3.466 \(\int \frac{1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=277 \[ -\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{7/2}}-\frac{d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{a^2 c^3 \sqrt{c+d x} (b c-a d)^3}-\frac{b (2 b c-a d)}{a^2 c (a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}} \]

[Out]

-(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2
)) - (b*(2*b*c - a*d))/(a^2*c*(b*c - a*d)*(a + b*x)*(c + d*x)^(3/2)) - 1/(a*c*x*
(a + b*x)*(c + d*x)^(3/2)) - (d*(2*b*c - a*d)*(b^2*c^2 - a*b*c*d + 5*a^2*d^2))/(
a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x]) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sq
rt[c]])/(a^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])
/Sqrt[b*c - a*d]])/(a^3*(b*c - a*d)^(7/2))

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Rubi [A]  time = 1.32076, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{7/2}}-\frac{d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{3 a^2 c^2 (c+d x)^{3/2} (b c-a d)^2}-\frac{d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{a^2 c^3 \sqrt{c+d x} (b c-a d)^3}-\frac{b (2 b c-a d)}{a^2 c (a+b x) (c+d x)^{3/2} (b c-a d)}-\frac{1}{a c x (a+b x) (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)^2*(c + d*x)^(5/2)),x]

[Out]

-(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(c + d*x)^(3/2
)) - (b*(2*b*c - a*d))/(a^2*c*(b*c - a*d)*(a + b*x)*(c + d*x)^(3/2)) - 1/(a*c*x*
(a + b*x)*(c + d*x)^(3/2)) - (d*(2*b*c - a*d)*(b^2*c^2 - a*b*c*d + 5*a^2*d^2))/(
a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x]) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sq
rt[c]])/(a^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])
/Sqrt[b*c - a*d]])/(a^3*(b*c - a*d)^(7/2))

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Rubi in Sympy [A]  time = 145.029, size = 252, normalized size = 0.91 \[ - \frac{b}{a x \left (a + b x\right ) \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{a d - 2 b c}{a^{2} c x \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{d \left (5 a^{2} d^{2} - 6 a b c d + 6 b^{2} c^{2}\right )}{3 a^{2} c^{2} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{d \left (a d - 2 b c\right ) \left (5 a^{2} d^{2} - a b c d + b^{2} c^{2}\right )}{a^{2} c^{3} \sqrt{c + d x} \left (a d - b c\right )^{3}} + \frac{b^{\frac{7}{2}} \left (9 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{3} \left (a d - b c\right )^{\frac{7}{2}}} + \frac{\left (5 a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{3} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x+a)**2/(d*x+c)**(5/2),x)

[Out]

-b/(a*x*(a + b*x)*(c + d*x)**(3/2)*(a*d - b*c)) - (a*d - 2*b*c)/(a**2*c*x*(c + d
*x)**(3/2)*(a*d - b*c)) - d*(5*a**2*d**2 - 6*a*b*c*d + 6*b**2*c**2)/(3*a**2*c**2
*(c + d*x)**(3/2)*(a*d - b*c)**2) - d*(a*d - 2*b*c)*(5*a**2*d**2 - a*b*c*d + b**
2*c**2)/(a**2*c**3*sqrt(c + d*x)*(a*d - b*c)**3) + b**(7/2)*(9*a*d - 4*b*c)*atan
(sqrt(b)*sqrt(c + d*x)/sqrt(a*d - b*c))/(a**3*(a*d - b*c)**(7/2)) + (5*a*d + 4*b
*c)*atanh(sqrt(c + d*x)/sqrt(c))/(a**3*c**(7/2))

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Mathematica [A]  time = 1.46908, size = 200, normalized size = 0.72 \[ -\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{7/2}}+\sqrt{c+d x} \left (\frac{b^4}{a^2 (a+b x) (a d-b c)^3}-\frac{1}{a^2 c^3 x}+\frac{4 d^3 (a d-2 b c)}{c^3 (c+d x) (b c-a d)^3}-\frac{2 d^3}{3 c^2 (c+d x)^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x)^2*(c + d*x)^(5/2)),x]

[Out]

Sqrt[c + d*x]*(-(1/(a^2*c^3*x)) + b^4/(a^2*(-(b*c) + a*d)^3*(a + b*x)) - (2*d^3)
/(3*c^2*(b*c - a*d)^2*(c + d*x)^2) + (4*d^3*(-2*b*c + a*d))/(c^3*(b*c - a*d)^3*(
c + d*x))) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a^3*c^(7/2)) - (b
^(7/2)*(4*b*c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a^3*(b
*c - a*d)^(7/2))

