3.7.38 \(\int \frac {1}{x^3 (a+b \sqrt {c+d x})} \, dx\) [638]

Optimal. Leaf size=204 \[ -\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}-\frac {b \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 c^{3/2} \left (a^2-b^2 c\right )^3}+\frac {a b^4 d^2 \log (x)}{\left (a^2-b^2 c\right )^3}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3} \]

[Out]

-1/4*b*(-3*b^4*c^2-6*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)^3+a*b^4*d^2*ln(x)/
(-b^2*c+a^2)^3-2*a*b^4*d^2*ln(a+b*(d*x+c)^(1/2))/(-b^2*c+a^2)^3+1/2*(-a+b*(d*x+c)^(1/2))/(-b^2*c+a^2)/x^2-1/4*
b*d*(4*a*b*c-(3*b^2*c+a^2)*(d*x+c)^(1/2))/c/(-b^2*c+a^2)^2/x

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Rubi [A]
time = 0.20, antiderivative size = 204, 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 \sqrt {c+d x}}{2 x^2 \left (a^2-b^2 c\right )}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c x \left (a^2-b^2 c\right )^2}+\frac {a b^4 d^2 \log (x)}{\left (a^2-b^2 c\right )^3}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3}-\frac {b d^2 \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 c^{3/2} \left (a^2-b^2 c\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*Sqrt[c + d*x])),x]

[Out]

-1/2*(a - b*Sqrt[c + d*x])/((a^2 - b^2*c)*x^2) - (b*d*(4*a*b*c - (a^2 + 3*b^2*c)*Sqrt[c + d*x]))/(4*c*(a^2 - b
^2*c)^2*x) - (b*(a^4 - 6*a^2*b^2*c - 3*b^4*c^2)*d^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(4*c^(3/2)*(a^2 - b^2*c)^3
) + (a*b^4*d^2*Log[x])/(a^2 - b^2*c)^3 - (2*a*b^4*d^2*Log[a + b*Sqrt[c + d*x]])/(a^2 - b^2*c)^3

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 )} \, dx &=d^2 \text {Subst}\left (\int \frac {1}{\left (a+b \sqrt {x}\right ) (-c+x)^3} \, dx,x,c+d x\right )\\ &=\left (2 d^2\right ) \text {Subst}\left (\int \frac {x}{(a+b x) \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}+\frac {d^2 \text {Subst}\left (\int \frac {-a b c+3 b^2 c x}{(a+b x) \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}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}+\frac {d^2 \text {Subst}\left (\int \frac {a b c \left (a^2-5 b^2 c\right )+b^2 c \left (a^2+3 b^2 c\right ) x}{(a+b x) \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}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}+\frac {d^2 \text {Subst}\left (\int \left (-\frac {8 a b^5 c^2}{\left (a^2-b^2 c\right ) (a+b x)}-\frac {b c \left (-a^4+6 a^2 b^2 c+3 b^4 c^2-8 a b^3 c x\right )}{\left (-a^2+b^2 c\right ) \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 \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3}+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {-a^4+6 a^2 b^2 c+3 b^4 c^2-8 a b^3 c x}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^3}\\ &=-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3}-\frac {\left (2 a b^4 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 )^3}-\frac {\left (b \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^3}\\ &=-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}-\frac {b \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 c^{3/2} \left (a^2-b^2 c\right )^3}+\frac {a b^4 d^2 \log (x)}{\left (a^2-b^2 c\right )^3}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 198, normalized size = 0.97 \begin {gather*} \frac {b \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) d^2 x^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+\sqrt {c} \left (\left (a^2-b^2 c\right ) \left (2 a^3 c-2 a b^2 c (c-2 d x)+b^3 c (2 c-3 d x) \sqrt {c+d x}-a^2 b \sqrt {c+d x} (2 c+d x)\right )-4 a b^4 c d^2 x^2 \log (-d x)+8 a b^4 c d^2 x^2 \log \left (a+b \sqrt {c+d x}\right )\right )}{4 c^{3/2} \left (-a^2+b^2 c\right )^3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(a^4 - 6*a^2*b^2*c - 3*b^4*c^2)*d^2*x^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]] + Sqrt[c]*((a^2 - b^2*c)*(2*a^3*c -
2*a*b^2*c*(c - 2*d*x) + b^3*c*(2*c - 3*d*x)*Sqrt[c + d*x] - a^2*b*Sqrt[c + d*x]*(2*c + d*x)) - 4*a*b^4*c*d^2*x
^2*Log[-(d*x)] + 8*a*b^4*c*d^2*x^2*Log[a + b*Sqrt[c + d*x]]))/(4*c^(3/2)*(-a^2 + b^2*c)^3*x^2)

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Maple [A]
time = 0.05, size = 227, normalized size = 1.11

method	result	size
derivativedivides	\(2 d^{2} \left (-\frac {\frac {-\frac {b \left (-3 b^{4} c^{2}+2 a^{2} b^{2} c +a^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{8 c}+\left (-\frac {1}{2} a \,b^{4} c +\frac {1}{2} b^{2} a^{3}\right ) \left (d x +c \right )+\left (\frac {3}{4} a^{2} b^{3} c -\frac {1}{8} b \,a^{4}-\frac {5}{8} c^{2} b^{5}\right ) \sqrt {d x +c}+\frac {3 a \,b^{4} c^{2}}{4}-a^{3} b^{2} c +\frac {a^{5}}{4}}{d^{2} x^{2}}+\frac {b \left (-4 a \,b^{3} c \ln \left (-d x \right )+\frac {\left (-3 b^{4} c^{2}-6 a^{2} b^{2} c +a^{4}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{8 c}}{\left (-b^{2} c +a^{2}\right )^{3}}-\frac {a \,b^{4} \ln \left (a +b \sqrt {d x +c}\right )}{\left (-b^{2} c +a^{2}\right )^{3}}\right )\)	\(227\)
default	\(2 d^{2} \left (-\frac {\frac {-\frac {b \left (-3 b^{4} c^{2}+2 a^{2} b^{2} c +a^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{8 c}+\left (-\frac {1}{2} a \,b^{4} c +\frac {1}{2} b^{2} a^{3}\right ) \left (d x +c \right )+\left (\frac {3}{4} a^{2} b^{3} c -\frac {1}{8} b \,a^{4}-\frac {5}{8} c^{2} b^{5}\right ) \sqrt {d x +c}+\frac {3 a \,b^{4} c^{2}}{4}-a^{3} b^{2} c +\frac {a^{5}}{4}}{d^{2} x^{2}}+\frac {b \left (-4 a \,b^{3} c \ln \left (-d x \right )+\frac {\left (-3 b^{4} c^{2}-6 a^{2} b^{2} c +a^{4}\right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{8 c}}{\left (-b^{2} c +a^{2}\right )^{3}}-\frac {a \,b^{4} \ln \left (a +b \sqrt {d x +c}\right )}{\left (-b^{2} c +a^{2}\right )^{3}}\right )\)	\(227\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d^2*(-1/(-b^2*c+a^2)^3*((-1/8*b*(-3*b^4*c^2+2*a^2*b^2*c+a^4)/c*(d*x+c)^(3/2)+(-1/2*a*b^4*c+1/2*b^2*a^3)*(d*x
+c)+(3/4*a^2*b^3*c-1/8*b*a^4-5/8*c^2*b^5)*(d*x+c)^(1/2)+3/4*a*b^4*c^2-a^3*b^2*c+1/4*a^5)/d^2/x^2+1/8*b/c*(-4*a
*b^3*c*ln(-d*x)+(-3*b^4*c^2-6*a^2*b^2*c+a^4)/c^(1/2)*arctanh((d*x+c)^(1/2)/c^(1/2))))-a*b^4/(-b^2*c+a^2)^3*ln(
a+b*(d*x+c)^(1/2)))

