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

Optimal. Leaf size=66 \[ \frac{1}{5} a^2 c^2 x^4 \sqrt{c x^2}+\frac{1}{3} a b c^2 x^5 \sqrt{c x^2}+\frac{1}{7} b^2 c^2 x^6 \sqrt{c x^2} \]

[Out]

(a^2*c^2*x^4*Sqrt[c*x^2])/5 + (a*b*c^2*x^5*Sqrt[c*x^2])/3 + (b^2*c^2*x^6*Sqrt[c*
x^2])/7

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Rubi [A]  time = 0.0414672, antiderivative size = 66, 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 \[ \frac{1}{5} a^2 c^2 x^4 \sqrt{c x^2}+\frac{1}{3} a b c^2 x^5 \sqrt{c x^2}+\frac{1}{7} b^2 c^2 x^6 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*c^2*x^4*Sqrt[c*x^2])/5 + (a*b*c^2*x^5*Sqrt[c*x^2])/3 + (b^2*c^2*x^6*Sqrt[c*
x^2])/7

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Rubi in Sympy [A]  time = 17.7404, size = 60, normalized size = 0.91 \[ \frac{a^{2} c^{2} x^{4} \sqrt{c x^{2}}}{5} + \frac{a b c^{2} x^{5} \sqrt{c x^{2}}}{3} + \frac{b^{2} c^{2} x^{6} \sqrt{c x^{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**2*x**4*sqrt(c*x**2)/5 + a*b*c**2*x**5*sqrt(c*x**2)/3 + b**2*c**2*x**6*sq
rt(c*x**2)/7

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Mathematica [A]  time = 0.00561474, size = 36, normalized size = 0.55 \[ \frac{1}{105} c x^2 \left (c x^2\right )^{3/2} \left (21 a^2+35 a b x+15 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(c*x^2*(c*x^2)^(3/2)*(21*a^2 + 35*a*b*x + 15*b^2*x^2))/105

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Maple [A]  time = 0.007, size = 29, normalized size = 0.4 \[{\frac{15\,{b}^{2}{x}^{2}+35\,abx+21\,{a}^{2}}{105} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*b^2*x^2+35*a*b*x+21*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 = 0.199887, size = 57, normalized size = 0.86 \[ \frac{1}{105} \,{\left (15 \, b^{2} c^{2} x^{6} + 35 \, a b c^{2} x^{5} + 21 \, a^{2} c^{2} x^{4}\right )} \sqrt{c x^{2}} \]

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]

1/105*(15*b^2*c^2*x^6 + 35*a*b*c^2*x^5 + 21*a^2*c^2*x^4)*sqrt(c*x^2)

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Sympy [A]  time = 6.94461, size = 54, normalized size = 0.82 \[ \frac{a^{2} c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{5} + \frac{a b c^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}}{3} + \frac{b^{2} c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209671, size = 59, normalized size = 0.89 \[ \frac{1}{105} \,{\left (15 \, b^{2} c^{2} x^{7}{\rm sign}\left (x\right ) + 35 \, a b c^{2} x^{6}{\rm sign}\left (x\right ) + 21 \, a^{2} c^{2} x^{5}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

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]

1/105*(15*b^2*c^2*x^7*sign(x) + 35*a*b*c^2*x^6*sign(x) + 21*a^2*c^2*x^5*sign(x))
*sqrt(c)