3.868 \(\int \frac{\left (c x^2\right )^{3/2}}{x^7 (a+b x)} \, dx\)

Optimal. Leaf size=112 \[ -\frac{b^3 c \sqrt{c x^2} \log (x)}{a^4 x}+\frac{b^3 c \sqrt{c x^2} \log (a+b x)}{a^4 x}-\frac{b^2 c \sqrt{c x^2}}{a^3 x^2}+\frac{b c \sqrt{c x^2}}{2 a^2 x^3}-\frac{c \sqrt{c x^2}}{3 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.0745695, antiderivative size = 112, 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 \[ -\frac{b^3 c \sqrt{c x^2} \log (x)}{a^4 x}+\frac{b^3 c \sqrt{c x^2} \log (a+b x)}{a^4 x}-\frac{b^2 c \sqrt{c x^2}}{a^3 x^2}+\frac{b c \sqrt{c x^2}}{2 a^2 x^3}-\frac{c \sqrt{c x^2}}{3 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)^(3/2)/(x^7*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.8768, size = 104, normalized size = 0.93 \[ - \frac{c \sqrt{c x^{2}}}{3 a x^{4}} + \frac{b c \sqrt{c x^{2}}}{2 a^{2} x^{3}} - \frac{b^{2} c \sqrt{c x^{2}}}{a^{3} x^{2}} - \frac{b^{3} c \sqrt{c x^{2}} \log{\left (x \right )}}{a^{4} x} + \frac{b^{3} c \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(3/2)/x**7/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.0400254, size = 65, normalized size = 0.58 \[ -\frac{\left (c x^2\right )^{3/2} \left (a \left (2 a^2-3 a b x+6 b^2 x^2\right )-6 b^3 x^3 \log (a+b x)+6 b^3 x^3 \log (x)\right )}{6 a^4 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x^2)^(3/2)/(x^7*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.007, size = 62, normalized size = 0.6 \[ -{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,a{b}^{2}{x}^{2}-3\,{a}^{2}bx+2\,{a}^{3}}{6\,{x}^{6}{a}^{4}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(3/2)/x^7/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.34792, size = 89, normalized size = 0.79 \[ \frac{b^{3} c^{\frac{3}{2}} \log \left (b x + a\right )}{a^{4}} - \frac{b^{3} c^{\frac{3}{2}} \log \left (x\right )}{a^{4}} - \frac{6 \, b^{2} c^{\frac{3}{2}} x^{2} - 3 \, a b c^{\frac{3}{2}} x + 2 \, a^{2} c^{\frac{3}{2}}}{6 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)/((b*x + a)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.222494, size = 80, normalized size = 0.71 \[ \frac{{\left (6 \, b^{3} c x^{3} \log \left (\frac{b x + a}{x}\right ) - 6 \, a b^{2} c x^{2} + 3 \, a^{2} b c x - 2 \, a^{3} c\right )} \sqrt{c x^{2}}}{6 \, a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(3/2)/((b*x + a)*x^7),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}}{x^{7} \left (a + b x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**(3/2)/x**7/(b*x+a),x)

[Out]

Integral((c*x**2)**(3/2)/(x**7*(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^7),x, algorithm="giac")

[Out]

Exception raised: TypeError