3.1387 \(\int (a+b x)^5 (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=158 \[ -\frac {10 b^4 (c+d x)^{13/2} (b c-a d)}{13 d^6}+\frac {20 b^3 (c+d x)^{11/2} (b c-a d)^2}{11 d^6}-\frac {20 b^2 (c+d x)^{9/2} (b c-a d)^3}{9 d^6}+\frac {10 b (c+d x)^{7/2} (b c-a d)^4}{7 d^6}-\frac {2 (c+d x)^{5/2} (b c-a d)^5}{5 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6} \]

[Out]

-2/5*(-a*d+b*c)^5*(d*x+c)^(5/2)/d^6+10/7*b*(-a*d+b*c)^4*(d*x+c)^(7/2)/d^6-20/9*b^2*(-a*d+b*c)^3*(d*x+c)^(9/2)/
d^6+20/11*b^3*(-a*d+b*c)^2*(d*x+c)^(11/2)/d^6-10/13*b^4*(-a*d+b*c)*(d*x+c)^(13/2)/d^6+2/15*b^5*(d*x+c)^(15/2)/
d^6

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \[ -\frac {10 b^4 (c+d x)^{13/2} (b c-a d)}{13 d^6}+\frac {20 b^3 (c+d x)^{11/2} (b c-a d)^2}{11 d^6}-\frac {20 b^2 (c+d x)^{9/2} (b c-a d)^3}{9 d^6}+\frac {10 b (c+d x)^{7/2} (b c-a d)^4}{7 d^6}-\frac {2 (c+d x)^{5/2} (b c-a d)^5}{5 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*(c + d*x)^(3/2),x]

[Out]

(-2*(b*c - a*d)^5*(c + d*x)^(5/2))/(5*d^6) + (10*b*(b*c - a*d)^4*(c + d*x)^(7/2))/(7*d^6) - (20*b^2*(b*c - a*d
)^3*(c + d*x)^(9/2))/(9*d^6) + (20*b^3*(b*c - a*d)^2*(c + d*x)^(11/2))/(11*d^6) - (10*b^4*(b*c - a*d)*(c + d*x
)^(13/2))/(13*d^6) + (2*b^5*(c + d*x)^(15/2))/(15*d^6)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (a+b x)^5 (c+d x)^{3/2} \, dx &=\int \left (\frac {(-b c+a d)^5 (c+d x)^{3/2}}{d^5}+\frac {5 b (b c-a d)^4 (c+d x)^{5/2}}{d^5}-\frac {10 b^2 (b c-a d)^3 (c+d x)^{7/2}}{d^5}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{9/2}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{11/2}}{d^5}+\frac {b^5 (c+d x)^{13/2}}{d^5}\right ) \, dx\\ &=-\frac {2 (b c-a d)^5 (c+d x)^{5/2}}{5 d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{7/2}}{7 d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{9/2}}{9 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{11/2}}{11 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{13/2}}{13 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 123, normalized size = 0.78 \[ \frac {2 (c+d x)^{5/2} \left (-17325 b^4 (c+d x)^4 (b c-a d)+40950 b^3 (c+d x)^3 (b c-a d)^2-50050 b^2 (c+d x)^2 (b c-a d)^3+32175 b (c+d x) (b c-a d)^4-9009 (b c-a d)^5+3003 b^5 (c+d x)^5\right )}{45045 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(5/2)*(-9009*(b*c - a*d)^5 + 32175*b*(b*c - a*d)^4*(c + d*x) - 50050*b^2*(b*c - a*d)^3*(c + d*x)^
2 + 40950*b^3*(b*c - a*d)^2*(c + d*x)^3 - 17325*b^4*(b*c - a*d)*(c + d*x)^4 + 3003*b^5*(c + d*x)^5))/(45045*d^
6)

