3.14.99 \(\int (a+b x)^5 (c+d x)^{5/2} \, dx\) [1399]

Optimal. Leaf size=158 \[ -\frac {2 (b c-a d)^5 (c+d x)^{7/2}}{7 d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{9/2}}{9 d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{11/2}}{11 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{13/2}}{13 d^6}-\frac {2 b^4 (b c-a d) (c+d x)^{15/2}}{3 d^6}+\frac {2 b^5 (c+d x)^{17/2}}{17 d^6} \]

[Out]

-2/7*(-a*d+b*c)^5*(d*x+c)^(7/2)/d^6+10/9*b*(-a*d+b*c)^4*(d*x+c)^(9/2)/d^6-20/11*b^2*(-a*d+b*c)^3*(d*x+c)^(11/2
)/d^6+20/13*b^3*(-a*d+b*c)^2*(d*x+c)^(13/2)/d^6-2/3*b^4*(-a*d+b*c)*(d*x+c)^(15/2)/d^6+2/17*b^5*(d*x+c)^(17/2)/
d^6

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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 {2 b^4 (c+d x)^{15/2} (b c-a d)}{3 d^6}+\frac {20 b^3 (c+d x)^{13/2} (b c-a d)^2}{13 d^6}-\frac {20 b^2 (c+d x)^{11/2} (b c-a d)^3}{11 d^6}+\frac {10 b (c+d x)^{9/2} (b c-a d)^4}{9 d^6}-\frac {2 (c+d x)^{7/2} (b c-a d)^5}{7 d^6}+\frac {2 b^5 (c+d x)^{17/2}}{17 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*(c + d*x)^(5/2),x]

[Out]

