3.1447 \(\int \frac{(a+b x)^5}{(a c+b c x)^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{9/2}}{9 b c^6} \]

[Out]

(2*(a*c + b*c*x)^(9/2))/(9*b*c^6)

_______________________________________________________________________________________

Rubi [A]  time = 0.0137135, 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 \[ \frac{2 (a c+b c x)^{9/2}}{9 b c^6} \]

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)^(9/2))/(9*b*c^6)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.3532, size = 19, normalized size = 0.86 \[ \frac{2 \left (a c + b c x\right )^{\frac{9}{2}}}{9 b c^{6}} \]

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)**(9/2)/(9*b*c**6)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0217454, size = 25, normalized size = 1.14 \[ \frac{2 (a+b x)^6}{9 b (c (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^6)/(9*b*(c*(a + b*x))^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \[{\frac{2\, \left ( bx+a \right ) ^{6}}{9\,b} \left ( bcx+ac \right ) ^{-{\frac{3}{2}}}} \]

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]

2/9*(b*x+a)^6/b/(b*c*x+a*c)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.32729, size = 24, normalized size = 1.09 \[ \frac{2 \,{\left (b c x + a c\right )}^{\frac{9}{2}}}{9 \, b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(b*c*x + a*c)^(9/2)/(b*c^6)

_______________________________________________________________________________________

Fricas [A]  time = 0.220685, size = 76, normalized size = 3.45 \[ \frac{2 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt{b c x + a c}}{9 \, b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*sqrt(b*c*x + a*c)/
(b*c^2)

_______________________________________________________________________________________

Sympy [A]  time = 2.05422, size = 83, normalized size = 3.77 \[ \begin{cases} \frac{2 b^{\frac{7}{2}} \left (\frac{a}{b} + x\right )^{\frac{9}{2}}}{9 c^{\frac{3}{2}}} & \text{for}\: \left (\frac{a}{b} + x > -1 \wedge \frac{a}{b} + x < 1\right ) \vee \frac{a}{b} + x > 1 \vee \frac{a}{b} + x < -1 \\\frac{b^{\frac{7}{2}}{G_{2, 2}^{1, 1}\left (\begin{matrix} 1 & \frac{11}{2} \\\frac{9}{2} & 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )}}{c^{\frac{3}{2}}} + \frac{b^{\frac{7}{2}}{G_{2, 2}^{0, 2}\left (\begin{matrix} \frac{11}{2}, 1 & \\ & \frac{9}{2}, 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )}}{c^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

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**(7/2)*(a/b + x)**(9/2)/(9*c**(3/2)), (a/b + x > 1) | (a/b + x <
-1) | ((a/b + x > -1) & (a/b + x < 1))), (b**(7/2)*meijerg(((1,), (11/2,)), ((9/
2,), (0,)), a/b + x)/c**(3/2) + b**(7/2)*meijerg(((11/2, 1), ()), ((), (9/2, 0))
, a/b + x)/c**(3/2), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219783, size = 440, normalized size = 20. \[ \frac{2 \,{\left (315 \, \sqrt{b c x + a c} a^{4} - \frac{420 \,{\left (3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}\right )} a^{3}}{c} + \frac{126 \,{\left (15 \, \sqrt{b c x + a c} a^{2} b^{8} c^{10} - 10 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a b^{8} c^{9} + 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}} b^{8} c^{8}\right )} a^{2}}{b^{8} c^{10}} - \frac{36 \,{\left (35 \, \sqrt{b c x + a c} a^{3} b^{18} c^{21} - 35 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{2} b^{18} c^{20} + 21 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a b^{18} c^{19} - 5 \,{\left (b c x + a c\right )}^{\frac{7}{2}} b^{18} c^{18}\right )} a}{b^{18} c^{21}} + \frac{315 \, \sqrt{b c x + a c} a^{4} b^{32} c^{36} - 420 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{3} b^{32} c^{35} + 378 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{2} b^{32} c^{34} - 180 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a b^{32} c^{33} + 35 \,{\left (b c x + a c\right )}^{\frac{9}{2}} b^{32} c^{32}}{b^{32} c^{36}}\right )}}{315 \, b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(315*sqrt(b*c*x + a*c)*a^4 - 420*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^
(3/2))*a^3/c + 126*(15*sqrt(b*c*x + a*c)*a^2*b^8*c^10 - 10*(b*c*x + a*c)^(3/2)*a
*b^8*c^9 + 3*(b*c*x + a*c)^(5/2)*b^8*c^8)*a^2/(b^8*c^10) - 36*(35*sqrt(b*c*x + a
*c)*a^3*b^18*c^21 - 35*(b*c*x + a*c)^(3/2)*a^2*b^18*c^20 + 21*(b*c*x + a*c)^(5/2
)*a*b^18*c^19 - 5*(b*c*x + a*c)^(7/2)*b^18*c^18)*a/(b^18*c^21) + (315*sqrt(b*c*x
+ a*c)*a^4*b^32*c^36 - 420*(b*c*x + a*c)^(3/2)*a^3*b^32*c^35 + 378*(b*c*x + a*c
)^(5/2)*a^2*b^32*c^34 - 180*(b*c*x + a*c)^(7/2)*a*b^32*c^33 + 35*(b*c*x + a*c)^(
9/2)*b^32*c^32)/(b^32*c^36))/(b*c^2)