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 418, normalized size = 2.65 \[ \frac {2 \, {\left (3003 \, b^{5} d^{7} x^{7} - 256 \, b^{5} c^{7} + 1920 \, a b^{4} c^{6} d - 6240 \, a^{2} b^{3} c^{5} d^{2} + 11440 \, a^{3} b^{2} c^{4} d^{3} - 12870 \, a^{4} b c^{3} d^{4} + 9009 \, a^{5} c^{2} d^{5} + 231 \, {\left (16 \, b^{5} c d^{6} + 75 \, a b^{4} d^{7}\right )} x^{6} + 63 \, {\left (b^{5} c^{2} d^{5} + 350 \, a b^{4} c d^{6} + 650 \, a^{2} b^{3} d^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} c^{3} d^{4} - 15 \, a b^{4} c^{2} d^{5} - 1560 \, a^{2} b^{3} c d^{6} - 1430 \, a^{3} b^{2} d^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} c^{4} d^{3} - 120 \, a b^{4} c^{3} d^{4} + 390 \, a^{2} b^{3} c^{2} d^{5} + 14300 \, a^{3} b^{2} c d^{6} + 6435 \, a^{4} b d^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} c^{5} d^{2} - 240 \, a b^{4} c^{4} d^{3} + 780 \, a^{2} b^{3} c^{3} d^{4} - 1430 \, a^{3} b^{2} c^{2} d^{5} - 17160 \, a^{4} b c d^{6} - 3003 \, a^{5} d^{7}\right )} x^{2} + {\left (128 \, b^{5} c^{6} d - 960 \, a b^{4} c^{5} d^{2} + 3120 \, a^{2} b^{3} c^{4} d^{3} - 5720 \, a^{3} b^{2} c^{3} d^{4} + 6435 \, a^{4} b c^{2} d^{5} + 18018 \, a^{5} c d^{6}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*b^5*d^7*x^7 - 256*b^5*c^7 + 1920*a*b^4*c^6*d - 6240*a^2*b^3*c^5*d^2 + 11440*a^3*b^2*c^4*d^3 - 12
870*a^4*b*c^3*d^4 + 9009*a^5*c^2*d^5 + 231*(16*b^5*c*d^6 + 75*a*b^4*d^7)*x^6 + 63*(b^5*c^2*d^5 + 350*a*b^4*c*d
^6 + 650*a^2*b^3*d^7)*x^5 - 35*(2*b^5*c^3*d^4 - 15*a*b^4*c^2*d^5 - 1560*a^2*b^3*c*d^6 - 1430*a^3*b^2*d^7)*x^4
+ 5*(16*b^5*c^4*d^3 - 120*a*b^4*c^3*d^4 + 390*a^2*b^3*c^2*d^5 + 14300*a^3*b^2*c*d^6 + 6435*a^4*b*d^7)*x^3 - 3*
(32*b^5*c^5*d^2 - 240*a*b^4*c^4*d^3 + 780*a^2*b^3*c^3*d^4 - 1430*a^3*b^2*c^2*d^5 - 17160*a^4*b*c*d^6 - 3003*a^
5*d^7)*x^2 + (128*b^5*c^6*d - 960*a*b^4*c^5*d^2 + 3120*a^2*b^3*c^4*d^3 - 5720*a^3*b^2*c^3*d^4 + 6435*a^4*b*c^2
*d^5 + 18018*a^5*c*d^6)*x)*sqrt(d*x + c)/d^6

________________________________________________________________________________________

giac [B]  time = 1.53, size = 1084, normalized size = 6.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(45045*sqrt(d*x + c)*a^5*c^2 + 30030*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^5*c + 75075*((d*x + c)^(3
/2) - 3*sqrt(d*x + c)*c)*a^4*b*c^2/d + 3003*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*
a^5 + 30030*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b^2*c^2/d^2 + 30030*(3*(d*x
+ c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^4*b*c/d + 12870*(5*(d*x + c)^(7/2) - 21*(d*x + c)^
(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^2*b^3*c^2/d^3 + 25740*(5*(d*x + c)^(7/2) - 21*(d*x
+ c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^3*b^2*c/d^2 + 6435*(5*(d*x + c)^(7/2) - 21*(d*
x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a^4*b/d + 715*(35*(d*x + c)^(9/2) - 180*(d*x +
c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*a*b^4*c^2/d^4 + 2860*
(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x
+ c)*c^4)*a^2*b^3*c/d^3 + 1430*(35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d
*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*a^3*b^2/d^2 + 65*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990
*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*b^5*c^2/d^
5 + 650*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 11
55*(d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*a*b^4*c/d^4 + 650*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c
+ 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d*x + c)*c^5)*a^2*
b^3/d^3 + 30*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*c^2 - 8580*(d*x + c)^(7/2)
*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x + c)*c^6)*b^5*c/d^5 + 75*(231*(d*x
+ c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*c^2 - 8580*(d*x + c)^(7/2)*c^3 + 9009*(d*x + c)^(
5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x + c)*c^6)*a*b^4/d^4 + 7*(429*(d*x + c)^(15/2) - 3465*(d*x
+ c)^(13/2)*c + 12285*(d*x + c)^(11/2)*c^2 - 25025*(d*x + c)^(9/2)*c^3 + 32175*(d*x + c)^(7/2)*c^4 - 27027*(d*
x + c)^(5/2)*c^5 + 15015*(d*x + c)^(3/2)*c^6 - 6435*sqrt(d*x + c)*c^7)*b^5/d^5)/d

