3.917 \(\int \frac{x^5}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx\)

Optimal. Leaf size=73 \[ -\frac{a^2 x}{b^3 c \sqrt{c x^2} (a+b x)}-\frac{2 a x \log (a+b x)}{b^3 c \sqrt{c x^2}}+\frac{x^2}{b^2 c \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0583041, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2 x}{b^3 c \sqrt{c x^2} (a+b x)}-\frac{2 a x \log (a+b x)}{b^3 c \sqrt{c x^2}}+\frac{x^2}{b^2 c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \sqrt{c x^{2}}}{b^{3} c^{2} x \left (a + b x\right )} - \frac{2 a \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{3} c^{2} x} + \frac{\sqrt{c x^{2}} \int \frac{1}{b^{2}}\, dx}{c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(c*x**2)/(b**3*c**2*x*(a + b*x)) - 2*a*sqrt(c*x**2)*log(a + b*x)/(b**3
*c**2*x) + sqrt(c*x**2)*Integral(b**(-2), x)/(c**2*x)

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Mathematica [A]  time = 0.0341972, size = 54, normalized size = 0.74 \[ \frac{x^3 \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \left (c x^2\right )^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(-a^2 + a*b*x + b^2*x^2 - 2*a*(a + b*x)*Log[a + b*x]))/(b^3*(c*x^2)^(3/2)*(
a + b*x))

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Maple [A]  time = 0.007, size = 62, normalized size = 0.9 \[ -{\frac{{x}^{3} \left ( 2\,\ln \left ( bx+a \right ) xab-{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -abx+{a}^{2} \right ) }{ \left ( bx+a \right ){b}^{3}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^5/((c*x^2)^(3/2)*(b*x + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218766, size = 85, normalized size = 1.16 \[ \frac{{\left (b^{2} x^{2} + a b x - a^{2} - 2 \,{\left (a b x + a^{2}\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{b^{4} c^{2} x^{2} + a b^{3} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*x^2 + a*b*x - a^2 - 2*(a*b*x + a^2)*log(b*x + a))*sqrt(c*x^2)/(b^4*c^2*x^2
+ a*b^3*c^2*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**5/((c*x**2)**(3/2)*(a + b*x)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^5/((c*x^2)^(3/2)*(b*x + a)^2), x)