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

Optimal. Leaf size=91 \[ -\frac{2 b c \sqrt{c x^2} \log (x)}{a^3 x}+\frac{2 b c \sqrt{c x^2} \log (a+b x)}{a^3 x}-\frac{b c \sqrt{c x^2}}{a^2 x (a+b x)}-\frac{c \sqrt{c x^2}}{a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0662371, antiderivative size = 91, 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 \[ -\frac{2 b c \sqrt{c x^2} \log (x)}{a^3 x}+\frac{2 b c \sqrt{c x^2} \log (a+b x)}{a^3 x}-\frac{b c \sqrt{c x^2}}{a^2 x (a+b x)}-\frac{c \sqrt{c x^2}}{a^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.9006, size = 85, normalized size = 0.93 \[ - \frac{b c \sqrt{c x^{2}}}{a^{2} x \left (a + b x\right )} - \frac{c \sqrt{c x^{2}}}{a^{2} x^{2}} - \frac{2 b c \sqrt{c x^{2}} \log{\left (x \right )}}{a^{3} x} + \frac{2 b c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0430611, size = 59, normalized size = 0.65 \[ -\frac{c^2 (a (a+2 b x)+2 b x \log (x) (a+b x)-2 b x (a+b x) \log (a+b x))}{a^3 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 74, normalized size = 0.8 \[ -{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,ab\ln \left ( x \right ) x-2\,\ln \left ( bx+a \right ) xab+2\,abx+{a}^{2}}{{a}^{3}{x}^{4} \left ( bx+a \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35947, size = 78, normalized size = 0.86 \[ \frac{2 \, b c^{\frac{3}{2}} \log \left (b x + a\right )}{a^{3}} - \frac{2 \, b c^{\frac{3}{2}} \log \left (x\right )}{a^{3}} - \frac{2 \, b c^{\frac{3}{2}} x + a c^{\frac{3}{2}}}{a^{2} b x^{2} + a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2201, size = 88, normalized size = 0.97 \[ -\frac{{\left (2 \, a b c x + a^{2} c - 2 \,{\left (b^{2} c x^{2} + a b c x\right )} \log \left (\frac{b x + a}{x}\right )\right )} \sqrt{c x^{2}}}{a^{3} b x^{3} + a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError