3.5.77 \(\int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx\) [477]

Optimal. Leaf size=216 \[ -\frac {d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right )}{a^2 c^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {b (2 b c-a d)}{a^2 c (b c-a d) (a+b x) \sqrt {c+d x}}-\frac {1}{a c x (a+b x) \sqrt {c+d x}}+\frac {(4 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{5/2}}-\frac {b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{5/2}} \]

[Out]

(3*a*d+4*b*c)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(5/2)-b^(5/2)*(-7*a*d+4*b*c)*arctanh(b^(1/2)*(d*x+c)^(1/2)/
(-a*d+b*c)^(1/2))/a^3/(-a*d+b*c)^(5/2)-d*(3*a^2*d^2-2*a*b*c*d+2*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/(d*x+c)^(1/2)-b*
(-a*d+2*b*c)/a^2/c/(-a*d+b*c)/(b*x+a)/(d*x+c)^(1/2)-1/a/c/x/(b*x+a)/(d*x+c)^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {105, 156, 157, 162, 65, 214} \begin {gather*} -\frac {b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{5/2}}+\frac {(3 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{5/2}}-\frac {d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{a^2 c^2 \sqrt {c+d x} (b c-a d)^2}-\frac {b (2 b c-a d)}{a^2 c (a+b x) \sqrt {c+d x} (b c-a d)}-\frac {1}{a c x (a+b x) \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*(2*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2))/(a^2*c^2*(b*c - a*d)^2*Sqrt[c + d*x])) - (b*(2*b*c - a*d))/(a^2*c*(b
*c - a*d)*(a + b*x)*Sqrt[c + d*x]) - 1/(a*c*x*(a + b*x)*Sqrt[c + d*x]) + ((4*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x
]/Sqrt[c]])/(a^3*c^(5/2)) - (b^(5/2)*(4*b*c - 7*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a^3*(b
*c - a*d)^(5/2))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 105

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

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 214

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

Rubi steps

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

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Mathematica [A]
time = 1.13, size = 202, normalized size = 0.94 \begin {gather*} \frac {-\frac {a \left (2 b^3 c^2 x (c+d x)+a^3 d^2 (c+3 d x)+a b^2 c \left (c^2-c d x-2 d^2 x^2\right )+a^2 b d \left (-2 c^2-c d x+3 d^2 x^2\right )\right )}{c^2 (b c-a d)^2 x (a+b x) \sqrt {c+d x}}+\frac {b^{5/2} (4 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}}+\frac {(4 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{5/2}}}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a*(2*b^3*c^2*x*(c + d*x) + a^3*d^2*(c + 3*d*x) + a*b^2*c*(c^2 - c*d*x - 2*d^2*x^2) + a^2*b*d*(-2*c^2 - c*d
*x + 3*d^2*x^2)))/(c^2*(b*c - a*d)^2*x*(a + b*x)*Sqrt[c + d*x])) + (b^(5/2)*(4*b*c - 7*a*d)*ArcTan[(Sqrt[b]*Sq
rt[c + d*x])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(5/2) + ((4*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/c^(5
/2))/a^3

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Maple [A]
time = 0.10, size = 174, normalized size = 0.81