(-2*(b*c - a*d)^5*(c + d*x)^(7/2))/(7*d^6) + (10*b*(b*c - a*d)^4*(c + d*x)^(9/2))/(9*d^6) - (20*b^2*(b*c - a*d
)^3*(c + d*x)^(11/2))/(11*d^6) + (20*b^3*(b*c - a*d)^2*(c + d*x)^(13/2))/(13*d^6) - (2*b^4*(b*c - a*d)*(c + d*
x)^(15/2))/(3*d^6) + (2*b^5*(c + d*x)^(17/2))/(17*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)^{5/2} \, dx &=\int \left (\frac {(-b c+a d)^5 (c+d x)^{5/2}}{d^5}+\frac {5 b (b c-a d)^4 (c+d x)^{7/2}}{d^5}-\frac {10 b^2 (b c-a d)^3 (c+d x)^{9/2}}{d^5}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{11/2}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{13/2}}{d^5}+\frac {b^5 (c+d x)^{15/2}}{d^5}\right ) \, dx\\ &=-\frac {2 (b c-a d)^5 (c+d x)^{7/2}}{7 d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{9/2}}{9 d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{11/2}}{11 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{13/2}}{13 d^6}-\frac {2 b^4 (b c-a d) (c+d x)^{15/2}}{3 d^6}+\frac {2 b^5 (c+d x)^{17/2}}{17 d^6}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 217, normalized size = 1.37 \begin {gather*} \frac {2 (c+d x)^{7/2} \left (21879 a^5 d^5+12155 a^4 b d^4 (-2 c+7 d x)+2210 a^3 b^2 d^3 \left (8 c^2-28 c d x+63 d^2 x^2\right )+510 a^2 b^3 d^2 \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )+17 a b^4 d \left (128 c^4-448 c^3 d x+1008 c^2 d^2 x^2-1848 c d^3 x^3+3003 d^4 x^4\right )+b^5 \left (-256 c^5+896 c^4 d x-2016 c^3 d^2 x^2+3696 c^2 d^3 x^3-6006 c d^4 x^4+9009 d^5 x^5\right )\right )}{153153 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(7/2)*(21879*a^5*d^5 + 12155*a^4*b*d^4*(-2*c + 7*d*x) + 2210*a^3*b^2*d^3*(8*c^2 - 28*c*d*x + 63*d
^2*x^2) + 510*a^2*b^3*d^2*(-16*c^3 + 56*c^2*d*x - 126*c*d^2*x^2 + 231*d^3*x^3) + 17*a*b^4*d*(128*c^4 - 448*c^3
*d*x + 1008*c^2*d^2*x^2 - 1848*c*d^3*x^3 + 3003*d^4*x^4) + b^5*(-256*c^5 + 896*c^4*d*x - 2016*c^3*d^2*x^2 + 36
96*c^2*d^3*x^3 - 6006*c*d^4*x^4 + 9009*d^5*x^5)))/(153153*d^6)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(548\) vs. \(2(158)=316\).
time = 38.50, size = 544, normalized size = 3.44 \begin {gather*} \frac {2 \left (21879 a^5 c^3 d^5+65637 a^5 c^2 d^6 x+65637 a^5 c d^7 x^2+21879 a^5 d^8 x^3-24310 a^4 b c^4 d^4+12155 a^4 b c^3 d^5 x+182325 a^4 b c^2 d^6 x^2+230945 a^4 b c d^7 x^3+85085 a^4 b d^8 x^4+17680 a^3 b^2 c^5 d^3-8840 a^3 b^2 c^4 d^4 x+6630 a^3 b^2 c^3 d^5 x^2+249730 a^3 b^2 c^2 d^6 x^3+355810 a^3 b^2 c d^7 x^4+139230 a^3 b^2 d^8 x^5-8160 a^2 b^3 c^6 d^2+4080 a^2 b^3 c^5 d^3 x-3060 a^2 b^3 c^4 d^4 x^2+2550 a^2 b^3 c^3 d^5 x^3+189210 a^2 b^3 c^2 d^6 x^4+289170 a^2 b^3 c d^7 x^5+117810 a^2 b^3 d^8 x^6+2176 a b^4 c^7 d-1088 a b^4 c^6 d^2 x+816 a b^4 c^5 d^3 x^2-680 a b^4 c^4 d^4 x^3+595 a b^4 c^3 d^5 x^4+76041 a b^4 c^2 d^6 x^5+121737 a b^4 c d^7 x^6+51051 a b^4 d^8 x^7-256 b^5 c^8+128 b^5 c^7 d x-96 b^5 c^6 d^2 x^2+80 b^5 c^5 d^3 x^3-70 b^5 c^4 d^4 x^4+63 b^5 c^3 d^5 x^5+12705 b^5 c^2 d^6 x^6+21021 b^5 c d^7 x^7+9009 b^5 d^8 x^8\right ) \sqrt {c+d x}}{153153 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2 (21879 a ^ 5 c ^ 3 d ^ 5 + 65637 a ^ 5 c ^ 2 d ^ 6 x + 65637 a ^ 5 c d ^ 7 x ^ 2 + 21879 a ^ 5 d ^ 8 x ^ 3 -
24310 a ^ 4 b c ^ 4 d ^ 4 + 12155 a ^ 4 b c ^ 3 d ^ 5 x + 182325 a ^ 4 b c ^ 2 d ^ 6 x ^ 2 + 230945 a ^ 4 b c
d ^ 7 x ^ 3 + 85085 a ^ 4 b d ^ 8 x ^ 4 + 17680 a ^ 3 b ^ 2 c ^ 5 d ^ 3 - 8840 a ^ 3 b ^ 2 c ^ 4 d ^ 4 x + 66
30 a ^ 3 b ^ 2 c ^ 3 d ^ 5 x ^ 2 + 249730 a ^ 3 b ^ 2 c ^ 2 d ^ 6 x ^ 3 + 355810 a ^ 3 b ^ 2 c d ^ 7 x ^ 4 + 1
39230 a ^ 3 b ^ 2 d ^ 8 x ^ 5 - 8160 a ^ 2 b ^ 3 c ^ 6 d ^ 2 + 4080 a ^ 2 b ^ 3 c ^ 5 d ^ 3 x - 3060 a ^ 2 b ^
3 c ^ 4 d ^ 4 x ^ 2 + 2550 a ^ 2 b ^ 3 c ^ 3 d ^ 5 x ^ 3 + 189210 a ^ 2 b ^ 3 c ^ 2 d ^ 6 x ^ 4 + 289170 a ^
2 b ^ 3 c d ^ 7 x ^ 5 + 117810 a ^ 2 b ^ 3 d ^ 8 x ^ 6 + 2176 a b ^ 4 c ^ 7 d - 1088 a b ^ 4 c ^ 6 d ^ 2 x + 8
16 a b ^ 4 c ^ 5 d ^ 3 x ^ 2 - 680 a b ^ 4 c ^ 4 d ^ 4 x ^ 3 + 595 a b ^ 4 c ^ 3 d ^ 5 x ^ 4 + 76041 a b ^ 4 c
^ 2 d ^ 6 x ^ 5 + 121737 a b ^ 4 c d ^ 7 x ^ 6 + 51051 a b ^ 4 d ^ 8 x ^ 7 - 256 b ^ 5 c ^ 8 + 128 b ^ 5 c ^
7 d x - 96 b ^ 5 c ^ 6 d ^ 2 x ^ 2 + 80 b ^ 5 c ^ 5 d ^ 3 x ^ 3 - 70 b ^ 5 c ^ 4 d ^ 4 x ^ 4 + 63 b ^ 5 c ^ 3
d ^ 5 x ^ 5 + 12705 b ^ 5 c ^ 2 d ^ 6 x ^ 6 + 21021 b ^ 5 c d ^ 7 x ^ 7 + 9009 b ^ 5 d ^ 8 x ^ 8) Sqrt[c + d x
] / (153153 d ^ 6)

