2)))/(4*c^3*(b^2*c - a^2)^4) + (a^2*b^5
*d^4*(11*b^2*c + a^2)^2*(c + d*x)^(1/2))/(4*c^2*(b^2*c - a^2)^6))*(4*b^6*c^4*d^2 + 20*a^2*b^4*c^3*d^2 + a^5*b*
d^2*(c^3)^(1/2) - 10*a^3*b^3*c*d^2*(c^3)^(1/2) - 15*a*b^5*c^2*d^2*(c^3)^(1/2)))/(4*(a^8*c^3 + b^8*c^7 - 4*a^6*
b^2*c^4 + 6*a^4*b^4*c^5 - 4*a^2*b^6*c^6)) + (log((a*b^4*d^4*(a^6 - 44*b^6*c^3 + 2*a^4*b^2*c - 103*a^2*b^4*c^2)
)/(4*c^2*(b^2*c - a^2)^6) - (b*d^2*((b^2*d^2*(a^2*(c + d*x)^(1/2) + 4*a*b*c + 3*b^2*c*(c + d*x)^(1/2))*(4*b^5*
c^4 - a^5*(c^3)^(1/2) + 20*a^2*b^3*c^3 + 10*a^3*b^2*c*(c^3)^(1/2) + 15*a*b^4*c^2*(c^3)^(1/2)))/(2*c^3*(b^2*c -
a^2)^4) - (b^3*d^2*(c + d*x)^(1/2)*(6*b^4*c^2 - a^4 + 19*a^2*b^2*c))/(c*(b^2*c - a^2)^3) + (a*b^2*d^2*(7*b^2*
c - a^2))/(2*c*(b^2*c - a^2)^2))*(4*b^5*c^4 - a^5*(c^3)^(1/2) + 20*a^2*b^3*c^3 + 10*a^3*b^2*c*(c^3)^(1/2) + 15
*a*b^4*c^2*(c^3)^(1/2)))/(4*c^3*(b^2*c - a^2)^4) + (a^2*b^5*d^4*(11*b^2*c + a^2)^2*(c + d*x)^(1/2))/(4*c^2*(b^
2*c - a^2)^6))*(4*b^6*c^4*d^2 + 20*a^2*b^4*c^3*d^2 - a^5*b*d^2*(c^3)^(1/2) + 10*a^3*b^3*c*d^2*(c^3)^(1/2) + 15
*a*b^5*c^2*d^2*(c^3)^(1/2)))/(4*(a^8*c^3 + b^8*c^7 - 4*a^6*b^2*c^4 + 6*a^4*b^4*c^5 - 4*a^2*b^6*c^6))

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