3.14.87 \(\int (a+b x)^5 (c+d x)^{3/2} \, dx\) [1387]

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

[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 = 0.04, 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 = {45} \begin {gather*} -\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} \end {gather*}

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 45

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 = 0.13, size = 217, normalized size = 1.37 \begin {gather*} \frac {2 (c+d x)^{5/2} \left (9009 a^5 d^5+6435 a^4 b d^4 (-2 c+5 d x)+1430 a^3 b^2 d^3 \left (8 c^2-20 c d x+35 d^2 x^2\right )+390 a^2 b^3 d^2 \left (-16 c^3+40 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )+15 a b^4 d \left (128 c^4-320 c^3 d x+560 c^2 d^2 x^2-840 c d^3 x^3+1155 d^4 x^4\right )+b^5 \left (-256 c^5+640 c^4 d x-1120 c^3 d^2 x^2+1680 c^2 d^3 x^3-2310 c d^4 x^4+3003 d^5 x^5\right )\right )}{45045 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(5/2)*(9009*a^5*d^5 + 6435*a^4*b*d^4*(-2*c + 5*d*x) + 1430*a^3*b^2*d^3*(8*c^2 - 20*c*d*x + 35*d^2
*x^2) + 390*a^2*b^3*d^2*(-16*c^3 + 40*c^2*d*x - 70*c*d^2*x^2 + 105*d^3*x^3) + 15*a*b^4*d*(128*c^4 - 320*c^3*d*
x + 560*c^2*d^2*x^2 - 840*c*d^3*x^3 + 1155*d^4*x^4) + b^5*(-256*c^5 + 640*c^4*d*x - 1120*c^3*d^2*x^2 + 1680*c^
2*d^3*x^3 - 2310*c*d^4*x^4 + 3003*d^5*x^5)))/(45045*d^6)

