3.909 \(\int \frac{x^5}{\sqrt{c x^2} (a+b x)^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac{a^4 x}{b^5 \sqrt{c x^2} (a+b x)}-\frac{4 a^3 x \log (a+b x)}{b^5 \sqrt{c x^2}}+\frac{3 a^2 x^2}{b^4 \sqrt{c x^2}}-\frac{a x^3}{b^3 \sqrt{c x^2}}+\frac{x^4}{3 b^2 \sqrt{c x^2}} \]

[Out]

(3*a^2*x^2)/(b^4*Sqrt[c*x^2]) - (a*x^3)/(b^3*Sqrt[c*x^2]) + x^4/(3*b^2*Sqrt[c*x^
2]) - (a^4*x)/(b^5*Sqrt[c*x^2]*(a + b*x)) - (4*a^3*x*Log[a + b*x])/(b^5*Sqrt[c*x
^2])

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Rubi [A]  time = 0.0885451, antiderivative size = 107, 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 \[ -\frac{a^4 x}{b^5 \sqrt{c x^2} (a+b x)}-\frac{4 a^3 x \log (a+b x)}{b^5 \sqrt{c x^2}}+\frac{3 a^2 x^2}{b^4 \sqrt{c x^2}}-\frac{a x^3}{b^3 \sqrt{c x^2}}+\frac{x^4}{3 b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*a^2*x^2)/(b^4*Sqrt[c*x^2]) - (a*x^3)/(b^3*Sqrt[c*x^2]) + x^4/(3*b^2*Sqrt[c*x^
2]) - (a^4*x)/(b^5*Sqrt[c*x^2]*(a + b*x)) - (4*a^3*x*Log[a + b*x])/(b^5*Sqrt[c*x
^2])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{4} \sqrt{c x^{2}}}{b^{5} c x \left (a + b x\right )} - \frac{4 a^{3} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{5} c x} + \frac{3 a^{2} \sqrt{c x^{2}}}{b^{4} c} - \frac{2 a \sqrt{c x^{2}} \int x\, dx}{b^{3} c x} + \frac{x^{2} \sqrt{c x^{2}}}{3 b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x+a)**2/(c*x**2)**(1/2),x)

[Out]

-a**4*sqrt(c*x**2)/(b**5*c*x*(a + b*x)) - 4*a**3*sqrt(c*x**2)*log(a + b*x)/(b**5
*c*x) + 3*a**2*sqrt(c*x**2)/(b**4*c) - 2*a*sqrt(c*x**2)*Integral(x, x)/(b**3*c*x
) + x**2*sqrt(c*x**2)/(3*b**2*c)

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Mathematica [A]  time = 0.0315065, size = 80, normalized size = 0.75 \[ \frac{x \left (-3 a^4+9 a^3 b x-12 a^3 (a+b x) \log (a+b x)+6 a^2 b^2 x^2-2 a b^3 x^3+b^4 x^4\right )}{3 b^5 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-3*a^4 + 9*a^3*b*x + 6*a^2*b^2*x^2 - 2*a*b^3*x^3 + b^4*x^4 - 12*a^3*(a + b*x
)*Log[a + b*x]))/(3*b^5*Sqrt[c*x^2]*(a + b*x))

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Maple [A]  time = 0.007, size = 86, normalized size = 0.8 \[ -{\frac{x \left ( -{x}^{4}{b}^{4}+2\,{x}^{3}a{b}^{3}+12\,\ln \left ( bx+a \right ) x{a}^{3}b-6\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) -9\,x{a}^{3}b+3\,{a}^{4} \right ) }{ \left ( 3\,bx+3\,a \right ){b}^{5}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x+a)^2/(c*x^2)^(1/2),x)

[Out]

-1/3*x*(-x^4*b^4+2*x^3*a*b^3+12*ln(b*x+a)*x*a^3*b-6*x^2*a^2*b^2+12*a^4*ln(b*x+a)
-9*x*a^3*b+3*a^4)/(c*x^2)^(1/2)/(b*x+a)/b^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217482, size = 115, normalized size = 1.07 \[ \frac{{\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \,{\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{3 \,{\left (b^{6} c x^{2} + a b^{5} c x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b^4*x^4 - 2*a*b^3*x^3 + 6*a^2*b^2*x^2 + 9*a^3*b*x - 3*a^4 - 12*(a^3*b*x + a
^4)*log(b*x + a))*sqrt(c*x^2)/(b^6*c*x^2 + a*b^5*c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x+a)**2/(c*x**2)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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