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Maple [A]
time = 0.15, size = 122, normalized size = 0.77

method	result	size
derivativedivides	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {17}{2}}}{17}+\frac {2 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {15}{2}}}{3}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{6}}\)	\(122\)
default	\(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {17}{2}}}{17}+\frac {2 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {15}{2}}}{3}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{6}}\)	\(122\)
gosper	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (9009 b^{5} x^{5} d^{5}+51051 a \,b^{4} d^{5} x^{4}-6006 b^{5} c \,d^{4} x^{4}+117810 a^{2} b^{3} d^{5} x^{3}-31416 a \,b^{4} c \,d^{4} x^{3}+3696 b^{5} c^{2} d^{3} x^{3}+139230 a^{3} b^{2} d^{5} x^{2}-64260 a^{2} b^{3} c \,d^{4} x^{2}+17136 a \,b^{4} c^{2} d^{3} x^{2}-2016 b^{5} c^{3} d^{2} x^{2}+85085 a^{4} b \,d^{5} x -61880 a^{3} b^{2} c \,d^{4} x +28560 a^{2} b^{3} c^{2} d^{3} x -7616 a \,b^{4} c^{3} d^{2} x +896 b^{5} c^{4} d x +21879 a^{5} d^{5}-24310 a^{4} b c \,d^{4}+17680 a^{3} b^{2} c^{2} d^{3}-8160 a^{2} b^{3} c^{3} d^{2}+2176 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{153153 d^{6}}\)	\(273\)
trager	\(\frac {2 \left (9009 b^{5} d^{8} x^{8}+51051 a \,b^{4} d^{8} x^{7}+21021 b^{5} c \,d^{7} x^{7}+117810 a^{2} b^{3} d^{8} x^{6}+121737 a \,b^{4} c \,d^{7} x^{6}+12705 b^{5} c^{2} d^{6} x^{6}+139230 a^{3} b^{2} d^{8} x^{5}+289170 a^{2} b^{3} c \,d^{7} x^{5}+76041 a \,b^{4} c^{2} d^{6} x^{5}+63 b^{5} c^{3} d^{5} x^{5}+85085 a^{4} b \,d^{8} x^{4}+355810 a^{3} b^{2} c \,d^{7} x^{4}+189210 a^{2} b^{3} c^{2} d^{6} x^{4}+595 a \,b^{4} c^{3} d^{5} x^{4}-70 b^{5} c^{4} d^{4} x^{4}+21879 a^{5} d^{8} x^{3}+230945 a^{4} b c \,d^{7} x^{3}+249730 a^{3} b^{2} c^{2} d^{6} x^{3}+2550 a^{2} b^{3} c^{3} d^{5} x^{3}-680 a \,b^{4} c^{4} d^{4} x^{3}+80 b^{5} c^{5} d^{3} x^{3}+65637 a^{5} c \,d^{7} x^{2}+182325 a^{4} b \,c^{2} d^{6} x^{2}+6630 a^{3} b^{2} c^{3} d^{5} x^{2}-3060 a^{2} b^{3} c^{4} d^{4} x^{2}+816 a \,b^{4} c^{5} d^{3} x^{2}-96 b^{5} c^{6} d^{2} x^{2}+65637 a^{5} c^{2} d^{6} x +12155 a^{4} b \,c^{3} d^{5} x -8840 a^{3} b^{2} c^{4} d^{4} x +4080 a^{2} b^{3} c^{5} d^{3} x -1088 a \,b^{4} c^{6} d^{2} x +128 b^{5} c^{7} d x +21879 a^{5} c^{3} d^{5}-24310 a^{4} b \,c^{4} d^{4}+17680 a^{3} b^{2} c^{5} d^{3}-8160 a^{2} b^{3} c^{6} d^{2}+2176 a \,b^{4} c^{7} d -256 b^{5} c^{8}\right ) \sqrt {d x +c}}{153153 d^{6}}\)	\(545\)
risch	\(\frac {2 \left (9009 b^{5} d^{8} x^{8}+51051 a \,b^{4} d^{8} x^{7}+21021 b^{5} c \,d^{7} x^{7}+117810 a^{2} b^{3} d^{8} x^{6}+121737 a \,b^{4} c \,d^{7} x^{6}+12705 b^{5} c^{2} d^{6} x^{6}+139230 a^{3} b^{2} d^{8} x^{5}+289170 a^{2} b^{3} c \,d^{7} x^{5}+76041 a \,b^{4} c^{2} d^{6} x^{5}+63 b^{5} c^{3} d^{5} x^{5}+85085 a^{4} b \,d^{8} x^{4}+355810 a^{3} b^{2} c \,d^{7} x^{4}+189210 a^{2} b^{3} c^{2} d^{6} x^{4}+595 a \,b^{4} c^{3} d^{5} x^{4}-70 b^{5} c^{4} d^{4} x^{4}+21879 a^{5} d^{8} x^{3}+230945 a^{4} b c \,d^{7} x^{3}+249730 a^{3} b^{2} c^{2} d^{6} x^{3}+2550 a^{2} b^{3} c^{3} d^{5} x^{3}-680 a \,b^{4} c^{4} d^{4} x^{3}+80 b^{5} c^{5} d^{3} x^{3}+65637 a^{5} c \,d^{7} x^{2}+182325 a^{4} b \,c^{2} d^{6} x^{2}+6630 a^{3} b^{2} c^{3} d^{5} x^{2}-3060 a^{2} b^{3} c^{4} d^{4} x^{2}+816 a \,b^{4} c^{5} d^{3} x^{2}-96 b^{5} c^{6} d^{2} x^{2}+65637 a^{5} c^{2} d^{6} x +12155 a^{4} b \,c^{3} d^{5} x -8840 a^{3} b^{2} c^{4} d^{4} x +4080 a^{2} b^{3} c^{5} d^{3} x -1088 a \,b^{4} c^{6} d^{2} x +128 b^{5} c^{7} d x +21879 a^{5} c^{3} d^{5}-24310 a^{4} b \,c^{4} d^{4}+17680 a^{3} b^{2} c^{5} d^{3}-8160 a^{2} b^{3} c^{6} d^{2}+2176 a \,b^{4} c^{7} d -256 b^{5} c^{8}\right ) \sqrt {d x +c}}{153153 d^{6}}\)	\(545\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d^6*(1/17*b^5*(d*x+c)^(17/2)+1/3*(a*d-b*c)*b^4*(d*x+c)^(15/2)+10/13*(a*d-b*c)^2*b^3*(d*x+c)^(13/2)+10/11*(a*
d-b*c)^3*b^2*(d*x+c)^(11/2)+5/9*(a*d-b*c)^4*b*(d*x+c)^(9/2)+1/7*(a*d-b*c)^5*(d*x+c)^(7/2))

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Maxima [A]
time = 0.29, size = 259, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (9009 \, {\left (d x + c\right )}^{\frac {17}{2}} b^{5} - 51051 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {15}{2}} + 117810 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {13}{2}} - 139230 \, {\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 {11}{2}} + 85085 \, {\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 {9}{2}} - 21879 \, {\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 {7}{2}}\right )}}{153153 \, d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153153*(9009*(d*x + c)^(17/2)*b^5 - 51051*(b^5*c - a*b^4*d)*(d*x + c)^(15/2) + 117810*(b^5*c^2 - 2*a*b^4*c*d
+ a^2*b^3*d^2)*(d*x + c)^(13/2) - 139230*(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)^
(11/2) + 85085*(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)^(9/2) - 2
1879*(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)^(
7/2))/d^6

