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

Optimal. Leaf size=92 \[ -\frac{a^3 c^2 \sqrt{c x^2} \log (a+b x)}{b^4 x}+\frac{a^2 c^2 \sqrt{c x^2}}{b^3}-\frac{a c^2 x \sqrt{c x^2}}{2 b^2}+\frac{c^2 x^2 \sqrt{c x^2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0651437, antiderivative size = 92, 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 \[ -\frac{a^3 c^2 \sqrt{c x^2} \log (a+b x)}{b^4 x}+\frac{a^2 c^2 \sqrt{c x^2}}{b^3}-\frac{a c^2 x \sqrt{c x^2}}{2 b^2}+\frac{c^2 x^2 \sqrt{c x^2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00850899, size = 54, normalized size = 0.59 \[ \frac{c \left (c x^2\right )^{3/2} \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 52, normalized size = 0.6 \[ -{\frac{-2\,{b}^{3}{x}^{3}+3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -6\,{a}^{2}bx}{6\,{x}^{5}{b}^{4}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(c*x^2)^(5/2)*(-2*b^3*x^3+3*a*b^2*x^2+6*a^3*ln(b*x+a)-6*a^2*b*x)/x^5/b^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((c*x^2)^(5/2)/((b*x + a)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.213613, size = 85, normalized size = 0.92 \[ \frac{{\left (2 \, b^{3} c^{2} x^{3} - 3 \, a b^{2} c^{2} x^{2} + 6 \, a^{2} b c^{2} x - 6 \, a^{3} c^{2} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{6 \, b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b^3*c^2*x^3 - 3*a*b^2*c^2*x^2 + 6*a^2*b*c^2*x - 6*a^3*c^2*log(b*x + a))*s
qrt(c*x^2)/(b^4*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232947, size = 113, normalized size = 1.23 \[ -\frac{1}{6} \,{\left (\frac{6 \, a^{3} c^{2}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{4}} - \frac{6 \, a^{3} c^{2}{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{4}} - \frac{2 \, b^{2} c^{2} x^{3}{\rm sign}\left (x\right ) - 3 \, a b c^{2} x^{2}{\rm sign}\left (x\right ) + 6 \, a^{2} c^{2} x{\rm sign}\left (x\right )}{b^{3}}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(6*a^3*c^2*ln(abs(b*x + a))*sign(x)/b^4 - 6*a^3*c^2*ln(abs(a))*sign(x)/b^4
- (2*b^2*c^2*x^3*sign(x) - 3*a*b*c^2*x^2*sign(x) + 6*a^2*c^2*x*sign(x))/b^3)*sqr
t(c)