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Maxima [A]
time = 0.52, size = 367, normalized size = 1.80 \begin {gather*} -\frac {1}{8} \, {\left (\frac {8 \, a b^{4} \log \left (d x\right )}{b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}} - \frac {16 \, a b^{4} \log \left (\sqrt {d x + c} b + a\right )}{b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}} - \frac {{\left (3 \, b^{5} c^{2} + 6 \, a^{2} b^{3} c - a^{4} b\right )} \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{{\left (b^{6} c^{4} - 3 \, a^{2} b^{4} c^{3} + 3 \, a^{4} b^{2} c^{2} - a^{6} c\right )} \sqrt {c}} + \frac {2 \, {\left (4 \, {\left (d x + c\right )} a b^{2} c - 6 \, a b^{2} c^{2} + 2 \, a^{3} c - {\left (3 \, b^{3} c + a^{2} b\right )} {\left (d x + c\right )}^{\frac {3}{2}} + {\left (5 \, b^{3} c^{2} - a^{2} b c\right )} \sqrt {d x + c}\right )}}{b^{4} c^{5} - 2 \, a^{2} b^{2} c^{4} + a^{4} c^{3} + {\left (b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c\right )} {\left (d x + c\right )}^{2} - 2 \, {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{3} + a^{4} c^{2}\right )} {\left (d x + c\right )}}\right )} d^{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)),x, algorithm="maxima")

[Out]

-1/8*(8*a*b^4*log(d*x)/(b^6*c^3 - 3*a^2*b^4*c^2 + 3*a^4*b^2*c - a^6) - 16*a*b^4*log(sqrt(d*x + c)*b + a)/(b^6*
c^3 - 3*a^2*b^4*c^2 + 3*a^4*b^2*c - a^6) - (3*b^5*c^2 + 6*a^2*b^3*c - a^4*b)*log((sqrt(d*x + c) - sqrt(c))/(sq
rt(d*x + c) + sqrt(c)))/((b^6*c^4 - 3*a^2*b^4*c^3 + 3*a^4*b^2*c^2 - a^6*c)*sqrt(c)) + 2*(4*(d*x + c)*a*b^2*c -
6*a*b^2*c^2 + 2*a^3*c - (3*b^3*c + a^2*b)*(d*x + c)^(3/2) + (5*b^3*c^2 - a^2*b*c)*sqrt(d*x + c))/(b^4*c^5 - 2
*a^2*b^2*c^4 + a^4*c^3 + (b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c)*(d*x + c)^2 - 2*(b^4*c^4 - 2*a^2*b^2*c^3 + a^4*c^2)
*(d*x + c)))*d^2

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Fricas [A]
time = 0.74, size = 534, normalized size = 2.62 \begin {gather*} \left [\frac {16 \, a b^{4} c^{2} d^{2} x^{2} \log \left (\sqrt {d x + c} b + a\right ) - 8 \, a b^{4} c^{2} d^{2} x^{2} \log \left (x\right ) + 4 \, a b^{4} c^{4} - 8 \, a^{3} b^{2} c^{3} + 4 \, a^{5} c^{2} + {\left (3 \, b^{5} c^{2} + 6 \, a^{2} b^{3} c - a^{4} b\right )} \sqrt {c} d^{2} x^{2} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 8 \, {\left (a b^{4} c^{3} - a^{3} b^{2} c^{2}\right )} d x - 2 \, {\left (2 \, b^{5} c^{4} - 4 \, a^{2} b^{3} c^{3} + 2 \, a^{4} b c^{2} - {\left (3 \, b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} - a^{4} b c\right )} d x\right )} \sqrt {d x + c}}{8 \, {\left (b^{6} c^{5} - 3 \, a^{2} b^{4} c^{4} + 3 \, a^{4} b^{2} c^{3} - a^{6} c^{2}\right )} x^{2}}, \frac {8 \, a b^{4} c^{2} d^{2} x^{2} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, a b^{4} c^{2} d^{2} x^{2} \log \left (x\right ) + 2 \, a b^{4} c^{4} - 4 \, a^{3} b^{2} c^{3} + 2 \, a^{5} c^{2} + {\left (3 \, b^{5} c^{2} + 6 \, a^{2} b^{3} c - a^{4} b\right )} \sqrt {-c} d^{2} x^{2} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) - 4 \, {\left (a b^{4} c^{3} - a^{3} b^{2} c^{2}\right )} d x - {\left (2 \, b^{5} c^{4} - 4 \, a^{2} b^{3} c^{3} + 2 \, a^{4} b c^{2} - {\left (3 \, b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} - a^{4} b c\right )} d x\right )} \sqrt {d x + c}}{4 \, {\left (b^{6} c^{5} - 3 \, a^{2} b^{4} c^{4} + 3 \, a^{4} b^{2} c^{3} - a^{6} c^{2}\right )} x^{2}}\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)),x, algorithm="fricas")

[Out]

[1/8*(16*a*b^4*c^2*d^2*x^2*log(sqrt(d*x + c)*b + a) - 8*a*b^4*c^2*d^2*x^2*log(x) + 4*a*b^4*c^4 - 8*a^3*b^2*c^3
+ 4*a^5*c^2 + (3*b^5*c^2 + 6*a^2*b^3*c - a^4*b)*sqrt(c)*d^2*x^2*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x)
- 8*(a*b^4*c^3 - a^3*b^2*c^2)*d*x - 2*(2*b^5*c^4 - 4*a^2*b^3*c^3 + 2*a^4*b*c^2 - (3*b^5*c^3 - 2*a^2*b^3*c^2 -
a^4*b*c)*d*x)*sqrt(d*x + c))/((b^6*c^5 - 3*a^2*b^4*c^4 + 3*a^4*b^2*c^3 - a^6*c^2)*x^2), 1/4*(8*a*b^4*c^2*d^2*x
^2*log(sqrt(d*x + c)*b + a) - 4*a*b^4*c^2*d^2*x^2*log(x) + 2*a*b^4*c^4 - 4*a^3*b^2*c^3 + 2*a^5*c^2 + (3*b^5*c^
2 + 6*a^2*b^3*c - a^4*b)*sqrt(-c)*d^2*x^2*arctan(sqrt(d*x + c)*sqrt(-c)/c) - 4*(a*b^4*c^3 - a^3*b^2*c^2)*d*x -
(2*b^5*c^4 - 4*a^2*b^3*c^3 + 2*a^4*b*c^2 - (3*b^5*c^3 - 2*a^2*b^3*c^2 - a^4*b*c)*d*x)*sqrt(d*x + c))/((b^6*c^
5 - 3*a^2*b^4*c^4 + 3*a^4*b^2*c^3 - a^6*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 )}\, 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)),x)

[Out]