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (134) = 268\).
time = 0.30, size = 497, normalized size = 3.15 \begin {gather*} \frac {2 \, {\left (9009 \, b^{5} d^{8} x^{8} - 256 \, b^{5} c^{8} + 2176 \, a b^{4} c^{7} d - 8160 \, a^{2} b^{3} c^{6} d^{2} + 17680 \, a^{3} b^{2} c^{5} d^{3} - 24310 \, a^{4} b c^{4} d^{4} + 21879 \, a^{5} c^{3} d^{5} + 3003 \, {\left (7 \, b^{5} c d^{7} + 17 \, a b^{4} d^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} c^{2} d^{6} + 527 \, a b^{4} c d^{7} + 510 \, a^{2} b^{3} d^{8}\right )} x^{6} + 63 \, {\left (b^{5} c^{3} d^{5} + 1207 \, a b^{4} c^{2} d^{6} + 4590 \, a^{2} b^{3} c d^{7} + 2210 \, a^{3} b^{2} d^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} c^{4} d^{4} - 17 \, a b^{4} c^{3} d^{5} - 5406 \, a^{2} b^{3} c^{2} d^{6} - 10166 \, a^{3} b^{2} c d^{7} - 2431 \, a^{4} b d^{8}\right )} x^{4} + {\left (80 \, b^{5} c^{5} d^{3} - 680 \, a b^{4} c^{4} d^{4} + 2550 \, a^{2} b^{3} c^{3} d^{5} + 249730 \, a^{3} b^{2} c^{2} d^{6} + 230945 \, a^{4} b c d^{7} + 21879 \, a^{5} d^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} c^{6} d^{2} - 272 \, a b^{4} c^{5} d^{3} + 1020 \, a^{2} b^{3} c^{4} d^{4} - 2210 \, a^{3} b^{2} c^{3} d^{5} - 60775 \, a^{4} b c^{2} d^{6} - 21879 \, a^{5} c d^{7}\right )} x^{2} + {\left (128 \, b^{5} c^{7} d - 1088 \, a b^{4} c^{6} d^{2} + 4080 \, a^{2} b^{3} c^{5} d^{3} - 8840 \, a^{3} b^{2} c^{4} d^{4} + 12155 \, a^{4} b c^{3} d^{5} + 65637 \, a^{5} c^{2} d^{6}\right )} x\right )} \sqrt {d x + c}}{153153 \, d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153153*(9009*b^5*d^8*x^8 - 256*b^5*c^8 + 2176*a*b^4*c^7*d - 8160*a^2*b^3*c^6*d^2 + 17680*a^3*b^2*c^5*d^3 - 2
4310*a^4*b*c^4*d^4 + 21879*a^5*c^3*d^5 + 3003*(7*b^5*c*d^7 + 17*a*b^4*d^8)*x^7 + 231*(55*b^5*c^2*d^6 + 527*a*b
^4*c*d^7 + 510*a^2*b^3*d^8)*x^6 + 63*(b^5*c^3*d^5 + 1207*a*b^4*c^2*d^6 + 4590*a^2*b^3*c*d^7 + 2210*a^3*b^2*d^8
)*x^5 - 35*(2*b^5*c^4*d^4 - 17*a*b^4*c^3*d^5 - 5406*a^2*b^3*c^2*d^6 - 10166*a^3*b^2*c*d^7 - 2431*a^4*b*d^8)*x^
4 + (80*b^5*c^5*d^3 - 680*a*b^4*c^4*d^4 + 2550*a^2*b^3*c^3*d^5 + 249730*a^3*b^2*c^2*d^6 + 230945*a^4*b*c*d^7 +
21879*a^5*d^8)*x^3 - 3*(32*b^5*c^6*d^2 - 272*a*b^4*c^5*d^3 + 1020*a^2*b^3*c^4*d^4 - 2210*a^3*b^2*c^3*d^5 - 60
775*a^4*b*c^2*d^6 - 21879*a^5*c*d^7)*x^2 + (128*b^5*c^7*d - 1088*a*b^4*c^6*d^2 + 4080*a^2*b^3*c^5*d^3 - 8840*a
^3*b^2*c^4*d^4 + 12155*a^4*b*c^3*d^5 + 65637*a^5*c^2*d^6)*x)*sqrt(d*x + c)/d^6

