3.2795 \(\int \left (c (a+b x)^2\right )^{5/2} \, dx\)

Optimal. Leaf size=30 \[ \frac{c^2 (a+b x)^5 \sqrt{c (a+b x)^2}}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.0323324, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{c^2 (a+b x)^5 \sqrt{c (a+b x)^2}}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*(a + b*x)^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.7465, size = 36, normalized size = 1.2 \[ \frac{\left (2 a + 2 b x\right ) \left (a^{2} c + 2 a b c x + b^{2} c x^{2}\right )^{\frac{5}{2}}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a + 2*b*x)*(a**2*c + 2*a*b*c*x + b**2*c*x**2)**(5/2)/(12*b)

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Mathematica [A]  time = 0.0352266, size = 25, normalized size = 0.83 \[ \frac{(a+b x) \left (c (a+b x)^2\right )^{5/2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 73, normalized size = 2.4 \[{\frac{x \left ({b}^{5}{x}^{5}+6\,a{b}^{4}{x}^{4}+15\,{a}^{2}{b}^{3}{x}^{3}+20\,{a}^{3}{b}^{2}{x}^{2}+15\,{a}^{4}bx+6\,{a}^{5} \right ) }{6\, \left ( bx+a \right ) ^{5}} \left ( c \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*(b*x+a)^2)^(5/2),x)

[Out]

1/6*x*(b^5*x^5+6*a*b^4*x^4+15*a^2*b^3*x^3+20*a^3*b^2*x^2+15*a^4*b*x+6*a^5)*(c*(b
*x+a)^2)^(5/2)/(b*x+a)^5

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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(((b*x + a)^2*c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.210082, size = 139, normalized size = 4.63 \[ \frac{{\left (b^{5} c^{2} x^{6} + 6 \, a b^{4} c^{2} x^{5} + 15 \, a^{2} b^{3} c^{2} x^{4} + 20 \, a^{3} b^{2} c^{2} x^{3} + 15 \, a^{4} b c^{2} x^{2} + 6 \, a^{5} c^{2} x\right )} \sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{6 \,{\left (b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^5*c^2*x^6 + 6*a*b^4*c^2*x^5 + 15*a^2*b^3*c^2*x^4 + 20*a^3*b^2*c^2*x^3 + 1
5*a^4*b*c^2*x^2 + 6*a^5*c^2*x)*sqrt(b^2*c*x^2 + 2*a*b*c*x + a^2*c)/(b*x + a)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218288, size = 174, normalized size = 5.8 \[ \frac{1}{6} \,{\left (b^{5} c^{2} x^{6}{\rm sign}\left (b x + a\right ) + 6 \, a b^{4} c^{2} x^{5}{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b^{3} c^{2} x^{4}{\rm sign}\left (b x + a\right ) + 20 \, a^{3} b^{2} c^{2} x^{3}{\rm sign}\left (b x + a\right ) + 15 \, a^{4} b c^{2} x^{2}{\rm sign}\left (b x + a\right ) + 6 \, a^{5} c^{2} x{\rm sign}\left (b x + a\right ) + \frac{a^{6} c^{2}{\rm sign}\left (b x + a\right )}{b}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(b^5*c^2*x^6*sign(b*x + a) + 6*a*b^4*c^2*x^5*sign(b*x + a) + 15*a^2*b^3*c^2*
x^4*sign(b*x + a) + 20*a^3*b^2*c^2*x^3*sign(b*x + a) + 15*a^4*b*c^2*x^2*sign(b*x
+ a) + 6*a^5*c^2*x*sign(b*x + a) + a^6*c^2*sign(b*x + a)/b)*sqrt(c)