3.773 \(\int x^2 \left (c x^2\right )^{5/2} (a+b x) \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{8} a c^2 x^7 \sqrt{c x^2}+\frac{1}{9} b c^2 x^8 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0363945, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{8} a c^2 x^7 \sqrt{c x^2}+\frac{1}{9} b c^2 x^8 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(c*x^2)^(5/2)*(a + b*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.65219, size = 36, normalized size = 0.88 \[ \frac{a c^{2} x^{7} \sqrt{c x^{2}}}{8} + \frac{b c^{2} x^{8} \sqrt{c x^{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(c*x**2)**(5/2)*(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.0103236, size = 24, normalized size = 0.59 \[ \frac{1}{72} x^3 \left (c x^2\right )^{5/2} (9 a+8 b x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(c*x^2)^(5/2)*(a + b*x),x]

[Out]

(x^3*(c*x^2)^(5/2)*(9*a + 8*b*x))/72

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Maple [A]  time = 0.006, size = 21, normalized size = 0.5 \[{\frac{{x}^{3} \left ( 8\,bx+9\,a \right ) }{72} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^2)^(5/2)*(b*x+a),x)

[Out]

1/72*x^3*(8*b*x+9*a)*(c*x^2)^(5/2)

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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)^(5/2)*(b*x + a)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.201994, size = 38, normalized size = 0.93 \[ \frac{1}{72} \,{\left (8 \, b c^{2} x^{8} + 9 \, a c^{2} x^{7}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)*x^2,x, algorithm="fricas")

[Out]

1/72*(8*b*c^2*x^8 + 9*a*c^2*x^7)*sqrt(c*x^2)

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Sympy [A]  time = 8.17725, size = 36, normalized size = 0.88 \[ \frac{a c^{\frac{5}{2}} x^{3} \left (x^{2}\right )^{\frac{5}{2}}}{8} + \frac{b c^{\frac{5}{2}} x^{4} \left (x^{2}\right )^{\frac{5}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(c*x**2)**(5/2)*(b*x+a),x)

[Out]

a*c**(5/2)*x**3*(x**2)**(5/2)/8 + b*c**(5/2)*x**4*(x**2)**(5/2)/9

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GIAC/XCAS [A]  time = 0.210167, size = 38, normalized size = 0.93 \[ \frac{1}{72} \,{\left (8 \, b c^{2} x^{9}{\rm sign}\left (x\right ) + 9 \, a c^{2} x^{8}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)*x^2,x, algorithm="giac")

[Out]

1/72*(8*b*c^2*x^9*sign(x) + 9*a*c^2*x^8*sign(x))*sqrt(c)