3.908 \(\int \frac{\left (c x^2\right )^{3/2}}{x^6 (a+b x)^2} \, dx\)

Optimal. Leaf size=117 \[ \frac{3 b^2 c \sqrt{c x^2} \log (x)}{a^4 x}-\frac{3 b^2 c \sqrt{c x^2} \log (a+b x)}{a^4 x}+\frac{b^2 c \sqrt{c x^2}}{a^3 x (a+b x)}+\frac{2 b c \sqrt{c x^2}}{a^3 x^2}-\frac{c \sqrt{c x^2}}{2 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0852988, antiderivative size = 117, 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 \[ \frac{3 b^2 c \sqrt{c x^2} \log (x)}{a^4 x}-\frac{3 b^2 c \sqrt{c x^2} \log (a+b x)}{a^4 x}+\frac{b^2 c \sqrt{c x^2}}{a^3 x (a+b x)}+\frac{2 b c \sqrt{c x^2}}{a^3 x^2}-\frac{c \sqrt{c x^2}}{2 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)^(3/2)/(x^6*(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.4734, size = 112, normalized size = 0.96 \[ - \frac{c \sqrt{c x^{2}}}{2 a^{2} x^{3}} + \frac{b^{2} c \sqrt{c x^{2}}}{a^{3} x \left (a + b x\right )} + \frac{2 b c \sqrt{c x^{2}}}{a^{3} x^{2}} + \frac{3 b^{2} c \sqrt{c x^{2}} \log{\left (x \right )}}{a^{4} x} - \frac{3 b^{2} c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(3/2)/x**6/(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0449586, size = 82, normalized size = 0.7 \[ \frac{\left (c x^2\right )^{3/2} \left (a \left (-a^2+3 a b x+6 b^2 x^2\right )+6 b^2 x^2 \log (x) (a+b x)-6 b^2 x^2 (a+b x) \log (a+b x)\right )}{2 a^4 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x^2)^(3/2)/(x^6*(a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.007, size = 95, normalized size = 0.8 \[{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,\ln \left ( x \right ){x}^{2}a{b}^{2}-6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}+6\,a{b}^{2}{x}^{2}+3\,{a}^{2}bx-{a}^{3}}{2\,{x}^{5}{a}^{4} \left ( bx+a \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(3/2)/x^6/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.35834, size = 107, normalized size = 0.91 \[ -\frac{3 \, b^{2} c^{\frac{3}{2}} \log \left (b x + a\right )}{a^{4}} + \frac{3 \, b^{2} c^{\frac{3}{2}} \log \left (x\right )}{a^{4}} + \frac{6 \, b^{2} c^{\frac{3}{2}} x^{2} + 3 \, a b c^{\frac{3}{2}} x - a^{2} c^{\frac{3}{2}}}{2 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)/((b*x + a)^2*x^6),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224114, size = 111, normalized size = 0.95 \[ \frac{{\left (6 \, a b^{2} c x^{2} + 3 \, a^{2} b c x - a^{3} c + 6 \,{\left (b^{3} c x^{3} + a b^{2} c x^{2}\right )} \log \left (\frac{x}{b x + a}\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)/((b*x + a)^2*x^6),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}}{x^{6} \left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**(3/2)/x**6/(b*x+a)**2,x)

[Out]

Integral((c*x**2)**(3/2)/(x**6*(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^6),x, algorithm="giac")

[Out]

Exception raised: TypeError