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]

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

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \[ \frac {2 (a c+b c x)^{15/2}}{15 b c^6} \]

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)

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 (a+b x)^5 (a c+b c x)^{3/2} \, dx &=\frac {\int (a c+b c x)^{13/2} \, dx}{c^5}\\ &=\frac {2 (a c+b c x)^{15/2}}{15 b c^6}\\ \end {align*}

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Mathematica [A]  time = 0.02, 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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fricas [B]  time = 0.42, size = 95, normalized size = 4.32 \[ \frac {2 \, {\left (b^{7} c x^{7} + 7 \, a b^{6} c x^{6} + 21 \, a^{2} b^{5} c x^{5} + 35 \, a^{3} b^{4} c x^{4} + 35 \, a^{4} b^{3} c x^{3} + 21 \, a^{5} b^{2} c x^{2} + 7 \, a^{6} b c x + a^{7} c\right )} \sqrt {b c x + a c}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^(3/2),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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giac [B]  time = 1.26, size = 637, normalized size = 28.95 \[ \frac {2 \, {\left (6435 \, \sqrt {b c x + a c} a^{7} c - 15015 \, {\left (3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}\right )} a^{6} + \frac {9009 \, {\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^{5}}{c} - \frac {6435 \, {\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^{4}}{c^{2}} + \frac {715 \, {\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^{3}}{c^{3}} - \frac {195 \, {\left (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}}\right )} a^{2}}{c^{4}} + \frac {15 \, {\left (3003 \, \sqrt {b c x + a c} a^{6} c^{6} - 6006 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{5} c^{5} + 9009 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{4} c^{4} - 8580 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{3} c^{3} + 5005 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{2} c^{2} - 1638 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a c + 231 \, {\left (b c x + a c\right )}^{\frac {13}{2}}\right )} a}{c^{5}} - \frac {6435 \, \sqrt {b c x + a c} a^{7} c^{7} - 15015 \, {\left (b c x + a c\right )}^{\frac {3}{2}} a^{6} c^{6} + 27027 \, {\left (b c x + a c\right )}^{\frac {5}{2}} a^{5} c^{5} - 32175 \, {\left (b c x + a c\right )}^{\frac {7}{2}} a^{4} c^{4} + 25025 \, {\left (b c x + a c\right )}^{\frac {9}{2}} a^{3} c^{3} - 12285 \, {\left (b c x + a c\right )}^{\frac {11}{2}} a^{2} c^{2} + 3465 \, {\left (b c x + a c\right )}^{\frac {13}{2}} a c - 429 \, {\left (b c x + a c\right )}^{\frac {15}{2}}}{c^{6}}\right )}}{6435 \, b} \]

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]

2/6435*(6435*sqrt(b*c*x + a*c)*a^7*c - 15015*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))*a^6 + 9009*(15*sq
rt(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^5/c - 6435*(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^4/c^2 + 71
5*(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^3/c^3 - 195*(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))*a^2/c^4 + 15*(3003*sqrt(b*c*x + a*c)*a^6*c^6 - 6006*(b*c*x + a*c)^(3/2)*a^5*c
^5 + 9009*(b*c*x + a*c)^(5/2)*a^4*c^4 - 8580*(b*c*x + a*c)^(7/2)*a^3*c^3 + 5005*(b*c*x + a*c)^(9/2)*a^2*c^2 -
1638*(b*c*x + a*c)^(11/2)*a*c + 231*(b*c*x + a*c)^(13/2))*a/c^5 - (6435*sqrt(b*c*x + a*c)*a^7*c^7 - 15015*(b*c
*x + a*c)^(3/2)*a^6*c^6 + 27027*(b*c*x + a*c)^(5/2)*a^5*c^5 - 32175*(b*c*x + a*c)^(7/2)*a^4*c^4 + 25025*(b*c*x
+ a*c)^(9/2)*a^3*c^3 - 12285*(b*c*x + a*c)^(11/2)*a^2*c^2 + 3465*(b*c*x + a*c)^(13/2)*a*c - 429*(b*c*x + a*c)
^(15/2))/c^6)/b

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maple [A]  time = 0.00, size = 23, normalized size = 1.05 \[ \frac {2 \left (b x +a \right )^{6} \left (b c x +a c \right )^{\frac {3}{2}}}{15 b} \]

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 = 1.37, size = 18, normalized size = 0.82 \[ \frac {2 \, {\left (b c x + a c\right )}^{\frac {15}{2}}}{15 \, b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(b*c*x + a*c)^(15/2)/(b*c^6)

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mupad [B]  time = 0.05, size = 17, normalized size = 0.77 \[ \frac {2\,{\left (c\,\left (a+b\,x\right )\right )}^{15/2}}{15\,b\,c^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(c*(a + b*x))^(15/2))/(15*b*c^6)

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sympy [A]  time = 1.21, size = 66, normalized size = 3.00 \[ \begin {cases} \frac {2 b^{\frac {13}{2}} c^{\frac {3}{2}} \left (\frac {a}{b} + x\right )^{\frac {15}{2}}}{15} & \text {for}\: \left |{\frac {a}{b} + x}\right | < 1 \\b^{\frac {13}{2}} c^{\frac {3}{2}} {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & \frac {17}{2} \\\frac {15}{2} & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} + b^{\frac {13}{2}} c^{\frac {3}{2}} {G_{2, 2}^{0, 2}\left (\begin {matrix} \frac {17}{2}, 1 & \\ & \frac {15}{2}, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} & \text {otherwise} \end {cases} \]

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,), (1
7/2,)), ((15/2,), (0,)), a/b + x) + b**(13/2)*c**(3/2)*meijerg(((17/2, 1), ()), ((), (15/2, 0)), a/b + x), Tru
e))

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