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

Optimal. Leaf size=75 \[ \frac{2 \sqrt{c+d x} (a d+b c)}{d \sqrt{a+b x} (b c-a d)^2}-\frac{2 c}{d \sqrt{a+b x} \sqrt{c+d x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.101357, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 \sqrt{c+d x} (a d+b c)}{d \sqrt{a+b x} (b c-a d)^2}-\frac{2 c}{d \sqrt{a+b x} \sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.47844, size = 63, normalized size = 0.84 \[ \frac{2 c}{d \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )} + \frac{2 \sqrt{c + d x} \left (a d + b c\right )}{d \sqrt{a + b x} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0743807, size = 43, normalized size = 0.57 \[ \frac{2 (2 a c+a d x+b c x)}{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 53, normalized size = 0.7 \[ 2\,{\frac{adx+bcx+2\,ac}{\sqrt{bx+a}\sqrt{dx+c} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*d*x+b*c*x+2*a*c)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(a^2*d^2-2*a*b*c*d+b^2*c^2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288115, size = 171, normalized size = 2.28 \[ \frac{2 \,{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.246935, size = 198, normalized size = 2.64 \[ \frac{2 \,{\left (\frac{\sqrt{b x + a} b^{3} c}{{\left (b^{2} c^{2}{\left | b \right |} - 2 \, a b c d{\left | b \right |} + a^{2} d^{2}{\left | b \right |}\right )} \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{2 \, \sqrt{b d} a b^{2}}{{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\left (b c{\left | b \right |} - a d{\left | b \right |}\right )}}\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(sqrt(b*x + a)*b^3*c/((b^2*c^2*abs(b) - 2*a*b*c*d*abs(b) + a^2*d^2*abs(b))*sqr
t(b^2*c + (b*x + a)*b*d - a*b*d)) + 2*sqrt(b*d)*a*b^2/((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)*(b*c*abs(b) - a*d*abs
(b))))/b