method	result	size
derivativedivides	\(2 d^{3} \left (-\frac {b^{3} \left (\frac {a d \sqrt {d x +c}}{2 b \left (d x +c \right )+2 a d -2 b c}+\frac {\left (7 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3} \left (a d -b c \right )^{2}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (3 a d +4 b c \right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{c^{2} a^{3} d^{3}}-\frac {1}{c^{2} \left (a d -b c \right )^{2} \sqrt {d x +c}}\right )\)	\(174\)
default	\(2 d^{3} \left (-\frac {b^{3} \left (\frac {a d \sqrt {d x +c}}{2 b \left (d x +c \right )+2 a d -2 b c}+\frac {\left (7 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3} \left (a d -b c \right )^{2}}+\frac {-\frac {a \sqrt {d x +c}}{2 x}+\frac {\left (3 a d +4 b c \right ) \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{c^{2} a^{3} d^{3}}-\frac {1}{c^{2} \left (a d -b c \right )^{2} \sqrt {d x +c}}\right )\)	\(174\)
risch	\(-\frac {\sqrt {d x +c}}{c^{2} a^{2} x}-\frac {2 d^{3}}{c^{2} \left (a d -b c \right )^{2} \sqrt {d x +c}}-\frac {d \,b^{3} \sqrt {d x +c}}{a^{2} \left (a d -b c \right )^{2} \left (b d x +a d \right )}-\frac {7 d \,b^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{2} \left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}}+\frac {4 c \,b^{4} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} \left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}}+\frac {3 d \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{c^{\frac {5}{2}} a^{2}}+\frac {4 \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) b}{c^{\frac {3}{2}} a^{3}}\)	\(229\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d^3*(-b^3/a^3/d^3/(a*d-b*c)^2*(1/2*a*d*(d*x+c)^(1/2)/(b*(d*x+c)+a*d-b*c)+1/2*(7*a*d-4*b*c)/((a*d-b*c)*b)^(1/
2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)))+1/c^2/a^3/d^3*(-1/2*a*(d*x+c)^(1/2)/x+1/2*(3*a*d+4*b*c)/c^(1/2
)*arctanh((d*x+c)^(1/2)/c^(1/2)))-1/c^2/(a*d-b*c)^2/(d*x+c)^(1/2))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (194) = 388\).
time = 1.63, size = 2312, normalized size = 10.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^2 + (4*a*b^3*c
^5 - 7*a^2*b^2*c^4*d)*x)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d + 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*
c - a*d)))/(b*x + a)) - ((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*
b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^2 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^
2 + 3*a^4*c*d^3)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^
4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^2 + (2*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2
*d^2 + 3*a^4*c*d^3)*x)*sqrt(d*x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6
- a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x), -1/2*(2*(
(4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^2 + (4*a*b^3*c^5 - 7*a
^2*b^2*c^4*d)*x)*sqrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) -
((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^
2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^2 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt
(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3
*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^2 + (2*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*s
qrt(d*x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*
c^4*d^2 + a^6*c^3*d^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x), -1/2*(2*((4*b^4*c^3*d - 5*a*b^3*c
^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a
^4*d^4)*x^2 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(-c)*arctan(sqrt(d*x + c)
*sqrt(-c)/c) + ((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^2 + (4
*a*b^3*c^5 - 7*a^2*b^2*c^4*d)*x)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d + 2*(b*c - a*d)*sqrt(d*x + c)*sq
rt(b/(b*c - a*d)))/(b*x + a)) + 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*
d^2 + 3*a^3*b*c*d^3)*x^2 + (2*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(d*x + c))/((a^3
*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d
^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x), -(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^3 + (4*b^4*c^5
- 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^2 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*d)*x)*sqrt(-b/(b*c - a*d))*arctan(-(b*
c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) + ((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3
+ 3*a^3*b*d^4)*x^3 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^2 + (4*a*b^3*c
^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) + (a^2*b^2*
c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^2 + (2*a*b^3*c^4 - a
^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x)*sqrt(d*x + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*
d^3)*x^3 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^2 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^
6*c^4*d^2)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.49, size = 338, normalized size = 1.56 \begin {gather*} \frac {{\left (4 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x + c\right )}^{2} b^{3} c^{2} d - 2 \, {\left (d x + c\right )} b^{3} c^{3} d - 2 \, {\left (d x + c\right )}^{2} a b^{2} c d^{2} + 3 \, {\left (d x + c\right )} a b^{2} c^{2} d^{2} + 3 \, {\left (d x + c\right )}^{2} a^{2} b d^{3} - 7 \, {\left (d x + c\right )} a^{2} b c d^{3} + 2 \, a^{2} b c^{2} d^{3} + 3 \, {\left (d x + c\right )} a^{3} d^{4} - 2 \, a^{3} c d^{4}}{{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left ({\left (d x + c\right )}^{\frac {5}{2}} b - 2 \, {\left (d x + c\right )}^{\frac {3}{2}} b c + \sqrt {d x + c} b c^{2} + {\left (d x + c\right )}^{\frac {3}{2}} a d - \sqrt {d x + c} a c d\right )}} - \frac {{\left (4 \, b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2)*sqrt
(-b^2*c + a*b*d)) - (2*(d*x + c)^2*b^3*c^2*d - 2*(d*x + c)*b^3*c^3*d - 2*(d*x + c)^2*a*b^2*c*d^2 + 3*(d*x + c)
*a*b^2*c^2*d^2 + 3*(d*x + c)^2*a^2*b*d^3 - 7*(d*x + c)*a^2*b*c*d^3 + 2*a^2*b*c^2*d^3 + 3*(d*x + c)*a^3*d^4 - 2
*a^3*c*d^4)/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*((d*x + c)^(5/2)*b - 2*(d*x + c)^(3/2)*b*c + sqrt(d*x
+ c)*b*c^2 + (d*x + c)^(3/2)*a*d - sqrt(d*x + c)*a*c*d)) - (4*b*c + 3*a*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^
3*sqrt(-c)*c^2)

