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

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

[Out]

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

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Rubi [A]  time = 0.041474, 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 \[ \frac{1}{4} a^2 c^2 x^3 \sqrt{c x^2}+\frac{2}{5} a b c^2 x^4 \sqrt{c x^2}+\frac{1}{6} b^2 c^2 x^5 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.6533, size = 61, normalized size = 0.92 \[ \frac{a^{2} c^{2} x^{3} \sqrt{c x^{2}}}{4} + \frac{2 a b c^{2} x^{4} \sqrt{c x^{2}}}{5} + \frac{b^{2} c^{2} x^{5} \sqrt{c x^{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00753272, size = 34, normalized size = 0.52 \[ \frac{1}{60} c x \left (c x^2\right )^{3/2} \left (15 a^2+24 a b x+10 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(c*x^2)^(3/2)*(15*a^2 + 24*a*b*x + 10*b^2*x^2))/60

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Maple [A]  time = 0.006, size = 32, normalized size = 0.5 \[{\frac{10\,{b}^{2}{x}^{2}+24\,abx+15\,{a}^{2}}{60\,x} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60/x*(10*b^2*x^2+24*a*b*x+15*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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.201348, size = 57, normalized size = 0.86 \[ \frac{1}{60} \,{\left (10 \, b^{2} c^{2} x^{5} + 24 \, a b c^{2} x^{4} + 15 \, a^{2} c^{2} x^{3}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*b^2*c^2*x^5 + 24*a*b*c^2*x^4 + 15*a^2*c^2*x^3)*sqrt(c*x^2)

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Sympy [A]  time = 6.83661, size = 54, normalized size = 0.82 \[ \frac{a^{2} c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{4 x} + \frac{2 a b c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{5} + \frac{b^{2} c^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**(5/2)*(x**2)**(5/2)/(4*x) + 2*a*b*c**(5/2)*(x**2)**(5/2)/5 + b**2*c**(5/
2)*x*(x**2)**(5/2)/6

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GIAC/XCAS [A]  time = 0.20637, size = 59, normalized size = 0.89 \[ \frac{1}{60} \,{\left (10 \, b^{2} c^{2} x^{6}{\rm sign}\left (x\right ) + 24 \, a b c^{2} x^{5}{\rm sign}\left (x\right ) + 15 \, a^{2} c^{2} x^{4}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*b^2*c^2*x^6*sign(x) + 24*a*b*c^2*x^5*sign(x) + 15*a^2*c^2*x^4*sign(x))*
sqrt(c)