________________________________________________________________________________________

Maple [A]
time = 0.16, size = 122, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {15}{2}}}{15}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\)	\(122\)
default	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {15}{2}}}{15}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\)	\(122\)
gosper	\(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (3003 b^{5} x^{5} d^{5}+17325 a \,b^{4} d^{5} x^{4}-2310 b^{5} c \,d^{4} x^{4}+40950 a^{2} b^{3} d^{5} x^{3}-12600 a \,b^{4} c \,d^{4} x^{3}+1680 b^{5} c^{2} d^{3} x^{3}+50050 a^{3} b^{2} d^{5} x^{2}-27300 a^{2} b^{3} c \,d^{4} x^{2}+8400 a \,b^{4} c^{2} d^{3} x^{2}-1120 b^{5} c^{3} d^{2} x^{2}+32175 a^{4} b \,d^{5} x -28600 a^{3} b^{2} c \,d^{4} x +15600 a^{2} b^{3} c^{2} d^{3} x -4800 a \,b^{4} c^{3} d^{2} x +640 b^{5} c^{4} d x +9009 a^{5} d^{5}-12870 a^{4} b c \,d^{4}+11440 a^{3} b^{2} c^{2} d^{3}-6240 a^{2} b^{3} c^{3} d^{2}+1920 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{45045 d^{6}}\)	\(273\)
trager	\(\frac {2 \left (3003 b^{5} d^{7} x^{7}+17325 a \,b^{4} d^{7} x^{6}+3696 b^{5} c \,d^{6} x^{6}+40950 a^{2} b^{3} d^{7} x^{5}+22050 a \,b^{4} c \,d^{6} x^{5}+63 b^{5} c^{2} d^{5} x^{5}+50050 a^{3} b^{2} d^{7} x^{4}+54600 a^{2} b^{3} c \,d^{6} x^{4}+525 a \,b^{4} c^{2} d^{5} x^{4}-70 b^{5} c^{3} d^{4} x^{4}+32175 a^{4} b \,d^{7} x^{3}+71500 a^{3} b^{2} c \,d^{6} x^{3}+1950 a^{2} b^{3} c^{2} d^{5} x^{3}-600 a \,b^{4} c^{3} d^{4} x^{3}+80 b^{5} c^{4} d^{3} x^{3}+9009 a^{5} d^{7} x^{2}+51480 a^{4} b c \,d^{6} x^{2}+4290 a^{3} b^{2} c^{2} d^{5} x^{2}-2340 a^{2} b^{3} c^{3} d^{4} x^{2}+720 a \,b^{4} c^{4} d^{3} x^{2}-96 b^{5} c^{5} d^{2} x^{2}+18018 a^{5} c \,d^{6} x +6435 a^{4} b \,c^{2} d^{5} x -5720 a^{3} b^{2} c^{3} d^{4} x +3120 a^{2} b^{3} c^{4} d^{3} x -960 a \,b^{4} c^{5} d^{2} x +128 b^{5} c^{6} d x +9009 a^{5} c^{2} d^{5}-12870 a^{4} b \,c^{3} d^{4}+11440 a^{3} b^{2} c^{4} d^{3}-6240 a^{2} b^{3} c^{5} d^{2}+1920 a \,b^{4} c^{6} d -256 b^{5} c^{7}\right ) \sqrt {d x +c}}{45045 d^{6}}\)	\(453\)
risch	\(\frac {2 \left (3003 b^{5} d^{7} x^{7}+17325 a \,b^{4} d^{7} x^{6}+3696 b^{5} c \,d^{6} x^{6}+40950 a^{2} b^{3} d^{7} x^{5}+22050 a \,b^{4} c \,d^{6} x^{5}+63 b^{5} c^{2} d^{5} x^{5}+50050 a^{3} b^{2} d^{7} x^{4}+54600 a^{2} b^{3} c \,d^{6} x^{4}+525 a \,b^{4} c^{2} d^{5} x^{4}-70 b^{5} c^{3} d^{4} x^{4}+32175 a^{4} b \,d^{7} x^{3}+71500 a^{3} b^{2} c \,d^{6} x^{3}+1950 a^{2} b^{3} c^{2} d^{5} x^{3}-600 a \,b^{4} c^{3} d^{4} x^{3}+80 b^{5} c^{4} d^{3} x^{3}+9009 a^{5} d^{7} x^{2}+51480 a^{4} b c \,d^{6} x^{2}+4290 a^{3} b^{2} c^{2} d^{5} x^{2}-2340 a^{2} b^{3} c^{3} d^{4} x^{2}+720 a \,b^{4} c^{4} d^{3} x^{2}-96 b^{5} c^{5} d^{2} x^{2}+18018 a^{5} c \,d^{6} x +6435 a^{4} b \,c^{2} d^{5} x -5720 a^{3} b^{2} c^{3} d^{4} x +3120 a^{2} b^{3} c^{4} d^{3} x -960 a \,b^{4} c^{5} d^{2} x +128 b^{5} c^{6} d x +9009 a^{5} c^{2} d^{5}-12870 a^{4} b \,c^{3} d^{4}+11440 a^{3} b^{2} c^{4} d^{3}-6240 a^{2} b^{3} c^{5} d^{2}+1920 a \,b^{4} c^{6} d -256 b^{5} c^{7}\right ) \sqrt {d x +c}}{45045 d^{6}}\)	\(453\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d^6*(1/15*b^5*(d*x+c)^(15/2)+5/13*(a*d-b*c)*b^4*(d*x+c)^(13/2)+10/11*(a*d-b*c)^2*b^3*(d*x+c)^(11/2)+10/9*(a*
d-b*c)^3*b^2*(d*x+c)^(9/2)+5/7*(a*d-b*c)^4*b*(d*x+c)^(7/2)+1/5*(a*d-b*c)^5*(d*x+c)^(5/2))

