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

Optimal. Leaf size=41 \[ \frac{1}{7} a c^2 x^6 \sqrt{c x^2}+\frac{1}{8} b c^2 x^7 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.030822, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{7} a c^2 x^6 \sqrt{c x^2}+\frac{1}{8} b c^2 x^7 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(c*x**2)**(5/2)*(a + b*x), x)

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Mathematica [A]  time = 0.00945454, size = 24, normalized size = 0.59 \[ \frac{1}{56} x^2 \left (c x^2\right )^{5/2} (8 a+7 b x) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(c*x^2)^(5/2)*(8*a + 7*b*x))/56

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Maple [A]  time = 0.005, size = 21, normalized size = 0.5 \[{\frac{{x}^{2} \left ( 7\,bx+8\,a \right ) }{56} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^2)^(5/2)*(b*x+a),x)

[Out]

1/56*x^2*(7*b*x+8*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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.208879, size = 38, normalized size = 0.93 \[ \frac{1}{56} \,{\left (7 \, b c^{2} x^{7} + 8 \, a c^{2} x^{6}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/56*(7*b*c^2*x^7 + 8*a*c^2*x^6)*sqrt(c*x^2)

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Sympy [A]  time = 6.39168, size = 36, normalized size = 0.88 \[ \frac{a c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}}{7} + \frac{b c^{\frac{5}{2}} x^{3} \left (x^{2}\right )^{\frac{5}{2}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**(5/2)*x**2*(x**2)**(5/2)/7 + b*c**(5/2)*x**3*(x**2)**(5/2)/8

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GIAC/XCAS [A]  time = 0.20429, size = 38, normalized size = 0.93 \[ \frac{1}{56} \,{\left (7 \, b c^{2} x^{8}{\rm sign}\left (x\right ) + 8 \, a c^{2} x^{7}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/56*(7*b*c^2*x^8*sign(x) + 8*a*c^2*x^7*sign(x))*sqrt(c)