3.2796 \(\int \left (c (a+b x)^2\right )^{3/2} \, dx\)

Optimal. Leaf size=28 \[ \frac{c (a+b x)^3 \sqrt{c (a+b x)^2}}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0286164, antiderivative size = 28, 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 \[ \frac{c (a+b x)^3 \sqrt{c (a+b x)^2}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*(a + b*x)^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.7658, size = 36, normalized size = 1.29 \[ \frac{\left (2 a + 2 b x\right ) \left (a^{2} c + 2 a b c x + b^{2} c x^{2}\right )^{\frac{3}{2}}}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*(b*x+a)**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0186441, size = 25, normalized size = 0.89 \[ \frac{(a+b x) \left (c (a+b x)^2\right )^{3/2}}{4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*(a + b*x)^2)^(3/2),x]

[Out]

((a + b*x)*(c*(a + b*x)^2)^(3/2))/(4*b)

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Maple [B]  time = 0.005, size = 51, normalized size = 1.8 \[{\frac{x \left ({b}^{3}{x}^{3}+4\,a{b}^{2}{x}^{2}+6\,{a}^{2}bx+4\,{a}^{3} \right ) }{4\, \left ( bx+a \right ) ^{3}} \left ( c \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*(b*x+a)^2)^(3/2),x)

[Out]

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

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.212155, size = 90, normalized size = 3.21 \[ \frac{{\left (b^{3} c x^{4} + 4 \, a b^{2} c x^{3} + 6 \, a^{2} b c x^{2} + 4 \, a^{3} c x\right )} \sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{4 \,{\left (b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^2*c)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (c \left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*(b*x+a)**2)**(3/2),x)

[Out]

Integral((c*(a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.217388, size = 28, normalized size = 1. \[ \frac{{\left (b x + a\right )}^{4} c^{\frac{3}{2}}{\rm sign}\left (b x + a\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^2*c)^(3/2),x, algorithm="giac")

[Out]

1/4*(b*x + a)^4*c^(3/2)*sign(b*x + a)/b