3.1445 \(\int (a+b x)^5 \sqrt{a c+b c x} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{13/2}}{13 b c^6} \]

[Out]

(2*(a*c + b*c*x)^(13/2))/(13*b*c^6)

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Rubi [A]  time = 0.0136806, 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 \[ \frac{2 (a c+b c x)^{13/2}}{13 b c^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5*Sqrt[a*c + b*c*x],x]

[Out]

(2*(a*c + b*c*x)^(13/2))/(13*b*c^6)

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Rubi in Sympy [A]  time = 4.38896, size = 19, normalized size = 0.86 \[ \frac{2 \left (a c + b c x\right )^{\frac{13}{2}}}{13 b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5*(b*c*x+a*c)**(1/2),x)

[Out]

2*(a*c + b*c*x)**(13/2)/(13*b*c**6)

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Mathematica [A]  time = 0.0174487, size = 25, normalized size = 1.14 \[ \frac{2 (a+b x)^6 \sqrt{c (a+b x)}}{13 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^6*Sqrt[c*(a + b*x)])/(13*b)

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Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \[{\frac{2\, \left ( bx+a \right ) ^{6}}{13\,b}\sqrt{bcx+ac}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(b*c*x+a*c)^(1/2),x)

[Out]

2/13*(b*x+a)^6*(b*c*x+a*c)^(1/2)/b

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Maxima [A]  time = 1.32059, size = 24, normalized size = 1.09 \[ \frac{2 \,{\left (b c x + a c\right )}^{\frac{13}{2}}}{13 \, b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*(b*c*x + a*c)^(13/2)/(b*c^6)

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Fricas [A]  time = 0.210378, size = 101, normalized size = 4.59 \[ \frac{2 \,{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \sqrt{b c x + a c}}{13 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*(b^6*x^6 + 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 +
6*a^5*b*x + a^6)*sqrt(b*c*x + a*c)/b

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Sympy [A]  time = 1.53358, size = 66, normalized size = 3. \[ \begin{cases} \frac{2 b^{\frac{11}{2}} \sqrt{c} \left (\frac{a}{b} + x\right )^{\frac{13}{2}}}{13} & \text{for}\: \left |{\frac{a}{b} + x}\right | < 1 \\b^{\frac{11}{2}} \sqrt{c}{G_{2, 2}^{1, 1}\left (\begin{matrix} 1 & \frac{15}{2} \\\frac{13}{2} & 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )} + b^{\frac{11}{2}} \sqrt{c}{G_{2, 2}^{0, 2}\left (\begin{matrix} \frac{15}{2}, 1 & \\ & \frac{13}{2}, 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5*(b*c*x+a*c)**(1/2),x)

[Out]

Piecewise((2*b**(11/2)*sqrt(c)*(a/b + x)**(13/2)/13, Abs(a/b + x) < 1), (b**(11/
2)*sqrt(c)*meijerg(((1,), (15/2,)), ((13/2,), (0,)), a/b + x) + b**(11/2)*sqrt(c
)*meijerg(((15/2, 1), ()), ((), (13/2, 0)), a/b + x), True))

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GIAC/XCAS [A]  time = 0.22327, size = 621, normalized size = 28.23 \[ \frac{2 \,{\left (3003 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{5} - \frac{3003 \,{\left (5 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}\right )} a^{4}}{c} + \frac{858 \,{\left (35 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{2} b^{12} c^{14} - 42 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a b^{12} c^{13} + 15 \,{\left (b c x + a c\right )}^{\frac{7}{2}} b^{12} c^{12}\right )} a^{3}}{b^{12} c^{14}} - \frac{286 \,{\left (105 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{3} b^{24} c^{27} - 189 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{2} b^{24} c^{26} + 135 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a b^{24} c^{25} - 35 \,{\left (b c x + a c\right )}^{\frac{9}{2}} b^{24} c^{24}\right )} a^{2}}{b^{24} c^{27}} + \frac{13 \,{\left (1155 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{4} b^{40} c^{44} - 2772 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{3} b^{40} c^{43} + 2970 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a^{2} b^{40} c^{42} - 1540 \,{\left (b c x + a c\right )}^{\frac{9}{2}} a b^{40} c^{41} + 315 \,{\left (b c x + a c\right )}^{\frac{11}{2}} b^{40} c^{40}\right )} a}{b^{40} c^{44}} - \frac{3003 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{5} b^{60} c^{65} - 9009 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{4} b^{60} c^{64} + 12870 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a^{3} b^{60} c^{63} - 10010 \,{\left (b c x + a c\right )}^{\frac{9}{2}} a^{2} b^{60} c^{62} + 4095 \,{\left (b c x + a c\right )}^{\frac{11}{2}} a b^{60} c^{61} - 693 \,{\left (b c x + a c\right )}^{\frac{13}{2}} b^{60} c^{60}}{b^{60} c^{65}}\right )}}{9009 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9009*(3003*(b*c*x + a*c)^(3/2)*a^5 - 3003*(5*(b*c*x + a*c)^(3/2)*a*c - 3*(b*c*
x + a*c)^(5/2))*a^4/c + 858*(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^3/(b^12*c^14) - 286
*(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^2/(b
^24*c^27) + 13*(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)*a/(b^40*c^44) - (3003*(b*c
*x + a*c)^(3/2)*a^5*b^60*c^65 - 9009*(b*c*x + a*c)^(5/2)*a^4*b^60*c^64 + 12870*(
b*c*x + a*c)^(7/2)*a^3*b^60*c^63 - 10010*(b*c*x + a*c)^(9/2)*a^2*b^60*c^62 + 409
5*(b*c*x + a*c)^(11/2)*a*b^60*c^61 - 693*(b*c*x + a*c)^(13/2)*b^60*c^60)/(b^60*c
^65))/(b*c)