3.1444 \(\int (a+b x)^5 (a c+b c x)^{3/2} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{15/2}}{15 b c^6} \]

[Out]

(2*(a*c + b*c*x)^(15/2))/(15*b*c^6)

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Rubi [A]  time = 0.0145839, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 (a c+b c x)^{15/2}}{15 b c^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*c + b*c*x)^(15/2))/(15*b*c^6)

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Rubi in Sympy [A]  time = 4.26134, size = 19, normalized size = 0.86 \[ \frac{2 \left (a c + b c x\right )^{\frac{15}{2}}}{15 b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a*c + b*c*x)**(15/2)/(15*b*c**6)

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Mathematica [A]  time = 0.0312204, size = 25, normalized size = 1.14 \[ \frac{2 (a+b x)^6 (c (a+b x))^{3/2}}{15 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^6*(c*(a + b*x))^(3/2))/(15*b)

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Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \[{\frac{2\, \left ( bx+a \right ) ^{6}}{15\,b} \left ( bcx+ac \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(b*x+a)^6*(b*c*x+a*c)^(3/2)/b

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Maxima [A]  time = 1.35065, size = 24, normalized size = 1.09 \[ \frac{2 \,{\left (b c x + a c\right )}^{\frac{15}{2}}}{15 \, b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(b*c*x + a*c)^(15/2)/(b*c^6)

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Fricas [A]  time = 0.2041, size = 128, normalized size = 5.82 \[ \frac{2 \,{\left (b^{7} c x^{7} + 7 \, a b^{6} c x^{6} + 21 \, a^{2} b^{5} c x^{5} + 35 \, a^{3} b^{4} c x^{4} + 35 \, a^{4} b^{3} c x^{3} + 21 \, a^{5} b^{2} c x^{2} + 7 \, a^{6} b c x + a^{7} c\right )} \sqrt{b c x + a c}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(b^7*c*x^7 + 7*a*b^6*c*x^6 + 21*a^2*b^5*c*x^5 + 35*a^3*b^4*c*x^4 + 35*a^4*b
^3*c*x^3 + 21*a^5*b^2*c*x^2 + 7*a^6*b*c*x + a^7*c)*sqrt(b*c*x + a*c)/b

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Sympy [A]  time = 2.04953, size = 66, normalized size = 3. \[ \begin{cases} \frac{2 b^{\frac{13}{2}} c^{\frac{3}{2}} \left (\frac{a}{b} + x\right )^{\frac{15}{2}}}{15} & \text{for}\: \left |{\frac{a}{b} + x}\right | < 1 \\b^{\frac{13}{2}} c^{\frac{3}{2}}{G_{2, 2}^{1, 1}\left (\begin{matrix} 1 & \frac{17}{2} \\\frac{15}{2} & 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )} + b^{\frac{13}{2}} c^{\frac{3}{2}}{G_{2, 2}^{0, 2}\left (\begin{matrix} \frac{17}{2}, 1 & \\ & \frac{15}{2}, 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*b**(13/2)*c**(3/2)*(a/b + x)**(15/2)/15, Abs(a/b + x) < 1), (b**(13
/2)*c**(3/2)*meijerg(((1,), (17/2,)), ((15/2,), (0,)), a/b + x) + b**(13/2)*c**(
3/2)*meijerg(((17/2, 1), ()), ((), (15/2, 0)), a/b + x), True))

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GIAC/XCAS [A]  time = 0.231747, size = 1, normalized size = 0.05 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done