3.1446 \(\int \frac{(a+b x)^5}{\sqrt{a c+b c x}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{11/2}}{11 b c^6} \]

[Out]

(2*(a*c + b*c*x)^(11/2))/(11*b*c^6)

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Rubi [A]  time = 0.0135094, 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 \[ \frac{2 (a c+b c x)^{11/2}}{11 b c^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*c + b*c*x)^(11/2))/(11*b*c^6)

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Rubi in Sympy [A]  time = 4.40205, size = 19, normalized size = 0.86 \[ \frac{2 \left (a c + b c x\right )^{\frac{11}{2}}}{11 b c^{6}} \]

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)**(11/2)/(11*b*c**6)

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Mathematica [A]  time = 0.0199615, size = 25, normalized size = 1.14 \[ \frac{2 (a+b x)^6}{11 b \sqrt{c (a+b x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^6)/(11*b*Sqrt[c*(a + b*x)])

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Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \[{\frac{2\, \left ( bx+a \right ) ^{6}}{11\,b}{\frac{1}{\sqrt{bcx+ac}}}} \]

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]

2/11*(b*x+a)^6/b/(b*c*x+a*c)^(1/2)

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Maxima [A]  time = 1.35638, size = 505, normalized size = 22.95 \[ \frac{2 \,{\left (693 \, \sqrt{b c x + a c} a^{5} - \frac{1155 \,{\left (3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}\right )} a^{4}}{c} + \frac{462 \,{\left (15 \, \sqrt{b c x + a c} a^{2} c^{2} - 10 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c + 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}\right )} a^{3}}{c^{2}} - \frac{198 \,{\left (35 \, \sqrt{b c x + a c} a^{3} c^{3} - 35 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{2} c^{2} + 21 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a c - 5 \,{\left (b c x + a c\right )}^{\frac{7}{2}}\right )} a^{2}}{c^{3}} + \frac{11 \,{\left (315 \, \sqrt{b c x + a c} a^{4} c^{4} - 420 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{3} c^{3} + 378 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{2} c^{2} - 180 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a c + 35 \,{\left (b c x + a c\right )}^{\frac{9}{2}}\right )} a}{c^{4}} - \frac{693 \, \sqrt{b c x + a c} a^{5} c^{5} - 1155 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{4} c^{4} + 1386 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{3} c^{3} - 990 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a^{2} c^{2} + 385 \,{\left (b c x + a c\right )}^{\frac{9}{2}} a c - 63 \,{\left (b c x + a c\right )}^{\frac{11}{2}}}{c^{5}}\right )}}{693 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*(693*sqrt(b*c*x + a*c)*a^5 - 1155*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)
^(3/2))*a^4/c + 462*(15*sqrt(b*c*x + a*c)*a^2*c^2 - 10*(b*c*x + a*c)^(3/2)*a*c +
3*(b*c*x + a*c)^(5/2))*a^3/c^2 - 198*(35*sqrt(b*c*x + a*c)*a^3*c^3 - 35*(b*c*x
+ a*c)^(3/2)*a^2*c^2 + 21*(b*c*x + a*c)^(5/2)*a*c - 5*(b*c*x + a*c)^(7/2))*a^2/c
^3 + 11*(315*sqrt(b*c*x + a*c)*a^4*c^4 - 420*(b*c*x + a*c)^(3/2)*a^3*c^3 + 378*(
b*c*x + a*c)^(5/2)*a^2*c^2 - 180*(b*c*x + a*c)^(7/2)*a*c + 35*(b*c*x + a*c)^(9/2
))*a/c^4 - (693*sqrt(b*c*x + a*c)*a^5*c^5 - 1155*(b*c*x + a*c)^(3/2)*a^4*c^4 + 1
386*(b*c*x + a*c)^(5/2)*a^3*c^3 - 990*(b*c*x + a*c)^(7/2)*a^2*c^2 + 385*(b*c*x +
a*c)^(9/2)*a*c - 63*(b*c*x + a*c)^(11/2))/c^5)/(b*c)

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Fricas [A]  time = 0.215835, size = 90, normalized size = 4.09 \[ \frac{2 \,{\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \sqrt{b c x + a c}}{11 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5)
*sqrt(b*c*x + a*c)/(b*c)