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Maple [A]  time = 0.039, size = 280, normalized size = 1. \[ -{\frac{2\,{d}^{3}}{3\,{c}^{2} \left ( ad-bc \right ) ^{2}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}-4\,{\frac{{d}^{4}a}{{c}^{3} \left ( ad-bc \right ) ^{3}\sqrt{dx+c}}}+8\,{\frac{{d}^{3}b}{{c}^{2} \left ( ad-bc \right ) ^{3}\sqrt{dx+c}}}-{\frac{1}{{a}^{2}{c}^{3}x}\sqrt{dx+c}}+5\,{\frac{d}{{a}^{2}{c}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+4\,{\frac{b}{{a}^{3}{c}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+{\frac{d{b}^{4}}{{a}^{2} \left ( ad-bc \right ) ^{3} \left ( bdx+ad \right ) }\sqrt{dx+c}}+9\,{\frac{d{b}^{4}}{{a}^{2} \left ( ad-bc \right ) ^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-4\,{\frac{{b}^{5}c}{{a}^{3} \left ( ad-bc \right ) ^{3}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^2/(d*x+c)^(5/2),x)

[Out]

-2/3*d^3/c^2/(a*d-b*c)^2/(d*x+c)^(3/2)-4*d^4/c^3/(a*d-b*c)^3/(d*x+c)^(1/2)*a+8*d
^3/c^2/(a*d-b*c)^3/(d*x+c)^(1/2)*b-1/a^2/c^3*(d*x+c)^(1/2)/x+5*d/a^2/c^(7/2)*arc
tanh((d*x+c)^(1/2)/c^(1/2))+4/a^3/c^(5/2)*arctanh((d*x+c)^(1/2)/c^(1/2))*b+d*b^4
/a^2/(a*d-b*c)^3*(d*x+c)^(1/2)/(b*d*x+a*d)+9*d*b^4/a^2/(a*d-b*c)^3/((a*d-b*c)*b)
^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))-4*b^5/a^3/(a*d-b*c)^3/((a*d-b
*c)*b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.69936, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^(5/2)*x^2),x, algorithm="fricas")

[Out]

