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 = 0.0044284, 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, Rules used = {21, 32} \[ \frac{2 (a c+b c x)^{11/2}}{11 b c^6} \]

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)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^5}{\sqrt{a c+b c x}} \, dx &=\frac{\int (a c+b c x)^{9/2} \, dx}{c^5}\\ &=\frac{2 (a c+b c x)^{11/2}}{11 b c^6}\\ \end{align*}

Mathematica [A]  time = 0.0134179, 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 = 0.001, size = 23, normalized size = 1.1 \begin{align*}{\frac{2\, \left ( bx+a \right ) ^{6}}{11\,b}{\frac{1}{\sqrt{bcx+ac}}}} \end{align*}

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 [B]  time = 0.964648, size = 505, normalized size = 22.95 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(1/2),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*(3
15*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 + 1386*(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 [B]  time = 2.02566, size = 143, normalized size = 6.5 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(1/2),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 = 1.29558, size = 73, normalized size = 3.32 \begin{align*} \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}\right | > 1 \vee \left |{\frac{a}{b} + x}\right | < 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} \end{align*}

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)), (Abs(a/b + x) > 1) | (Abs(a/b + x) < 1)), (b**(9/2)*meij
erg(((1,), (13/2,)), ((11/2,), (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 [B]  time = 1.0683, size = 505, normalized size = 22.95 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(1/2),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*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*(3
15*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 + 1386*(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)