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Sympy [A]
time = 21.93, size = 1292, normalized size = 8.18

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*c**2*Piecewise((sqrt(c)*x, Eq(d, 0)), (2*(c + d*x)**(3/2)/(3*d), True)) + 4*a**5*c*(-c*(c + d*x)**(3/2)/3
+ (c + d*x)**(5/2)/5)/d + 2*a**5*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d +
10*a**4*b*c**2*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + 20*a**4*b*c*(c**2*(c + d*x)**(3/2)/3 - 2*c*
(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**2 + 10*a**4*b*(-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**2 + 20*a**3*b**2*c**2*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c +
d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**3 + 40*a**3*b**2*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**3 + 20*a**3*b**2*(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**3 + 20*a**2*b**3*c
**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**4
+ 40*a**2*b**3*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**4 + 20*a**2*b**3*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 1
0*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**4
+ 10*a*b**4*c**2*(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 + 20*a*b**4*c*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 1
0*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
+ 10*a*b**4*(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**5 + 2*b**5*
c**2*(-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 + 4*b**5*c*(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 + 2*b**5*(-c**7*(c + d*x)**(3/2)/3 + 7*c**6*(c + d*x)**(5/2)/5
- 3*c**5*(c + d*x)**(7/2) + 35*c**4*(c + d*x)**(9/2)/9 - 35*c**3*(c + d*x)**(11/2)/11 + 21*c**2*(c + d*x)**(1
3/2)/13 - 7*c*(c + d*x)**(15/2)/15 + (c + d*x)**(17/2)/17)/d**6

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1599 vs. \(2 (134) = 268\).
time = 0.02, size = 2696, normalized size = 17.06

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(765765*sqrt(d*x + c)*a^5*c^3 + 765765*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^5*c^2 + 1276275*((d*x
+ c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*b*c^3/d + 153153*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x +
c)*c^2)*a^5*c + 510510*(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^3/d^2 + 76
5765*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^4*b*c^2/d + 21879*(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^5 + 218790*(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^3/d^3 + 656370*(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^2/d^2 + 328185*(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*c/d + 12155*(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^3/d^4 + 72930*(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^2/d^3 + 72930*(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*c/d^2 + 12155*(35*(d*x + c)^(9/2) - 1
80*(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^4*b/d + 11
05*(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^3/d^5 + 16575*(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*b^4*c
^2/d^4 + 33150*(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*c/d^3 + 11050*(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^3*b^2/d^2 + 765*(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^2/d^5 +
3825*(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*c/d^4 + 2550*(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^2*b^3/d^3 + 357*(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*c/d^5 + 595*(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)*a*b^4/d^4 + 7*(6435*(d*x +
c)^(17/2) - 58344*(d*x + c)^(15/2)*c + 235620*(d*x + c)^(13/2)*c^2 - 556920*(d*x + c)^(11/2)*c^3 + 850850*(d*x
+ c)^(9/2)*c^4 - 875160*(d*x + c)^(7/2)*c^5 + 612612*(d*x + c)^(5/2)*c^6 - 291720*(d*x + c)^(3/2)*c^7 + 10939
5*sqrt(d*x + c)*c^8)*b^5/d^5)/d

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Mupad [B]
time = 0.27, size = 137, normalized size = 0.87 \begin {gather*} \frac {2\,b^5\,{\left (c+d\,x\right )}^{17/2}}{17\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{15/2}}{15\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6}+\frac {10\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5*(c + d*x)^(5/2),x)

[Out]

(2*b^5*(c + d*x)^(17/2))/(17*d^6) - ((10*b^5*c - 10*a*b^4*d)*(c + d*x)^(15/2))/(15*d^6) + (2*(a*d - b*c)^5*(c
+ d*x)^(7/2))/(7*d^6) + (20*b^2*(a*d - b*c)^3*(c + d*x)^(11/2))/(11*d^6) + (20*b^3*(a*d - b*c)^2*(c + d*x)^(13
/2))/(13*d^6) + (10*b*(a*d - b*c)^4*(c + d*x)^(9/2))/(9*d^6)

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