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

Optimal. Leaf size=66 \[ \frac{1}{6} a^2 c^2 x^5 \sqrt{c x^2}+\frac{2}{7} a b c^2 x^6 \sqrt{c x^2}+\frac{1}{8} b^2 c^2 x^7 \sqrt{c x^2} \]

[Out]

(a^2*c^2*x^5*Sqrt[c*x^2])/6 + (2*a*b*c^2*x^6*Sqrt[c*x^2])/7 + (b^2*c^2*x^7*Sqrt[
c*x^2])/8

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Rubi [A]  time = 0.0445365, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{1}{6} a^2 c^2 x^5 \sqrt{c x^2}+\frac{2}{7} a b c^2 x^6 \sqrt{c x^2}+\frac{1}{8} b^2 c^2 x^7 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*c^2*x^5*Sqrt[c*x^2])/6 + (2*a*b*c^2*x^6*Sqrt[c*x^2])/7 + (b^2*c^2*x^7*Sqrt[
c*x^2])/8

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Rubi in Sympy [A]  time = 12.5922, size = 61, normalized size = 0.92 \[ \frac{a^{2} c^{2} x^{5} \sqrt{c x^{2}}}{6} + \frac{2 a b c^{2} x^{6} \sqrt{c x^{2}}}{7} + \frac{b^{2} c^{2} x^{7} \sqrt{c x^{2}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**2*x**5*sqrt(c*x**2)/6 + 2*a*b*c**2*x**6*sqrt(c*x**2)/7 + b**2*c**2*x**7*
sqrt(c*x**2)/8

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Mathematica [A]  time = 0.0125648, size = 33, normalized size = 0.5 \[ \frac{1}{168} x \left (c x^2\right )^{5/2} \left (28 a^2+48 a b x+21 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*x^2)^(5/2)*(28*a^2 + 48*a*b*x + 21*b^2*x^2))/168

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Maple [A]  time = 0.007, size = 30, normalized size = 0.5 \[{\frac{x \left ( 21\,{b}^{2}{x}^{2}+48\,abx+28\,{a}^{2} \right ) }{168} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(5/2)*(b*x+a)^2,x)

[Out]

1/168*x*(21*b^2*x^2+48*a*b*x+28*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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.205562, size = 57, normalized size = 0.86 \[ \frac{1}{168} \,{\left (21 \, b^{2} c^{2} x^{7} + 48 \, a b c^{2} x^{6} + 28 \, a^{2} c^{2} x^{5}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/168*(21*b^2*c^2*x^7 + 48*a*b*c^2*x^6 + 28*a^2*c^2*x^5)*sqrt(c*x^2)

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Sympy [A]  time = 6.81288, size = 60, normalized size = 0.91 \[ \frac{a^{2} c^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}}{6} + \frac{2 a b c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}}{7} + \frac{b^{2} c^{\frac{5}{2}} x^{3} \left (x^{2}\right )^{\frac{5}{2}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**(5/2)*x*(x**2)**(5/2)/6 + 2*a*b*c**(5/2)*x**2*(x**2)**(5/2)/7 + b**2*c**
(5/2)*x**3*(x**2)**(5/2)/8

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GIAC/XCAS [A]  time = 0.20669, size = 59, normalized size = 0.89 \[ \frac{1}{168} \,{\left (21 \, b^{2} c^{2} x^{8}{\rm sign}\left (x\right ) + 48 \, a b c^{2} x^{7}{\rm sign}\left (x\right ) + 28 \, a^{2} c^{2} x^{6}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/168*(21*b^2*c^2*x^8*sign(x) + 48*a*b*c^2*x^7*sign(x) + 28*a^2*c^2*x^6*sign(x))
*sqrt(c)