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

Optimal. Leaf size=49 \[ \frac{a x}{b^2 c \sqrt{c x^2} (a+b x)}+\frac{x \log (a+b x)}{b^2 c \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.040467, antiderivative size = 49, 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 \[ \frac{a x}{b^2 c \sqrt{c x^2} (a+b x)}+\frac{x \log (a+b x)}{b^2 c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.0922, size = 46, normalized size = 0.94 \[ \frac{a \sqrt{c x^{2}}}{b^{2} c^{2} x \left (a + b x\right )} + \frac{\sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{2} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0194758, size = 37, normalized size = 0.76 \[ \frac{x^3 ((a+b x) \log (a+b x)+a)}{b^2 \left (c x^2\right )^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 41, normalized size = 0.8 \[{\frac{{x}^{3} \left ( b\ln \left ( bx+a \right ) x+a\ln \left ( bx+a \right ) +a \right ) }{ \left ( bx+a \right ){b}^{2}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21606, size = 59, normalized size = 1.2 \[ \frac{\sqrt{c x^{2}}{\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} c^{2} x^{2} + a b^{2} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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