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

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Giac [A]
time = 4.13, size = 375, normalized size = 1.84 \begin {gather*} \frac {2 \, a b^{5} d^{2} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{7} c^{3} - 3 \, a^{2} b^{5} c^{2} + 3 \, a^{4} b^{3} c - a^{6} b} - \frac {a b^{4} d^{2} \log \left (d x\right )}{b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}} + \frac {{\left (3 \, b^{5} c^{2} d^{2} + 6 \, a^{2} b^{3} c d^{2} - a^{4} b d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{4 \, {\left (b^{6} c^{4} - 3 \, a^{2} b^{4} c^{3} + 3 \, a^{4} b^{2} c^{2} - a^{6} c\right )} \sqrt {-c}} + \frac {6 \, a b^{4} c^{3} d^{2} - 8 \, a^{3} b^{2} c^{2} d^{2} + 2 \, a^{5} c d^{2} + {\left (3 \, b^{5} c^{2} d^{2} - 2 \, a^{2} b^{3} c d^{2} - a^{4} b d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 4 \, {\left (a b^{4} c^{2} d^{2} - a^{3} b^{2} c d^{2}\right )} {\left (d x + c\right )} - {\left (5 \, b^{5} c^{3} d^{2} - 6 \, a^{2} b^{3} c^{2} d^{2} + a^{4} b c d^{2}\right )} \sqrt {d x + c}}{4 \, {\left (b^{2} c - a^{2}\right )}^{3} 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)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 5.01, size = 1094, normalized size = 5.36 \begin {gather*} \frac {\ln \left (\frac {b^5\,d^4\,{\left (a^2+3\,c\,b^2\right )}^2\,\sqrt {c+d\,x}}{16\,c^2\,{\left (b^2\,c-a^2\right )}^4}-\frac {a\,b^4\,d^4\,\left (-a^4+2\,a^2\,b^2\,c+15\,b^4\,c^2\right )}{16\,c^2\,{\left (b^2\,c-a^2\right )}^4}-\frac {b\,d^2\,\sqrt {c^3}\,\left (\frac {b^2\,d^2\,\left (3\,b^2\,c-a^2\right )}{4\,c\,\left (b^2\,c-a^2\right )}+\frac {b^2\,d^2\,\sqrt {c^3}\,\left (a^2\,\sqrt {c+d\,x}+4\,a\,b\,c+3\,b^2\,c\,\sqrt {c+d\,x}\right )\,\left (3\,b^4\,c^2-a^4+6\,a^2\,b^2\,c+8\,a\,b^3\,\sqrt {c^3}\right )}{4\,c^3\,{\left (b^2\,c-a^2\right )}^3}-\frac {a\,b^3\,d^2\,\left (9\,b^2\,c-a^2\right )\,\sqrt {c+d\,x}}{2\,c\,{\left (b^2\,c-a^2\right )}^2}\right )\,\left (3\,b^4\,c^2-a^4+6\,a^2\,b^2\,c+8\,a\,b^3\,\sqrt {c^3}\right )}{8\,c^3\,{\left (b^2\,c-a^2\right )}^3}\right )\,\left (8\,a\,b^4\,c^3\,d^2-a^4\,b\,d^2\,\sqrt {c^3}+3\,b^5\,c^2\,d^2\,\sqrt {c^3}+6\,a^2\,b^3\,c\,d^2\,\sqrt {c^3}\right )}{8\,\left (a^6\,c^3-3\,a^4\,b^2\,c^4+3\,a^2\,b^4\,c^5-b^6\,c^6\right )}-\frac {\frac {a^3\,d^2-3\,a\,b^2\,c\,d^2}{2\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}-\frac {\left (a^2\,b\,d^2+3\,c\,b^3\,d^2\right )\,{\left (c+d\,x\right )}^{3/2}}{4\,c\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}+\frac {b\,d^2\,\left (5\,b^2\,c-a^2\right )\,\sqrt {c+d\,x}}{4\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}+\frac {a\,b^2\,d^2\,\left (c+d\,x\right )}{a^4-2\,a^2\,b^2\,c+b^4\,c^2}}{{\left (c+d\,x\right )}^2-2\,c\,\left (c+d\,x\right )+c^2}+\frac {\ln \left (\frac {b^5\,d^4\,{\left (a^2+3\,c\,b^2\right )}^2\,\sqrt {c+d\,x}}{16\,c^2\,{\left (b^2\,c-a^2\right )}^4}-\frac {a\,b^4\,d^4\,\left (-a^4+2\,a^2\,b^2\,c+15\,b^4\,c^2\right )}{16\,c^2\,{\left (b^2\,c-a^2\right )}^4}-\frac {b\,d^2\,\sqrt {c^3}\,\left (\frac {b^2\,d^2\,\left (3\,b^2\,c-a^2\right )}{4\,c\,\left (b^2\,c-a^2\right )}+\frac {b^2\,d^2\,\sqrt {c^3}\,\left (a^2\,\sqrt {c+d\,x}+4\,a\,b\,c+3\,b^2\,c\,\sqrt {c+d\,x}\right )\,\left (a^4-3\,b^4\,c^2-6\,a^2\,b^2\,c+8\,a\,b^3\,\sqrt {c^3}\right )}{4\,c^3\,{\left (b^2\,c-a^2\right )}^3}-\frac {a\,b^3\,d^2\,\left (9\,b^2\,c-a^2\right )\,\sqrt {c+d\,x}}{2\,c\,{\left (b^2\,c-a^2\right )}^2}\right )\,\left (a^4-3\,b^4\,c^2-6\,a^2\,b^2\,c+8\,a\,b^3\,\sqrt {c^3}\right )}{8\,c^3\,{\left (b^2\,c-a^2\right )}^3}\right )\,\left (8\,a\,b^4\,c^3\,d^2+a^4\,b\,d^2\,\sqrt {c^3}-3\,b^5\,c^2\,d^2\,\sqrt {c^3}-6\,a^2\,b^3\,c\,d^2\,\sqrt {c^3}\right )}{8\,\left (a^6\,c^3-3\,a^4\,b^2\,c^4+3\,a^2\,b^4\,c^5-b^6\,c^6\right )}+\frac {2\,a\,b^4\,d^2\,\ln \left (a+b\,\sqrt {c+d\,x}\right )}{{\left (b^2\,c-a^2\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((b^5*d^4*(3*b^2*c + a^2)^2*(c + d*x)^(1/2))/(16*c^2*(b^2*c - a^2)^4) - (a*b^4*d^4*(15*b^4*c^2 - a^4 + 2*a
^2*b^2*c))/(16*c^2*(b^2*c - a^2)^4) - (b*d^2*(c^3)^(1/2)*((b^2*d^2*(3*b^2*c - a^2))/(4*c*(b^2*c - a^2)) + (b^2
*d^2*(c^3)^(1/2)*(a^2*(c + d*x)^(1/2) + 4*a*b*c + 3*b^2*c*(c + d*x)^(1/2))*(3*b^4*c^2 - a^4 + 6*a^2*b^2*c + 8*
a*b^3*(c^3)^(1/2)))/(4*c^3*(b^2*c - a^2)^3) - (a*b^3*d^2*(9*b^2*c - a^2)*(c + d*x)^(1/2))/(2*c*(b^2*c - a^2)^2
))*(3*b^4*c^2 - a^4 + 6*a^2*b^2*c + 8*a*b^3*(c^3)^(1/2)))/(8*c^3*(b^2*c - a^2)^3))*(8*a*b^4*c^3*d^2 - a^4*b*d^
2*(c^3)^(1/2) + 3*b^5*c^2*d^2*(c^3)^(1/2) + 6*a^2*b^3*c*d^2*(c^3)^(1/2)))/(8*(a^6*c^3 - b^6*c^6 - 3*a^4*b^2*c^
4 + 3*a^2*b^4*c^5)) - ((a^3*d^2 - 3*a*b^2*c*d^2)/(2*(a^4 + b^4*c^2 - 2*a^2*b^2*c)) - ((a^2*b*d^2 + 3*b^3*c*d^2
)*(c + d*x)^(3/2))/(4*c*(a^4 + b^4*c^2 - 2*a^2*b^2*c)) + (b*d^2*(5*b^2*c - a^2)*(c + d*x)^(1/2))/(4*(a^4 + b^4
*c^2 - 2*a^2*b^2*c)) + (a*b^2*d^2*(c + d*x))/(a^4 + b^4*c^2 - 2*a^2*b^2*c))/((c + d*x)^2 - 2*c*(c + d*x) + c^2
) + (log((b^5*d^4*(3*b^2*c + a^2)^2*(c + d*x)^(1/2))/(16*c^2*(b^2*c - a^2)^4) - (a*b^4*d^4*(15*b^4*c^2 - a^4 +
2*a^2*b^2*c))/(16*c^2*(b^2*c - a^2)^4) - (b*d^2*(c^3)^(1/2)*((b^2*d^2*(3*b^2*c - a^2))/(4*c*(b^2*c - a^2)) +
(b^2*d^2*(c^3)^(1/2)*(a^2*(c + d*x)^(1/2) + 4*a*b*c + 3*b^2*c*(c + d*x)^(1/2))*(a^4 - 3*b^4*c^2 - 6*a^2*b^2*c
+ 8*a*b^3*(c^3)^(1/2)))/(4*c^3*(b^2*c - a^2)^3) - (a*b^3*d^2*(9*b^2*c - a^2)*(c + d*x)^(1/2))/(2*c*(b^2*c - a^
2)^2))*(a^4 - 3*b^4*c^2 - 6*a^2*b^2*c + 8*a*b^3*(c^3)^(1/2)))/(8*c^3*(b^2*c - a^2)^3))*(8*a*b^4*c^3*d^2 + a^4*
b*d^2*(c^3)^(1/2) - 3*b^5*c^2*d^2*(c^3)^(1/2) - 6*a^2*b^3*c*d^2*(c^3)^(1/2)))/(8*(a^6*c^3 - b^6*c^6 - 3*a^4*b^
2*c^4 + 3*a^2*b^4*c^5)) + (2*a*b^4*d^2*log(a + b*(c + d*x)^(1/2)))/(b^2*c - a^2)^3

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