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 259, normalized size = 1.64 \begin {gather*} \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}} \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (134) = 268\).
time = 0.49, size = 418, normalized size = 2.65 \begin {gather*} \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}} \end {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 13.83, size = 763, normalized size = 4.83 \begin {gather*} a^{5} c \left (\begin {cases} \sqrt {c} x & \text {for}\: d = 0 \\\frac {2 \left (c + d x\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) + \frac {2 a^{5} \left (- \frac {c \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d} + \frac {10 a^{4} b c \left (- \frac {c \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} + \frac {10 a^{4} b \left (\frac {c^{2} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{2}} + \frac {20 a^{3} b^{2} c \left (\frac {c^{2} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} + \frac {20 a^{3} b^{2} \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{3}} + \frac {20 a^{2} b^{3} c \left (- \frac {c^{3} \left (c + d x\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {\left (c + d x\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} + \frac {20 a^{2} b^{3} \left (\frac {c^{4} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {\left (c + d x\right )^{\frac {11}{2}}}{11}\right )}{d^{4}} + \frac {10 a b^{4} c \left (\frac {c^{4} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {\left (c + d x\right )^{\frac {11}{2}}}{11}\right )}{d^{5}} + \frac {10 a b^{4} \left (- \frac {c^{5} \left (c + d x\right )^{\frac {3}{2}}}{3} + c^{4} \left (c + d x\right )^{\frac {5}{2}} - \frac {10 c^{3} \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {10 c^{2} \left (c + d x\right )^{\frac {9}{2}}}{9} - \frac {5 c \left (c + d x\right )^{\frac {11}{2}}}{11} + \frac {\left (c + d x\right )^{\frac {13}{2}}}{13}\right )}{d^{5}} + \frac {2 b^{5} c \left (- \frac {c^{5} \left (c + d x\right )^{\frac {3}{2}}}{3} + c^{4} \left (c + d x\right )^{\frac {5}{2}} - \frac {10 c^{3} \left (c + d x\right )^{\frac {7}{2}}}{7} + \frac {10 c^{2} \left (c + d x\right )^{\frac {9}{2}}}{9} - \frac {5 c \left (c + d x\right )^{\frac {11}{2}}}{11} + \frac {\left (c + d x\right )^{\frac {13}{2}}}{13}\right )}{d^{6}} + \frac {2 b^{5} \left (\frac {c^{6} \left (c + d x\right )^{\frac {3}{2}}}{3} - \frac {6 c^{5} \left (c + d x\right )^{\frac {5}{2}}}{5} + \frac {15 c^{4} \left (c + d x\right )^{\frac {7}{2}}}{7} - \frac {20 c^{3} \left (c + d x\right )^{\frac {9}{2}}}{9} + \frac {15 c^{2} \left (c + d x\right )^{\frac {11}{2}}}{11} - \frac {6 c \left (c + d x\right )^{\frac {13}{2}}}{13} + \frac {\left (c + d x\right )^{\frac {15}{2}}}{15}\right )}{d^{6}} \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1084 vs. \(2 (134) = 268\).
time = 1.46, size = 1084, normalized size = 6.86 \begin {gather*} \frac {2 \, {\left (45045 \, \sqrt {d x + c} a^{5} c^{2} + 30030 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{5} c + \frac {75075 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{4} b c^{2}}{d} + 3003 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{5} + \frac {30030 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{3} b^{2} c^{2}}{d^{2}} + \frac {30030 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{4} b c}{d} + \frac {12870 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a^{2} b^{3} c^{2}}{d^{3}} + \frac {25740 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a^{3} b^{2} c}{d^{2}} + \frac {6435 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a^{4} b}{d} + \frac {715 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a b^{4} c^{2}}{d^{4}} + \frac {2860 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a^{2} b^{3} c}{d^{3}} + \frac {1430 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a^{3} b^{2}}{d^{2}} + \frac {65 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b^{5} c^{2}}{d^{5}} + \frac {650 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} a b^{4} c}{d^{4}} + \frac {650 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} a^{2} b^{3}}{d^{3}} + \frac {30 \, {\left (231 \, {\left (d x + c\right )}^{\frac {13}{2}} - 1638 \, {\left (d x + c\right )}^{\frac {11}{2}} c + 5005 \, {\left (d x + c\right )}^{\frac {9}{2}} c^{2} - 8580 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{3} + 9009 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{4} - 6006 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{5} + 3003 \, \sqrt {d x + c} c^{6}\right )} b^{5} c}{d^{5}} + \frac {75 \, {\left (231 \, {\left (d x + c\right )}^{\frac {13}{2}} - 1638 \, {\left (d x + c\right )}^{\frac {11}{2}} c + 5005 \, {\left (d x + c\right )}^{\frac {9}{2}} c^{2} - 8580 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{3} + 9009 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{4} - 6006 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{5} + 3003 \, \sqrt {d x + c} c^{6}\right )} a b^{4}}{d^{4}} + \frac {7 \, {\left (429 \, {\left (d x + c\right )}^{\frac {15}{2}} - 3465 \, {\left (d x + c\right )}^{\frac {13}{2}} c + 12285 \, {\left (d x + c\right )}^{\frac {11}{2}} c^{2} - 25025 \, {\left (d x + c\right )}^{\frac {9}{2}} c^{3} + 32175 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{4} - 27027 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{5} + 15015 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{6} - 6435 \, \sqrt {d x + c} c^{7}\right )} b^{5}}{d^{5}}\right )}}{45045 \, d} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.24, size = 137, normalized size = 0.87 \begin {gather*} \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} \end {gather*}

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)

________________________________________________________________________________________