3.852 \(\int \frac{x^3 \sqrt{c x^2}}{a+b x} \, dx\)

Optimal. Leaf size=102 \[ \frac{a^4 \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a^3 \sqrt{c x^2}}{b^4}+\frac{a^2 x \sqrt{c x^2}}{2 b^3}-\frac{a x^2 \sqrt{c x^2}}{3 b^2}+\frac{x^3 \sqrt{c x^2}}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0800921, antiderivative size = 102, 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 \[ \frac{a^4 \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a^3 \sqrt{c x^2}}{b^4}+\frac{a^2 x \sqrt{c x^2}}{2 b^3}-\frac{a x^2 \sqrt{c x^2}}{3 b^2}+\frac{x^3 \sqrt{c x^2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{4} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{5} x} + \frac{a^{2} \sqrt{c x^{2}} \int x\, dx}{b^{3} x} - \frac{a x^{2} \sqrt{c x^{2}}}{3 b^{2}} + \frac{x^{3} \sqrt{c x^{2}}}{4 b} - \frac{\sqrt{c x^{2}} \int a^{3}\, dx}{b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(c*x**2)**(1/2)/(b*x+a),x)

[Out]

a**4*sqrt(c*x**2)*log(a + b*x)/(b**5*x) + a**2*sqrt(c*x**2)*Integral(x, x)/(b**3
*x) - a*x**2*sqrt(c*x**2)/(3*b**2) + x**3*sqrt(c*x**2)/(4*b) - sqrt(c*x**2)*Inte
gral(a**3, x)/(b**4*x)

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Mathematica [A]  time = 0.0280353, size = 63, normalized size = 0.62 \[ \frac{c x \left (12 a^4 \log (a+b x)+b x \left (-12 a^3+6 a^2 b x-4 a b^2 x^2+3 b^3 x^3\right )\right )}{12 b^5 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(b*x*(-12*a^3 + 6*a^2*b*x - 4*a*b^2*x^2 + 3*b^3*x^3) + 12*a^4*Log[a + b*x])
)/(12*b^5*Sqrt[c*x^2])

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Maple [A]  time = 0.008, size = 63, normalized size = 0.6 \[{\frac{3\,{x}^{4}{b}^{4}-4\,{x}^{3}a{b}^{3}+6\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) -12\,x{a}^{3}b}{12\,{b}^{5}x}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(c*x^2)^(1/2)/(b*x+a),x)

[Out]

1/12*(c*x^2)^(1/2)*(3*x^4*b^4-4*x^3*a*b^3+6*x^2*a^2*b^2+12*a^4*ln(b*x+a)-12*x*a^
3*b)/b^5/x

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.213199, size = 84, normalized size = 0.82 \[ \frac{{\left (3 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a^{3} b x + 12 \, a^{4} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{12 \, b^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*b^4*x^4 - 4*a*b^3*x^3 + 6*a^2*b^2*x^2 - 12*a^3*b*x + 12*a^4*log(b*x + a)
)*sqrt(c*x^2)/(b^5*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(c*x**2)**(1/2)/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.206193, size = 109, normalized size = 1.07 \[ \frac{1}{12} \, \sqrt{c}{\left (\frac{12 \, a^{4}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{5}} - \frac{12 \, a^{4}{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{5}} + \frac{3 \, b^{3} x^{4}{\rm sign}\left (x\right ) - 4 \, a b^{2} x^{3}{\rm sign}\left (x\right ) + 6 \, a^{2} b x^{2}{\rm sign}\left (x\right ) - 12 \, a^{3} x{\rm sign}\left (x\right )}{b^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(c)*(12*a^4*ln(abs(b*x + a))*sign(x)/b^5 - 12*a^4*ln(abs(a))*sign(x)/b^
5 + (3*b^3*x^4*sign(x) - 4*a*b^2*x^3*sign(x) + 6*a^2*b*x^2*sign(x) - 12*a^3*x*si
gn(x))/b^4)