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

Optimal. Leaf size=186 \[ \frac{4 \sqrt{a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b \sqrt{c+d x} (b c-a d)^4}+\frac{2 \sqrt{a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b^2 (c+d x)^{3/2} (b c-a d)^3}-\frac{2 a^2}{3 b^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}+\frac{4 a c}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.439923, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{4 \sqrt{a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b \sqrt{c+d x} (b c-a d)^4}+\frac{2 \sqrt{a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b^2 (c+d x)^{3/2} (b c-a d)^3}-\frac{2 a^2}{3 b^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}+\frac{4 a c}{b \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.7661, size = 172, normalized size = 0.92 \[ \frac{4 a c}{d \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2 c^{2}}{3 d^{2} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{4 \sqrt{c + d x} \left (a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{3 d \sqrt{a + b x} \left (a d - b c\right )^{4}} + \frac{2 \sqrt{c + d x} \left (a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{3 d^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a*c/(d*(a + b*x)**(3/2)*sqrt(c + d*x)*(a*d - b*c)**2) - 2*c**2/(3*d**2*(a + b*
x)**(3/2)*(c + d*x)**(3/2)*(a*d - b*c)) + 4*sqrt(c + d*x)*(a**2*d**2 + 6*a*b*c*d
+ b**2*c**2)/(3*d*sqrt(a + b*x)*(a*d - b*c)**4) + 2*sqrt(c + d*x)*(a**2*d**2 +
6*a*b*c*d + b**2*c**2)/(3*d**2*(a + b*x)**(3/2)*(a*d - b*c)**3)

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Mathematica [A]  time = 0.329903, size = 107, normalized size = 0.58 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{a^2 (a d-b c)}{(a+b x)^2}+\frac{c^2 (b c-a d)}{(c+d x)^2}+\frac{2 a (a d+3 b c)}{a+b x}+\frac{2 c (3 a d+b c)}{c+d x}\right )}{3 (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 203, normalized size = 1.1 \[{\frac{4\,{a}^{2}b{d}^{3}{x}^{3}+24\,a{b}^{2}c{d}^{2}{x}^{3}+4\,{b}^{3}{c}^{2}d{x}^{3}+6\,{a}^{3}{d}^{3}{x}^{2}+42\,{a}^{2}bc{d}^{2}{x}^{2}+42\,a{b}^{2}{c}^{2}d{x}^{2}+6\,{b}^{3}{c}^{3}{x}^{2}+24\,{a}^{3}c{d}^{2}x+48\,{a}^{2}b{c}^{2}dx+24\,a{b}^{2}{c}^{3}x+16\,{a}^{3}{c}^{2}d+16\,{a}^{2}b{c}^{3}}{3\,{a}^{4}{d}^{4}-12\,{a}^{3}bc{d}^{3}+18\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-12\,a{b}^{3}{c}^{3}d+3\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(2*a^2*b*d^3*x^3+12*a*b^2*c*d^2*x^3+2*b^3*c^2*d*x^3+3*a^3*d^3*x^2+21*a^2*b*c
*d^2*x^2+21*a*b^2*c^2*d*x^2+3*b^3*c^3*x^2+12*a^3*c*d^2*x+24*a^2*b*c^2*d*x+12*a*b
^2*c^3*x+8*a^3*c^2*d+8*a^2*b*c^3)/(b*x+a)^(3/2)/(d*x+c)^(3/2)/(a^4*d^4-4*a^3*b*c
*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)

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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(x^2/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.754237, size = 632, normalized size = 3.4 \[ \frac{2 \,{\left (8 \, a^{2} b c^{3} + 8 \, a^{3} c^{2} d + 2 \,{\left (b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 3 \,{\left (b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2} + 12 \,{\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(8*a^2*b*c^3 + 8*a^3*c^2*d + 2*(b^3*c^2*d + 6*a*b^2*c*d^2 + a^2*b*d^3)*x^3 +
3*(b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 + a^3*d^3)*x^2 + 12*(a*b^2*c^3 + 2*a
^2*b*c^2*d + a^3*c*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^2*b^4*c^6 - 4*a^3*b^3*
c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b
^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5
*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 +
a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*
c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*
a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.505282, size = 852, normalized size = 4.58 \[ -\frac{\sqrt{b x + a}{\left (\frac{2 \,{\left (b^{7} c^{5} d^{2}{\left | b \right |} - 6 \, a^{2} b^{5} c^{3} d^{4}{\left | b \right |} + 8 \, a^{3} b^{4} c^{2} d^{5}{\left | b \right |} - 3 \, a^{4} b^{3} c d^{6}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} + \frac{3 \,{\left (b^{8} c^{6} d{\left | b \right |} - 2 \, a b^{7} c^{5} d^{2}{\left | b \right |} - 2 \, a^{2} b^{6} c^{4} d^{3}{\left | b \right |} + 8 \, a^{3} b^{5} c^{3} d^{4}{\left | b \right |} - 7 \, a^{4} b^{4} c^{2} d^{5}{\left | b \right |} + 2 \, a^{5} b^{3} c d^{6}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{24 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{8 \,{\left (3 \, \sqrt{b d} a b^{6} c^{3} - 5 \, \sqrt{b d} a^{2} b^{5} c^{2} d + \sqrt{b d} a^{3} b^{4} c d^{2} + \sqrt{b d} a^{4} b^{3} d^{3} - 6 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} c^{2} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} c d + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{2} d^{2} + 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{2} c\right )}}{3 \,{\left (b^{3} c^{3}{\left | b \right |} - 3 \, a b^{2} c^{2} d{\left | b \right |} + 3 \, a^{2} b c d^{2}{\left | b \right |} - a^{3} d^{3}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*sqrt(b*x + a)*(2*(b^7*c^5*d^2*abs(b) - 6*a^2*b^5*c^3*d^4*abs(b) + 8*a^3*b^
4*c^2*d^5*abs(b) - 3*a^4*b^3*c*d^6*abs(b))*(b*x + a)/(b^8*c^2*d^4 - 2*a*b^7*c*d^
5 + a^2*b^6*d^6) + 3*(b^8*c^6*d*abs(b) - 2*a*b^7*c^5*d^2*abs(b) - 2*a^2*b^6*c^4*
d^3*abs(b) + 8*a^3*b^5*c^3*d^4*abs(b) - 7*a^4*b^4*c^2*d^5*abs(b) + 2*a^5*b^3*c*d
^6*abs(b))/(b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6))/(b^2*c + (b*x + a)*b*d -
a*b*d)^(3/2) + 8/3*(3*sqrt(b*d)*a*b^6*c^3 - 5*sqrt(b*d)*a^2*b^5*c^2*d + sqrt(b*
d)*a^3*b^4*c*d^2 + sqrt(b*d)*a^4*b^3*d^3 - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a)
- sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^4*c^2 + 3*sqrt(b*d)*(sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^3*c*d + 3*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^2*d^2 + 3*s
qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^2
*c)/((b^3*c^3*abs(b) - 3*a*b^2*c^2*d*abs(b) + 3*a^2*b*c*d^2*abs(b) - a^3*d^3*abs
(b))*(b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*
b*d))^2)^3)