[1/6*(3*((4*b^5*c^4*d - 9*a*b^4*c^3*d^2)*x^3 + (4*b^5*c^5 - 5*a*b^4*c^4*d - 9*a^
2*b^3*c^3*d^2)*x^2 + (4*a*b^4*c^5 - 9*a^2*b^3*c^4*d)*x)*sqrt(d*x + c)*sqrt(c)*sq
rt(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)) + 3*((4*b^5*c^4*d - 7*a*b^4*c^3*d^2 - 3*a^2*b^3*c^2*d^3
+ 11*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^3 + (4*b^5*c^5 - 3*a*b^4*c^4*d - 10*a^2*b^3
*c^3*d^2 + 8*a^3*b^2*c^2*d^3 + 6*a^4*b*c*d^4 - 5*a^5*d^5)*x^2 + (4*a*b^4*c^5 - 7
*a^2*b^3*c^4*d - 3*a^3*b^2*c^3*d^2 + 11*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x)*sqrt(d*x
+ c)*log(((d*x + 2*c)*sqrt(c) + 2*sqrt(d*x + c)*c)/x) - 2*(3*a^2*b^3*c^5 - 9*a^
3*b^2*c^4*d + 9*a^4*b*c^3*d^2 - 3*a^5*c^2*d^3 + 3*(2*a*b^4*c^3*d^2 - 3*a^2*b^3*c
^2*d^3 + 11*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^3 + (12*a*b^4*c^4*d - 15*a^2*b^3*c^3*
d^2 + 35*a^3*b^2*c^2*d^3 + 13*a^4*b*c*d^4 - 15*a^5*d^5)*x^2 + (6*a*b^4*c^5 - 3*a
^2*b^3*c^4*d - 9*a^3*b^2*c^3*d^2 + 41*a^4*b*c^2*d^3 - 20*a^5*c*d^4)*x)*sqrt(c))/
(((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^3 +
(a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x^2 + (a^4*b^3*c
^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x)*sqrt(d*x + c)*sqrt(c)),
-1/6*(6*((4*b^5*c^4*d - 9*a*b^4*c^3*d^2)*x^3 + (4*b^5*c^5 - 5*a*b^4*c^4*d - 9*a
^2*b^3*c^3*d^2)*x^2 + (4*a*b^4*c^5 - 9*a^2*b^3*c^4*d)*x)*sqrt(d*x + c)*sqrt(c)*s
qrt(-b/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x + c)*b))
- 3*((4*b^5*c^4*d - 7*a*b^4*c^3*d^2 - 3*a^2*b^3*c^2*d^3 + 11*a^3*b^2*c*d^4 - 5*a
^4*b*d^5)*x^3 + (4*b^5*c^5 - 3*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 8*a^3*b^2*c^2*
d^3 + 6*a^4*b*c*d^4 - 5*a^5*d^5)*x^2 + (4*a*b^4*c^5 - 7*a^2*b^3*c^4*d - 3*a^3*b^
2*c^3*d^2 + 11*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x)*sqrt(d*x + c)*log(((d*x + 2*c)*sq
rt(c) + 2*sqrt(d*x + c)*c)/x) + 2*(3*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 9*a^4*b*c^3
*d^2 - 3*a^5*c^2*d^3 + 3*(2*a*b^4*c^3*d^2 - 3*a^2*b^3*c^2*d^3 + 11*a^3*b^2*c*d^4
- 5*a^4*b*d^5)*x^3 + (12*a*b^4*c^4*d - 15*a^2*b^3*c^3*d^2 + 35*a^3*b^2*c^2*d^3
+ 13*a^4*b*c*d^4 - 15*a^5*d^5)*x^2 + (6*a*b^4*c^5 - 3*a^2*b^3*c^4*d - 9*a^3*b^2*
c^3*d^2 + 41*a^4*b*c^2*d^3 - 20*a^5*c*d^4)*x)*sqrt(c))/(((a^3*b^4*c^6*d - 3*a^4*
b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^3 + (a^3*b^4*c^7 - 2*a^4*b^3*
c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x^2 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*
a^6*b*c^5*d^2 - a^7*c^4*d^3)*x)*sqrt(d*x + c)*sqrt(c)), 1/6*(3*((4*b^5*c^4*d - 9
*a*b^4*c^3*d^2)*x^3 + (4*b^5*c^5 - 5*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^2 + (4*a
*b^4*c^5 - 9*a^2*b^3*c^4*d)*x)*sqrt(d*x + c)*sqrt(-c)*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))
- 6*((4*b^5*c^4*d - 7*a*b^4*c^3*d^2 - 3*a^2*b^3*c^2*d^3 + 11*a^3*b^2*c*d^4 - 5*
a^4*b*d^5)*x^3 + (4*b^5*c^5 - 3*a*b^4*c^4*d - 10*a^2*b^3*c^3*d^2 + 8*a^3*b^2*c^2
*d^3 + 6*a^4*b*c*d^4 - 5*a^5*d^5)*x^2 + (4*a*b^4*c^5 - 7*a^2*b^3*c^4*d - 3*a^3*b
^2*c^3*d^2 + 11*a^4*b*c^2*d^3 - 5*a^5*c*d^4)*x)*sqrt(d*x + c)*arctan(c/(sqrt(d*x
+ c)*sqrt(-c))) - 2*(3*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 9*a^4*b*c^3*d^2 - 3*a^5*
c^2*d^3 + 3*(2*a*b^4*c^3*d^2 - 3*a^2*b^3*c^2*d^3 + 11*a^3*b^2*c*d^4 - 5*a^4*b*d^
5)*x^3 + (12*a*b^4*c^4*d - 15*a^2*b^3*c^3*d^2 + 35*a^3*b^2*c^2*d^3 + 13*a^4*b*c*
d^4 - 15*a^5*d^5)*x^2 + (6*a*b^4*c^5 - 3*a^2*b^3*c^4*d - 9*a^3*b^2*c^3*d^2 + 41*
a^4*b*c^2*d^3 - 20*a^5*c*d^4)*x)*sqrt(-c))/(((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d^2
