3.2807 \(\int \left (c (a+b x)^3\right )^{3/2} \, dx\)

Optimal. Leaf size=28 \[ \frac{2 c (a+b x)^4 \sqrt{c (a+b x)^3}}{11 b} \]

[Out]

(2*c*(a + b*x)^4*Sqrt[c*(a + b*x)^3])/(11*b)

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Rubi [A]  time = 0.0310163, 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 \[ \frac{2 c (a+b x)^4 \sqrt{c (a+b x)^3}}{11 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*(a + b*x)^3)^(3/2),x]

[Out]

(2*c*(a + b*x)^4*Sqrt[c*(a + b*x)^3])/(11*b)

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Rubi in Sympy [A]  time = 9.2625, size = 51, normalized size = 1.82 \[ \frac{2 \left (3 a + 3 b x\right ) \left (a^{3} c + 3 a^{2} b c x + 3 a b^{2} c x^{2} + b^{3} c x^{3}\right )^{\frac{3}{2}}}{33 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0229572, size = 25, normalized size = 0.89 \[ \frac{2 (a+b x) \left (c (a+b x)^3\right )^{3/2}}{11 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)*(c*(a + b*x)^3)^(3/2))/(11*b)

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Maple [A]  time = 0.004, size = 22, normalized size = 0.8 \[{\frac{2\,bx+2\,a}{11\,b} \left ( c \left ( bx+a \right ) ^{3} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*(b*x+a)^3)^(3/2),x)

[Out]

2/11*(b*x+a)*(c*(b*x+a)^3)^(3/2)/b

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Maxima [A]  time = 1.41236, size = 89, normalized size = 3.18 \[ \frac{2 \,{\left (b^{4} c^{\frac{3}{2}} x^{4} + 4 \, a b^{3} c^{\frac{3}{2}} x^{3} + 6 \, a^{2} b^{2} c^{\frac{3}{2}} x^{2} + 4 \, a^{3} b c^{\frac{3}{2}} x + a^{4} c^{\frac{3}{2}}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{11 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*(b^4*c^(3/2)*x^4 + 4*a*b^3*c^(3/2)*x^3 + 6*a^2*b^2*c^(3/2)*x^2 + 4*a^3*b*c^
(3/2)*x + a^4*c^(3/2))*(b*x + a)^(3/2)/b

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Fricas [A]  time = 0.213404, size = 112, normalized size = 4. \[ \frac{2 \,{\left (b^{4} c x^{4} + 4 \, a b^{3} c x^{3} + 6 \, a^{2} b^{2} c x^{2} + 4 \, a^{3} b c x + a^{4} c\right )} \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{11 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*(b^4*c*x^4 + 4*a*b^3*c*x^3 + 6*a^2*b^2*c*x^2 + 4*a^3*b*c*x + a^4*c)*sqrt(b^
3*c*x^3 + 3*a*b^2*c*x^2 + 3*a^2*b*c*x + a^3*c)/b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((c*(a + b*x)**3)**(3/2), x)

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GIAC/XCAS [A]  time = 0.225506, size = 477, normalized size = 17.04 \[ \frac{2 \,{\left (1155 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{4}{\rm sign}\left (b x + a\right ) - \frac{924 \,{\left (5 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}\right )} a^{3}{\rm sign}\left (b x + a\right )}{c} + \frac{198 \,{\left (35 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{2} b^{12} c^{14} - 42 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a b^{12} c^{13} + 15 \,{\left (b c x + a c\right )}^{\frac{7}{2}} b^{12} c^{12}\right )} a^{2}{\rm sign}\left (b x + a\right )}{b^{12} c^{14}} - \frac{44 \,{\left (105 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{3} b^{24} c^{27} - 189 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{2} b^{24} c^{26} + 135 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a b^{24} c^{25} - 35 \,{\left (b c x + a c\right )}^{\frac{9}{2}} b^{24} c^{24}\right )} a{\rm sign}\left (b x + a\right )}{b^{24} c^{27}} + \frac{{\left (1155 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{4} b^{40} c^{44} - 2772 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{3} b^{40} c^{43} + 2970 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a^{2} b^{40} c^{42} - 1540 \,{\left (b c x + a c\right )}^{\frac{9}{2}} a b^{40} c^{41} + 315 \,{\left (b c x + a c\right )}^{\frac{11}{2}} b^{40} c^{40}\right )}{\rm sign}\left (b x + a\right )}{b^{40} c^{44}}\right )}}{3465 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(1155*(b*c*x + a*c)^(3/2)*a^4*sign(b*x + a) - 924*(5*(b*c*x + a*c)^(3/2)*
a*c - 3*(b*c*x + a*c)^(5/2))*a^3*sign(b*x + a)/c + 198*(35*(b*c*x + a*c)^(3/2)*a
^2*b^12*c^14 - 42*(b*c*x + a*c)^(5/2)*a*b^12*c^13 + 15*(b*c*x + a*c)^(7/2)*b^12*
c^12)*a^2*sign(b*x + a)/(b^12*c^14) - 44*(105*(b*c*x + a*c)^(3/2)*a^3*b^24*c^27
- 189*(b*c*x + a*c)^(5/2)*a^2*b^24*c^26 + 135*(b*c*x + a*c)^(7/2)*a*b^24*c^25 -
35*(b*c*x + a*c)^(9/2)*b^24*c^24)*a*sign(b*x + a)/(b^24*c^27) + (1155*(b*c*x + a
*c)^(3/2)*a^4*b^40*c^44 - 2772*(b*c*x + a*c)^(5/2)*a^3*b^40*c^43 + 2970*(b*c*x +
a*c)^(7/2)*a^2*b^40*c^42 - 1540*(b*c*x + a*c)^(9/2)*a*b^40*c^41 + 315*(b*c*x +
a*c)^(11/2)*b^40*c^40)*sign(b*x + a)/(b^40*c^44))/b