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

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

[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))

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Rubi [A]  time = 0.917912, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{5/2}}-\frac{d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{a^2 c^2 \sqrt{c+d x} (b c-a d)^2}-\frac{b (2 b c-a d)}{a^2 c (a+b x) \sqrt{c+d x} (b c-a d)}-\frac{1}{a c x (a+b x) \sqrt{c+d x}} \]

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))

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Rubi in Sympy [A]  time = 99.5598, size = 194, normalized size = 0.9 \[ - \frac{b}{a x \left (a + b x\right ) \sqrt{c + d x} \left (a d - b c\right )} - \frac{a d - 2 b c}{a^{2} c x \sqrt{c + d x} \left (a d - b c\right )} - \frac{d \left (3 a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{a^{2} c^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{b^{\frac{5}{2}} \left (7 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{a^{3} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{\left (3 a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{3} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b/(a*x*(a + b*x)*sqrt(c + d*x)*(a*d - b*c)) - (a*d - 2*b*c)/(a**2*c*x*sqrt(c +
d*x)*(a*d - b*c)) - d*(3*a**2*d**2 - 2*a*b*c*d + 2*b**2*c**2)/(a**2*c**2*sqrt(c
+ d*x)*(a*d - b*c)**2) - b**(5/2)*(7*a*d - 4*b*c)*atan(sqrt(b)*sqrt(c + d*x)/sqr
t(a*d - b*c))/(a**3*(a*d - b*c)**(5/2)) + (3*a*d + 4*b*c)*atanh(sqrt(c + d*x)/sq
rt(c))/(a**3*c**(5/2))

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Mathematica [A]  time = 0.950488, size = 164, normalized size = 0.76 \[ -\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3 c^{5/2}}+\sqrt{c+d x} \left (-\frac{\frac{b^3}{(a+b x) (b c-a d)^2}+\frac{1}{c^2 x}}{a^2}-\frac{2 d^3}{c^2 (c+d x) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.033, size = 229, normalized size = 1.1 \[ -2\,{\frac{{d}^{3}}{{c}^{2} \left ( ad-bc \right ) ^{2}\sqrt{dx+c}}}-{\frac{1}{{c}^{2}{a}^{2}x}\sqrt{dx+c}}+3\,{\frac{d}{{c}^{5/2}{a}^{2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+4\,{\frac{b}{{c}^{3/2}{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{d{b}^{3}}{{a}^{2} \left ( ad-bc \right ) ^{2} \left ( bdx+ad \right ) }\sqrt{dx+c}}-7\,{\frac{d{b}^{3}}{{a}^{2} \left ( ad-bc \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+4\,{\frac{{b}^{4}c}{{a}^{3} \left ( ad-bc \right ) ^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d^3/c^2/(a*d-b*c)^2/(d*x+c)^(1/2)-1/c^2/a^2*(d*x+c)^(1/2)/x+3*d/c^(5/2)/a^2*a
rctanh((d*x+c)^(1/2)/c^(1/2))+4/c^(3/2)/a^3*arctanh((d*x+c)^(1/2)/c^(1/2))*b-d*b
^3/a^2/(a*d-b*c)^2*(d*x+c)^(1/2)/(b*d*x+a*d)-7*d*b^3/a^2/(a*d-b*c)^2/((a*d-b*c)*
b)^(1/2)*arctan((d*x+c)^(1/2)*b/((a*d-b*c)*b)^(1/2))+4*b^4/a^3/(a*d-b*c)^2/((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)^(3/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.846414, 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)^(3/2)*x^2),x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.22767, size = 456, normalized size = 2.11 \[ \frac{{\left (4 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x + c\right )}^{2} b^{3} c^{2} d - 2 \,{\left (d x + c\right )} b^{3} c^{3} d - 2 \,{\left (d x + c\right )}^{2} a b^{2} c d^{2} + 3 \,{\left (d x + c\right )} a b^{2} c^{2} d^{2} + 3 \,{\left (d x + c\right )}^{2} a^{2} b d^{3} - 7 \,{\left (d x + c\right )} a^{2} b c d^{3} + 2 \, a^{2} b c^{2} d^{3} + 3 \,{\left (d x + c\right )} a^{3} d^{4} - 2 \, a^{3} c d^{4}}{{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )}{\left ({\left (d x + c\right )}^{\frac{5}{2}} b - 2 \,{\left (d x + c\right )}^{\frac{3}{2}} b c + \sqrt{d x + c} b c^{2} +{\left (d x + c\right )}^{\frac{3}{2}} a d - \sqrt{d x + c} a c d\right )}} - \frac{{\left (4 \, b c + 3 \, a d\right )} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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