+ 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x^3 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^
6*b*c^4*d^3 - a^7*c^3*d^4)*x^2 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^
2 - a^7*c^4*d^3)*x)*sqrt(d*x + c)*sqrt(-c)), -1/3*(3*((4*b^5*c^4*d - 9*a*b^4*c^3
*d^2)*x^3 + (4*b^5*c^5 - 5*a*b^4*c^4*d - 9*a^2*b^3*c^3*d^2)*x^2 + (4*a*b^4*c^5 -
9*a^2*b^3*c^4*d)*x)*sqrt(d*x + c)*sqrt(-c)*sqrt(-b/(b*c - a*d))*arctan(-(b*c -
a*d)*sqrt(-b/(b*c - a*d))/(sqrt(d*x + c)*b)) + 3*((4*b^5*c^4*d - 7*a*b^4*c^3*d^2
- 3*a^2*b^3*c^2*d^3 + 11*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^3 + (4*b^5*c^5 - 3*a*b^
4*c^4*d - 10*a^2*b^3*c^3*d^2 + 8*a^3*b^2*c^2*d^3 + 6*a^4*b*c*d^4 - 5*a^5*d^5)*x^
2 + (4*a*b^4*c^5 - 7*a^2*b^3*c^4*d - 3*a^3*b^2*c^3*d^2 + 11*a^4*b*c^2*d^3 - 5*a^
5*c*d^4)*x)*sqrt(d*x + c)*arctan(c/(sqrt(d*x + c)*sqrt(-c))) + (3*a^2*b^3*c^5 -
9*a^3*b^2*c^4*d + 9*a^4*b*c^3*d^2 - 3*a^5*c^2*d^3 + 3*(2*a*b^4*c^3*d^2 - 3*a^2*b
^3*c^2*d^3 + 11*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^3 + (12*a*b^4*c^4*d - 15*a^2*b^3*
c^3*d^2 + 35*a^3*b^2*c^2*d^3 + 13*a^4*b*c*d^4 - 15*a^5*d^5)*x^2 + (6*a*b^4*c^5 -
3*a^2*b^3*c^4*d - 9*a^3*b^2*c^3*d^2 + 41*a^4*b*c^2*d^3 - 20*a^5*c*d^4)*x)*sqrt(
-c))/(((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4)*x
^3 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x^2 + (a^4*
b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x)*sqrt(d*x + c)*sqrt
(-c))]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x+a)**2/(d*x+c)**(5/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.259581, size = 635, normalized size = 2.29 \[ \frac{{\left (4 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{4} c^{3} d - 2 \, \sqrt{d x + c} b^{4} c^{4} d - 3 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{3} c^{2} d^{2} + 4 \, \sqrt{d x + c} a b^{3} c^{3} d^{2} + 3 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b^{2} c d^{3} - 6 \, \sqrt{d x + c} a^{2} b^{2} c^{2} d^{3} -{\left (d x + c\right )}^{\frac{3}{2}} a^{3} b d^{4} + 4 \, \sqrt{d x + c} a^{3} b c d^{4} - \sqrt{d x + c} a^{4} d^{5}}{{\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3}\right )}{\left ({\left (d x + c\right )}^{2} b - 2 \,{\left (d x + c\right )} b c + b c^{2} +{\left (d x + c\right )} a d - a c d\right )}} - \frac{2 \,{\left (12 \,{\left (d x + c\right )} b c d^{3} + b c^{2} d^{3} - 6 \,{\left (d x + c\right )} a d^{4} - a c d^{4}\right )}}{3 \,{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}{\left (d x + c\right )}^{\frac{3}{2}}} - \frac{{\left (4 \, b c + 5 \, a d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*(d*x + c)^(5/2)*x^2),x, algorithm="giac")

[Out]

(4*b^5*c - 9*a*b^4*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b^3*c^3
- 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*sqrt(-b^2*c + a*b*d)) - (2*(d*x +
c)^(3/2)*b^4*c^3*d - 2*sqrt(d*x + c)*b^4*c^4*d - 3*(d*x + c)^(3/2)*a*b^3*c^2*d^2
+ 4*sqrt(d*x + c)*a*b^3*c^3*d^2 + 3*(d*x + c)^(3/2)*a^2*b^2*c*d^3 - 6*sqrt(d*x
+ c)*a^2*b^2*c^2*d^3 - (d*x + c)^(3/2)*a^3*b*d^4 + 4*sqrt(d*x + c)*a^3*b*c*d^4 -
sqrt(d*x + c)*a^4*d^5)/((a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*
c^3*d^3)*((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*c*d)) - 2/
3*(12*(d*x + c)*b*c*d^3 + b*c^2*d^3 - 6*(d*x + c)*a*d^4 - a*c*d^4)/((b^3*c^6 - 3
*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*(d*x + c)^(3/2)) - (4*b*c + 5*a*d)
*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)*c^3)