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

Optimal. Leaf size=60 \[ \frac{1}{7} a^2 c x^6 \sqrt{c x^2}+\frac{1}{4} a b c x^7 \sqrt{c x^2}+\frac{1}{9} b^2 c x^8 \sqrt{c x^2} \]

[Out]

(a^2*c*x^6*Sqrt[c*x^2])/7 + (a*b*c*x^7*Sqrt[c*x^2])/4 + (b^2*c*x^8*Sqrt[c*x^2])/
9

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Rubi [A]  time = 0.0473152, antiderivative size = 60, 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 \[ \frac{1}{7} a^2 c x^6 \sqrt{c x^2}+\frac{1}{4} a b c x^7 \sqrt{c x^2}+\frac{1}{9} b^2 c x^8 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*c*x^6*Sqrt[c*x^2])/7 + (a*b*c*x^7*Sqrt[c*x^2])/4 + (b^2*c*x^8*Sqrt[c*x^2])/
9

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Rubi in Sympy [A]  time = 18.6254, size = 54, normalized size = 0.9 \[ \frac{a^{2} c x^{6} \sqrt{c x^{2}}}{7} + \frac{a b c x^{7} \sqrt{c x^{2}}}{4} + \frac{b^{2} c x^{8} \sqrt{c x^{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c*x**6*sqrt(c*x**2)/7 + a*b*c*x**7*sqrt(c*x**2)/4 + b**2*c*x**8*sqrt(c*x**2
)/9

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Mathematica [A]  time = 0.0122243, size = 35, normalized size = 0.58 \[ \frac{1}{252} x^4 \left (c x^2\right )^{3/2} \left (36 a^2+63 a b x+28 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^4*(c*x^2)^(3/2)*(36*a^2 + 63*a*b*x + 28*b^2*x^2))/252

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Maple [A]  time = 0.006, size = 32, normalized size = 0.5 \[{\frac{{x}^{4} \left ( 28\,{b}^{2}{x}^{2}+63\,abx+36\,{a}^{2} \right ) }{252} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/252*x^4*(28*b^2*x^2+63*a*b*x+36*a^2)*(c*x^2)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.208958, size = 49, normalized size = 0.82 \[ \frac{1}{252} \,{\left (28 \, b^{2} c x^{8} + 63 \, a b c x^{7} + 36 \, a^{2} c x^{6}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/252*(28*b^2*c*x^8 + 63*a*b*c*x^7 + 36*a^2*c*x^6)*sqrt(c*x^2)

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Sympy [A]  time = 5.46974, size = 60, normalized size = 1. \[ \frac{a^{2} c^{\frac{3}{2}} x^{4} \left (x^{2}\right )^{\frac{3}{2}}}{7} + \frac{a b c^{\frac{3}{2}} x^{5} \left (x^{2}\right )^{\frac{3}{2}}}{4} + \frac{b^{2} c^{\frac{3}{2}} x^{6} \left (x^{2}\right )^{\frac{3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**(3/2)*x**4*(x**2)**(3/2)/7 + a*b*c**(3/2)*x**5*(x**2)**(3/2)/4 + b**2*c*
*(3/2)*x**6*(x**2)**(3/2)/9

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GIAC/XCAS [A]  time = 0.20659, size = 47, normalized size = 0.78 \[ \frac{1}{252} \,{\left (28 \, b^{2} x^{9}{\rm sign}\left (x\right ) + 63 \, a b x^{8}{\rm sign}\left (x\right ) + 36 \, a^{2} x^{7}{\rm sign}\left (x\right )\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/252*(28*b^2*x^9*sign(x) + 63*a*b*x^8*sign(x) + 36*a^2*x^7*sign(x))*c^(3/2)