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Mupad [B]
time = 2.73, size = 2500, normalized size = 11.57 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((-b^5*(a*d - b*c)^5)^(1/2)*((c + d*x)^(1/2)*(64*a^6*b^15*c^18*d^2 - 576*a^7*b^14*c^17*d^3 + 2228*a^8*b
^13*c^16*d^4 - 4768*a^9*b^12*c^15*d^5 + 5960*a^10*b^11*c^14*d^6 - 3976*a^11*b^10*c^13*d^7 + 578*a^12*b^9*c^12*
d^8 + 1004*a^13*b^8*c^11*d^9 - 442*a^14*b^7*c^10*d^10 - 320*a^15*b^6*c^9*d^11 + 362*a^16*b^5*c^8*d^12 - 132*a^
17*b^4*c^7*d^13 + 18*a^18*b^3*c^6*d^14) + ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(8*a^10*b^13*c^19*d^3 -
76*a^11*b^12*c^18*d^4 + 300*a^12*b^11*c^17*d^5 - 612*a^13*b^10*c^16*d^6 + 576*a^14*b^9*c^15*d^7 + 168*a^15*b^8
*c^14*d^8 - 1176*a^16*b^7*c^13*d^9 + 1560*a^17*b^6*c^12*d^10 - 1128*a^18*b^5*c^11*d^11 + 484*a^19*b^4*c^10*d^1
2 - 116*a^20*b^3*c^9*d^13 + 12*a^21*b^2*c^8*d^14 - ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(c + d*x)^(1/2)
*(16*a^12*b^13*c^21*d^2 - 168*a^13*b^12*c^20*d^3 + 800*a^14*b^11*c^19*d^4 - 2280*a^15*b^10*c^18*d^5 + 4320*a^1
6*b^9*c^17*d^6 - 5712*a^17*b^8*c^16*d^7 + 5376*a^18*b^7*c^15*d^8 - 3600*a^19*b^6*c^14*d^9 + 1680*a^20*b^5*c^13
*d^10 - 520*a^21*b^4*c^12*d^11 + 96*a^22*b^3*c^11*d^12 - 8*a^23*b^2*c^10*d^13))/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*
a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*
b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)))*(7*a*d - 4*b*c)*1i)/(2*(a^8*d^5 - a^3*b
^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)) + ((-b^5*(a*d - b*c)^5)^(
1/2)*((c + d*x)^(1/2)*(64*a^6*b^15*c^18*d^2 - 576*a^7*b^14*c^17*d^3 + 2228*a^8*b^13*c^16*d^4 - 4768*a^9*b^12*c
^15*d^5 + 5960*a^10*b^11*c^14*d^6 - 3976*a^11*b^10*c^13*d^7 + 578*a^12*b^9*c^12*d^8 + 1004*a^13*b^8*c^11*d^9 -
442*a^14*b^7*c^10*d^10 - 320*a^15*b^6*c^9*d^11 + 362*a^16*b^5*c^8*d^12 - 132*a^17*b^4*c^7*d^13 + 18*a^18*b^3*
c^6*d^14) - ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(8*a^10*b^13*c^19*d^3 - 76*a^11*b^12*c^18*d^4 + 300*a^
12*b^11*c^17*d^5 - 612*a^13*b^10*c^16*d^6 + 576*a^14*b^9*c^15*d^7 + 168*a^15*b^8*c^14*d^8 - 1176*a^16*b^7*c^13
*d^9 + 1560*a^17*b^6*c^12*d^10 - 1128*a^18*b^5*c^11*d^11 + 484*a^19*b^4*c^10*d^12 - 116*a^20*b^3*c^9*d^13 + 12
*a^21*b^2*c^8*d^14 + ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(c + d*x)^(1/2)*(16*a^12*b^13*c^21*d^2 - 168*
a^13*b^12*c^20*d^3 + 800*a^14*b^11*c^19*d^4 - 2280*a^15*b^10*c^18*d^5 + 4320*a^16*b^9*c^17*d^6 - 5712*a^17*b^8
*c^16*d^7 + 5376*a^18*b^7*c^15*d^8 - 3600*a^19*b^6*c^14*d^9 + 1680*a^20*b^5*c^13*d^10 - 520*a^21*b^4*c^12*d^11
