3.865 \(\int \frac{\left (c x^2\right )^{3/2}}{x^4 (a+b x)} \, dx\)

Optimal. Leaf size=44 \[ \frac{c \sqrt{c x^2} \log (x)}{a x}-\frac{c \sqrt{c x^2} \log (a+b x)}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0226804, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{c \sqrt{c x^2} \log (x)}{a x}-\frac{c \sqrt{c x^2} \log (a+b x)}{a x} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)^(3/2)/(x^4*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.4984, size = 36, normalized size = 0.82 \[ \frac{c \sqrt{c x^{2}} \log{\left (x \right )}}{a x} - \frac{c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(3/2)/x**4/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.0124848, size = 27, normalized size = 0.61 \[ \frac{\left (c x^2\right )^{3/2} (\log (x)-\log (a+b x))}{a x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x^2)^(3/2)/(x^4*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.008, size = 26, normalized size = 0.6 \[{\frac{\ln \left ( x \right ) -\ln \left ( bx+a \right ) }{a{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(3/2)/x^4/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.36218, size = 32, normalized size = 0.73 \[ -\frac{c^{\frac{3}{2}} \log \left (b x + a\right )}{a} + \frac{c^{\frac{3}{2}} \log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)/((b*x + a)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.21877, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{c x^{2}} c \log \left (\frac{x}{b x + a}\right )}{a x}, -\frac{2 \, \sqrt{-c} c \arctan \left (\frac{2 \, b c x^{2} + a c x}{\sqrt{c x^{2}} a \sqrt{-c}}\right )}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)/((b*x + a)*x^4),x, algorithm="fricas")

[Out]

[sqrt(c*x^2)*c*log(x/(b*x + a))/(a*x), -2*sqrt(-c)*c*arctan((2*b*c*x^2 + a*c*x)/
(sqrt(c*x^2)*a*sqrt(-c)))/a]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**(3/2)/x**4/(b*x+a),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)/((b*x + a)*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError