3.795 \(\int \frac{x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=189 \[ \frac{4 c \sqrt{a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 b d \sqrt{c+d x} (b c-a d)^4}-\frac{4 c \sqrt{a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}-\frac{4 a^2 c}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac{2 x^3}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.43815, antiderivative size = 189, 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 \[ \frac{4 c \sqrt{a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 b d \sqrt{c+d x} (b c-a d)^4}-\frac{4 c \sqrt{a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}-\frac{4 a^2 c}{b^2 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac{2 x^3}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 41.8424, size = 173, normalized size = 0.92 \[ - \frac{4 a c^{2}}{d^{2} \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d - b c\right )^{2}} + \frac{4 a \sqrt{c + d x} \left (a^{2} d^{2} - 6 a b c d - 3 b^{2} c^{2}\right )}{3 b d \sqrt{a + b x} \left (a d - b c\right )^{4}} - \frac{4 a \sqrt{c + d x} \left (a^{2} d^{2} + 3 b^{2} c^{2}\right )}{3 b d^{2} \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}} - \frac{2 x^{3}}{3 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.259943, size = 125, normalized size = 0.66 \[ -\frac{2 \left (a^3 \left (16 c^3+24 c^2 d x+6 c d^2 x^2-d^3 x^3\right )+3 a^2 b c x \left (8 c^2+12 c d x+3 d^2 x^2\right )+3 a b^2 c^2 x^2 (2 c+3 d x)-b^3 c^3 x^3\right )}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((a + b*x)^(5/2)*(c + d*x)^(5/2)),x]

[Out]

(-2*(-(b^3*c^3*x^3) + 3*a*b^2*c^2*x^2*(2*c + 3*d*x) + 3*a^2*b*c*x*(8*c^2 + 12*c*
d*x + 3*d^2*x^2) + a^3*(16*c^3 + 24*c^2*d*x + 6*c*d^2*x^2 - d^3*x^3)))/(3*(b*c -
a*d)^4*(a + b*x)^(3/2)*(c + d*x)^(3/2))

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Maple [A]  time = 0.013, size = 182, normalized size = 1. \[ -{\frac{-2\,{a}^{3}{d}^{3}{x}^{3}+18\,{a}^{2}bc{d}^{2}{x}^{3}+18\,a{b}^{2}{c}^{2}d{x}^{3}-2\,{b}^{3}{c}^{3}{x}^{3}+12\,{a}^{3}c{d}^{2}{x}^{2}+72\,{a}^{2}b{c}^{2}d{x}^{2}+12\,a{b}^{2}{c}^{3}{x}^{2}+48\,{a}^{3}{c}^{2}dx+48\,{a}^{2}b{c}^{3}x+32\,{a}^{3}{c}^{3}}{3\,{a}^{4}{d}^{4}-12\,{a}^{3}bc{d}^{3}+18\,{a}^{2}{c}^{2}{b}^{2}{d}^{2}-12\,a{b}^{3}{c}^{3}d+3\,{b}^{4}{c}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)

[Out]

-2/3*(-a^3*d^3*x^3+9*a^2*b*c*d^2*x^3+9*a*b^2*c^2*d*x^3-b^3*c^3*x^3+6*a^3*c*d^2*x
^2+36*a^2*b*c^2*d*x^2+6*a*b^2*c^3*x^2+24*a^3*c^2*d*x+24*a^2*b*c^3*x+16*a^3*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^3/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.727447, size = 605, normalized size = 3.2 \[ -\frac{2 \,{\left (16 \, a^{3} c^{3} -{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3} + 6 \,{\left (a b^{2} c^{3} + 6 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{2} + 24 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{3 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

-2/3*(16*a^3*c^3 - (b^3*c^3 - 9*a*b^2*c^2*d - 9*a^2*b*c*d^2 + a^3*d^3)*x^3 + 6*(
a*b^2*c^3 + 6*a^2*b*c^2*d + a^3*c*d^2)*x^2 + 24*(a^2*b*c^3 + a^3*c^2*d)*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**3/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.500984, size = 855, normalized size = 4.52 \[ -\frac{\sqrt{b x + a}{\left (\frac{{\left (b^{7} c^{6} d{\left | b \right |} - 12 \, a b^{6} c^{5} d^{2}{\left | b \right |} + 30 \, a^{2} b^{5} c^{4} d^{3}{\left | b \right |} - 28 \, a^{3} b^{4} c^{3} d^{4}{\left | b \right |} + 9 \, a^{4} b^{3} c^{2} d^{5}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}} - \frac{9 \,{\left (a b^{7} c^{6} d{\left | b \right |} - 4 \, a^{2} b^{6} c^{5} d^{2}{\left | b \right |} + 6 \, a^{3} b^{5} c^{4} d^{3}{\left | b \right |} - 4 \, a^{4} b^{4} c^{3} d^{4}{\left | b \right |} + a^{5} b^{3} c^{2} d^{5}{\left | b \right |}\right )}}{b^{8} c^{2} d^{4} - 2 \, a b^{7} c d^{5} + a^{2} b^{6} d^{6}}\right )}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{4 \,{\left (9 \, \sqrt{b d} a^{2} b^{5} c^{3} - 19 \, \sqrt{b d} a^{3} b^{4} c^{2} d + 11 \, \sqrt{b d} a^{4} b^{3} c d^{2} - \sqrt{b d} a^{5} b^{2} d^{3} - 18 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} c^{2} + 18 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{2} c d + 9 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b c - 3 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} d\right )}}{3 \,{\left (b^{3} c^{3}{\left | b \right |} - 3 \, a b^{2} c^{2} d{\left | b \right |} + 3 \, a^{2} b c d^{2}{\left | b \right |} - a^{3} d^{3}{\left | b \right |}\right )}{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x + a)^(5/2)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

-1/12*sqrt(b*x + a)*((b^7*c^6*d*abs(b) - 12*a*b^6*c^5*d^2*abs(b) + 30*a^2*b^5*c^
4*d^3*abs(b) - 28*a^3*b^4*c^3*d^4*abs(b) + 9*a^4*b^3*c^2*d^5*abs(b))*(b*x + a)/(
b^8*c^2*d^4 - 2*a*b^7*c*d^5 + a^2*b^6*d^6) - 9*(a*b^7*c^6*d*abs(b) - 4*a^2*b^6*c
^5*d^2*abs(b) + 6*a^3*b^5*c^4*d^3*abs(b) - 4*a^4*b^4*c^3*d^4*abs(b) + a^5*b^3*c^
2*d^5*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) - 4/3*(9*sqrt(b*d)*a^2*b^5*c^3 - 19*sqrt(b*d)*a^3*b^4*c^2*d + 1
1*sqrt(b*d)*a^4*b^3*c*d^2 - sqrt(b*d)*a^5*b^2*d^3 - 18*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^2 + 18*sqrt(b*d)*(s
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^2*c*d + 9*
sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*
b*c - 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)
)^4*a^3*d)/((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)