+ 96*a^22*b^3*c^11*d^12 - 8*a^23*b^2*c^10*d^13))/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3
*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2
+ 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)))*(7*a*d - 4*b*c)*1i)/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*
a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)))/(64*a^4*b^15*c^16*d^3 - 512*a^5*b^14*c^15*d^4 + 1804*a
^6*b^13*c^14*d^5 - 3668*a^7*b^12*c^13*d^6 + 4606*a^8*b^11*c^12*d^7 - 3248*a^9*b^10*c^11*d^8 + 322*a^10*b^9*c^1
0*d^9 + 1756*a^11*b^8*c^9*d^10 - 1742*a^12*b^7*c^8*d^11 + 744*a^13*b^6*c^7*d^12 - 126*a^14*b^5*c^6*d^13 - ((-b
^5*(a*d - b*c)^5)^(1/2)*((c + d*x)^(1/2)*(64*a^6*b^15*c^18*d^2 - 576*a^7*b^14*c^17*d^3 + 2228*a^8*b^13*c^16*d^
4 - 4768*a^9*b^12*c^15*d^5 + 5960*a^10*b^11*c^14*d^6 - 3976*a^11*b^10*c^13*d^7 + 578*a^12*b^9*c^12*d^8 + 1004*
a^13*b^8*c^11*d^9 - 442*a^14*b^7*c^10*d^10 - 320*a^15*b^6*c^9*d^11 + 362*a^16*b^5*c^8*d^12 - 132*a^17*b^4*c^7*
d^13 + 18*a^18*b^3*c^6*d^14) + ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(8*a^10*b^13*c^19*d^3 - 76*a^11*b^1
2*c^18*d^4 + 300*a^12*b^11*c^17*d^5 - 612*a^13*b^10*c^16*d^6 + 576*a^14*b^9*c^15*d^7 + 168*a^15*b^8*c^14*d^8 -
1176*a^16*b^7*c^13*d^9 + 1560*a^17*b^6*c^12*d^10 - 1128*a^18*b^5*c^11*d^11 + 484*a^19*b^4*c^10*d^12 - 116*a^2
0*b^3*c^9*d^13 + 12*a^21*b^2*c^8*d^14 - ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(c + d*x)^(1/2)*(16*a^12*b
^13*c^21*d^2 - 168*a^13*b^12*c^20*d^3 + 800*a^14*b^11*c^19*d^4 - 2280*a^15*b^10*c^18*d^5 + 4320*a^16*b^9*c^17*
d^6 - 5712*a^17*b^8*c^16*d^7 + 5376*a^18*b^7*c^15*d^8 - 3600*a^19*b^6*c^14*d^9 + 1680*a^20*b^5*c^13*d^10 - 520
*a^21*b^4*c^12*d^11 + 96*a^22*b^3*c^11*d^12 - 8*a^23*b^2*c^10*d^13))/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4
*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d -
10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)))*(7*a*d - 4*b*c))/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4
*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)) + ((-b^5*(a*d - b*c)^5)^(1/2)*((c + d*x
)^(1/2)*(64*a^6*b^15*c^18*d^2 - 576*a^7*b^14*c^17*d^3 + 2228*a^8*b^13*c^16*d^4 - 4768*a^9*b^12*c^15*d^5 + 5960
*a^10*b^11*c^14*d^6 - 3976*a^11*b^10*c^13*d^7 + 578*a^12*b^9*c^12*d^8 + 1004*a^13*b^8*c^11*d^9 - 442*a^14*b^7*
c^10*d^10 - 320*a^15*b^6*c^9*d^11 + 362*a^16*b^5*c^8*d^12 - 132*a^17*b^4*c^7*d^13 + 18*a^18*b^3*c^6*d^14) - ((
-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(8*a^10*b^13*c^19*d^3 - 76*a^11*b^12*c^18*d^4 + 300*a^12*b^11*c^17*d
^5 - 612*a^13*b^10*c^16*d^6 + 576*a^14*b^9*c^15...

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