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

Optimal. Leaf size=66 \[ \frac{1}{7} a^2 c^2 x^6 \sqrt{c x^2}+\frac{1}{4} a b c^2 x^7 \sqrt{c x^2}+\frac{1}{9} b^2 c^2 x^8 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0466228, antiderivative size = 66, 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 \[ \frac{1}{7} a^2 c^2 x^6 \sqrt{c x^2}+\frac{1}{4} a b c^2 x^7 \sqrt{c x^2}+\frac{1}{9} b^2 c^2 x^8 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0130076, size = 35, normalized size = 0.53 \[ \frac{1}{252} x^2 \left (c x^2\right )^{5/2} \left (36 a^2+63 a b x+28 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(c*x^2)^(5/2)*(36*a^2 + 63*a*b*x + 28*b^2*x^2))/252

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Maple [A]  time = 0.007, size = 32, normalized size = 0.5 \[{\frac{{x}^{2} \left ( 28\,{b}^{2}{x}^{2}+63\,abx+36\,{a}^{2} \right ) }{252} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/252*x^2*(28*b^2*x^2+63*a*b*x+36*a^2)*(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)^2*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.205066, size = 57, normalized size = 0.86 \[ \frac{1}{252} \,{\left (28 \, b^{2} c^{2} x^{8} + 63 \, a b c^{2} x^{7} + 36 \, a^{2} c^{2} x^{6}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.44826, size = 60, normalized size = 0.91 \[ \frac{a^{2} c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}}{7} + \frac{a b c^{\frac{5}{2}} x^{3} \left (x^{2}\right )^{\frac{5}{2}}}{4} + \frac{b^{2} c^{\frac{5}{2}} x^{4} \left (x^{2}\right )^{\frac{5}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.205685, size = 59, normalized size = 0.89 \[ \frac{1}{252} \,{\left (28 \, b^{2} c^{2} x^{9}{\rm sign}\left (x\right ) + 63 \, a b c^{2} x^{8}{\rm sign}\left (x\right ) + 36 \, a^{2} c^{2} x^{7}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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