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Sympy [A]  time = 2.04256, size = 83, normalized size = 3.77 \[ \begin{cases} \frac{2 b^{\frac{9}{2}} \left (\frac{a}{b} + x\right )^{\frac{11}{2}}}{11 \sqrt{c}} & \text{for}\: \left (\frac{a}{b} + x > -1 \wedge \frac{a}{b} + x < 1\right ) \vee \frac{a}{b} + x > 1 \vee \frac{a}{b} + x < -1 \\\frac{b^{\frac{9}{2}}{G_{2, 2}^{1, 1}\left (\begin{matrix} 1 & \frac{13}{2} \\\frac{11}{2} & 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )}}{\sqrt{c}} + \frac{b^{\frac{9}{2}}{G_{2, 2}^{0, 2}\left (\begin{matrix} \frac{13}{2}, 1 & \\ & \frac{11}{2}, 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )}}{\sqrt{c}} & \text{otherwise} \end{cases} \]

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**(9/2)*(a/b + x)**(11/2)/(11*sqrt(c)), (a/b + x > 1) | (a/b + x <
-1) | ((a/b + x > -1) & (a/b + x < 1))), (b**(9/2)*meijerg(((1,), (13/2,)), ((1
1/2,), (0,)), a/b + x)/sqrt(c) + b**(9/2)*meijerg(((13/2, 1), ()), ((), (11/2, 0
)), a/b + x)/sqrt(c), True))

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GIAC/XCAS [A]  time = 0.227483, size = 621, normalized size = 28.23 \[ \frac{2 \,{\left (693 \, \sqrt{b c x + a c} a^{5} - \frac{1155 \,{\left (3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}\right )} a^{4}}{c} + \frac{462 \,{\left (15 \, \sqrt{b c x + a c} a^{2} b^{8} c^{10} - 10 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a b^{8} c^{9} + 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}} b^{8} c^{8}\right )} a^{3}}{b^{8} c^{10}} - \frac{198 \,{\left (35 \, \sqrt{b c x + a c} a^{3} b^{18} c^{21} - 35 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{2} b^{18} c^{20} + 21 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a b^{18} c^{19} - 5 \,{\left (b c x + a c\right )}^{\frac{7}{2}} b^{18} c^{18}\right )} a^{2}}{b^{18} c^{21}} + \frac{11 \,{\left (315 \, \sqrt{b c x + a c} a^{4} b^{32} c^{36} - 420 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{3} b^{32} c^{35} + 378 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{2} b^{32} c^{34} - 180 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a b^{32} c^{33} + 35 \,{\left (b c x + a c\right )}^{\frac{9}{2}} b^{32} c^{32}\right )} a}{b^{32} c^{36}} - \frac{693 \, \sqrt{b c x + a c} a^{5} b^{50} c^{55} - 1155 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{4} b^{50} c^{54} + 1386 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{3} b^{50} c^{53} - 990 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a^{2} b^{50} c^{52} + 385 \,{\left (b c x + a c\right )}^{\frac{9}{2}} a b^{50} c^{51} - 63 \,{\left (b c x + a c\right )}^{\frac{11}{2}} b^{50} c^{50}}{b^{50} c^{55}}\right )}}{693 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*(693*sqrt(b*c*x + a*c)*a^5 - 1155*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)
^(3/2))*a^4/c + 462*(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^3/(b^8*c^10) - 198*(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^2/(b^18*c^21) + 11*(315*sqr
t(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)*a/(b^32*c^36) - (693*sqrt(b*c*x + a*c)*a^5*b^50*c^55 - 11
55*(b*c*x + a*c)^(3/2)*a^4*b^50*c^54 + 1386*(b*c*x + a*c)^(5/2)*a^3*b^50*c^53 -
990*(b*c*x + a*c)^(7/2)*a^2*b^50*c^52 + 385*(b*c*x + a*c)^(9/2)*a*b^50*c^51 - 63
*(b*c*x + a*c)^(11/2)*b^50*c^50)/(b^50*c^55))/(b*c)