________________________________________________________________________________________

maple [B]  time = 0.01, size = 273, normalized size = 1.73 \[ \frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (3003 b^{5} x^{5} d^{5}+17325 a \,b^{4} d^{5} x^{4}-2310 b^{5} c \,d^{4} x^{4}+40950 a^{2} b^{3} d^{5} x^{3}-12600 a \,b^{4} c \,d^{4} x^{3}+1680 b^{5} c^{2} d^{3} x^{3}+50050 a^{3} b^{2} d^{5} x^{2}-27300 a^{2} b^{3} c \,d^{4} x^{2}+8400 a \,b^{4} c^{2} d^{3} x^{2}-1120 b^{5} c^{3} d^{2} x^{2}+32175 a^{4} b \,d^{5} x -28600 a^{3} b^{2} c \,d^{4} x +15600 a^{2} b^{3} c^{2} d^{3} x -4800 a \,b^{4} c^{3} d^{2} x +640 b^{5} c^{4} d x +9009 a^{5} d^{5}-12870 a^{4} b c \,d^{4}+11440 a^{3} b^{2} c^{2} d^{3}-6240 a^{2} b^{3} c^{3} d^{2}+1920 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{45045 d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(d*x+c)^(5/2)*(3003*b^5*d^5*x^5+17325*a*b^4*d^5*x^4-2310*b^5*c*d^4*x^4+40950*a^2*b^3*d^5*x^3-12600*a*b
^4*c*d^4*x^3+1680*b^5*c^2*d^3*x^3+50050*a^3*b^2*d^5*x^2-27300*a^2*b^3*c*d^4*x^2+8400*a*b^4*c^2*d^3*x^2-1120*b^
5*c^3*d^2*x^2+32175*a^4*b*d^5*x-28600*a^3*b^2*c*d^4*x+15600*a^2*b^3*c^2*d^3*x-4800*a*b^4*c^3*d^2*x+640*b^5*c^4
*d*x+9009*a^5*d^5-12870*a^4*b*c*d^4+11440*a^3*b^2*c^2*d^3-6240*a^2*b^3*c^3*d^2+1920*a*b^4*c^4*d-256*b^5*c^5)/d
^6

________________________________________________________________________________________

maxima [A]  time = 1.35, size = 259, normalized size = 1.64 \[ \frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} b^{5} - 17325 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {13}{2}} + 40950 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 50050 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 32175 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 9009 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{45045 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(d*x + c)^(15/2)*b^5 - 17325*(b^5*c - a*b^4*d)*(d*x + c)^(13/2) + 40950*(b^5*c^2 - 2*a*b^4*c*d +
a^2*b^3*d^2)*(d*x + c)^(11/2) - 50050*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*(d*x + c)^(9/
2) + 32175*(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*(d*x + c)^(7/2) - 9009*
(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*(d*x + c)^(5/2))
/d^6

________________________________________________________________________________________

mupad [B]  time = 0.24, size = 137, normalized size = 0.87 \[ \frac {2\,b^5\,{\left (c+d\,x\right )}^{15/2}}{15\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {10\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5*(c + d*x)^(3/2),x)

[Out]

(2*b^5*(c + d*x)^(15/2))/(15*d^6) - ((10*b^5*c - 10*a*b^4*d)*(c + d*x)^(13/2))/(13*d^6) + (2*(a*d - b*c)^5*(c
+ d*x)^(5/2))/(5*d^6) + (20*b^2*(a*d - b*c)^3*(c + d*x)^(9/2))/(9*d^6) + (20*b^3*(a*d - b*c)^2*(c + d*x)^(11/2
))/(11*d^6) + (10*b*(a*d - b*c)^4*(c + d*x)^(7/2))/(7*d^6)

________________________________________________________________________________________

sympy [A]  time = 26.42, size = 763, normalized size = 4.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(d*x+c)**(3/2),x)

[Out]

a**5*c*Piecewise((sqrt(c)*x, Eq(d, 0)), (2*(c + d*x)**(3/2)/(3*d), True)) + 2*a**5*(-c*(c + d*x)**(3/2)/3 + (c
+ d*x)**(5/2)/5)/d + 10*a**4*b*c*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + 10*a**4*b*(c**2*(c + d*x
)**(3/2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**2 + 20*a**3*b**2*c*(c**2*(c + d*x)**(3/2)/3 - 2*c
*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**3 + 20*a**3*b**2*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5
/2)/5 - 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**3 + 20*a**2*b**3*c*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*
(c + d*x)**(5/2)/5 - 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**4 + 20*a**2*b**3*(c**4*(c + d*x)**(3/2)/3
- 4*c**3*(c + d*x)**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2)/9 + (c + d*x)**(11/2)/11)/d**4
+ 10*a*b**4*c*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x
)**(9/2)/9 + (c + d*x)**(11/2)/11)/d**5 + 10*a*b**4*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 10*c**
3*(c + d*x)**(7/2)/7 + 10*c**2*(c + d*x)**(9/2)/9 - 5*c*(c + d*x)**(11/2)/11 + (c + d*x)**(13/2)/13)/d**5 + 2*
b**5*c*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 10*c**3*(c + d*x)**(7/2)/7 + 10*c**2*(c + d*x)**(9/
2)/9 - 5*c*(c + d*x)**(11/2)/11 + (c + d*x)**(13/2)/13)/d**6 + 2*b**5*(c**6*(c + d*x)**(3/2)/3 - 6*c**5*(c + d
*x)**(5/2)/5 + 15*c**4*(c + d*x)**(7/2)/7 - 20*c**3*(c + d*x)**(9/2)/9 + 15*c**2*(c + d*x)**(11/2)/11 - 6*c*(c
+ d*x)**(13/2)/13 + (c + d*x)**(15/2)/15)/d**6

________________________________________________________________________________________