3.2806 \(\int \left (c (a+b x)^3\right )^{5/2} \, dx\)

Optimal. Leaf size=30 \[ \frac{2 c^2 (a+b x)^7 \sqrt{c (a+b x)^3}}{17 b} \]

[Out]

(2*c^2*(a + b*x)^7*Sqrt[c*(a + b*x)^3])/(17*b)

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Rubi [A]  time = 0.0348145, 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 \[ \frac{2 c^2 (a+b x)^7 \sqrt{c (a+b x)^3}}{17 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*(a + b*x)^3)^(5/2),x]

[Out]

(2*c^2*(a + b*x)^7*Sqrt[c*(a + b*x)^3])/(17*b)

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Rubi in Sympy [A]  time = 9.25371, size = 51, normalized size = 1.7 \[ \frac{2 \left (3 a + 3 b x\right ) \left (a^{3} c + 3 a^{2} b c x + 3 a b^{2} c x^{2} + b^{3} c x^{3}\right )^{\frac{5}{2}}}{51 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*(b*x+a)**3)**(5/2),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)**(5/2)/(
51*b)

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Mathematica [A]  time = 0.051539, size = 25, normalized size = 0.83 \[ \frac{2 (a+b x) \left (c (a+b x)^3\right )^{5/2}}{17 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*(a + b*x)^3)^(5/2),x]

[Out]

(2*(a + b*x)*(c*(a + b*x)^3)^(5/2))/(17*b)

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Maple [A]  time = 0.006, size = 22, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{17\,b} \left ( c \left ( bx+a \right ) ^{3} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*(b*x+a)^3)^(5/2),x)

[Out]

2/17*(b*x+a)*(c*(b*x+a)^3)^(5/2)/b

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Maxima [A]  time = 1.42313, size = 146, normalized size = 4.87 \[ \frac{2 \,{\left (b^{7} c^{\frac{5}{2}} x^{7} + 7 \, a b^{6} c^{\frac{5}{2}} x^{6} + 21 \, a^{2} b^{5} c^{\frac{5}{2}} x^{5} + 35 \, a^{3} b^{4} c^{\frac{5}{2}} x^{4} + 35 \, a^{4} b^{3} c^{\frac{5}{2}} x^{3} + 21 \, a^{5} b^{2} c^{\frac{5}{2}} x^{2} + 7 \, a^{6} b c^{\frac{5}{2}} x + a^{7} c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{17 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^3*c)^(5/2),x, algorithm="maxima")

[Out]

2/17*(b^7*c^(5/2)*x^7 + 7*a*b^6*c^(5/2)*x^6 + 21*a^2*b^5*c^(5/2)*x^5 + 35*a^3*b^
4*c^(5/2)*x^4 + 35*a^4*b^3*c^(5/2)*x^3 + 21*a^5*b^2*c^(5/2)*x^2 + 7*a^6*b*c^(5/2
)*x + a^7*c^(5/2))*(b*x + a)^(3/2)/b

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Fricas [A]  time = 0.21494, size = 182, normalized size = 6.07 \[ \frac{2 \,{\left (b^{7} c^{2} x^{7} + 7 \, a b^{6} c^{2} x^{6} + 21 \, a^{2} b^{5} c^{2} x^{5} + 35 \, a^{3} b^{4} c^{2} x^{4} + 35 \, a^{4} b^{3} c^{2} x^{3} + 21 \, a^{5} b^{2} c^{2} x^{2} + 7 \, a^{6} b c^{2} x + a^{7} c^{2}\right )} \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{17 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^3*c)^(5/2),x, algorithm="fricas")

[Out]

2/17*(b^7*c^2*x^7 + 7*a*b^6*c^2*x^6 + 21*a^2*b^5*c^2*x^5 + 35*a^3*b^4*c^2*x^4 +
35*a^4*b^3*c^2*x^3 + 21*a^5*b^2*c^2*x^2 + 7*a^6*b*c^2*x + a^7*c^2)*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(-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)**3)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233734, size = 1, normalized size = 0.03 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x + a)^3*c)^(5/2),x, algorithm="giac")

[Out]

Done