3.353 \(\int (d+e x)^{7/2} (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=248 \[ \frac {6 c (d+e x)^{17/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{17 e^7}-\frac {2 (d+e x)^{15/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{15 e^7}+\frac {6 d (d+e x)^{13/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac {6 c^2 (d+e x)^{19/2} (2 c d-b e)}{19 e^7}+\frac {2 d^3 (d+e x)^{9/2} (c d-b e)^3}{9 e^7}-\frac {6 d^2 (d+e x)^{11/2} (c d-b e)^2 (2 c d-b e)}{11 e^7}+\frac {2 c^3 (d+e x)^{21/2}}{21 e^7} \]

[Out]

2/9*d^3*(-b*e+c*d)^3*(e*x+d)^(9/2)/e^7-6/11*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*(e*x+d)^(11/2)/e^7+6/13*d*(-b*e+c*d)
*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(13/2)/e^7-2/15*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(1
5/2)/e^7+6/17*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(17/2)/e^7-6/19*c^2*(-b*e+2*c*d)*(e*x+d)^(19/2)/e^7+2/21
*c^3*(e*x+d)^(21/2)/e^7

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Rubi [A]  time = 0.14, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {698} \[ \frac {6 c (d+e x)^{17/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{17 e^7}-\frac {2 (d+e x)^{15/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{15 e^7}+\frac {6 d (d+e x)^{13/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{13 e^7}-\frac {6 c^2 (d+e x)^{19/2} (2 c d-b e)}{19 e^7}-\frac {6 d^2 (d+e x)^{11/2} (c d-b e)^2 (2 c d-b e)}{11 e^7}+\frac {2 d^3 (d+e x)^{9/2} (c d-b e)^3}{9 e^7}+\frac {2 c^3 (d+e x)^{21/2}}{21 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)*(b*x + c*x^2)^3,x]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(9/2))/(9*e^7) - (6*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^7)
+ (6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2
- 10*b*c*d*e + b^2*e^2)*(d + e*x)^(15/2))/(15*e^7) + (6*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(17/2))
/(17*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(19/2))/(19*e^7) + (2*c^3*(d + e*x)^(21/2))/(21*e^7)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^{7/2} \left (b x+c x^2\right )^3 \, dx &=\int \left (\frac {d^3 (c d-b e)^3 (d+e x)^{7/2}}{e^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{9/2}}{e^6}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{13/2}}{e^6}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{15/2}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{17/2}}{e^6}+\frac {c^3 (d+e x)^{19/2}}{e^6}\right ) \, dx\\ &=\frac {2 d^3 (c d-b e)^3 (d+e x)^{9/2}}{9 e^7}-\frac {6 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac {6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{15/2}}{15 e^7}+\frac {6 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{17/2}}{17 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{19/2}}{19 e^7}+\frac {2 c^3 (d+e x)^{21/2}}{21 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 206, normalized size = 0.83 \[ \frac {2 (d+e x)^{9/2} \left (2567565 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-969969 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+3357585 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-2297295 c^2 (d+e x)^5 (2 c d-b e)+1616615 d^3 (c d-b e)^3-3968055 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+692835 c^3 (d+e x)^6\right )}{14549535 e^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)*(b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(9/2)*(1616615*d^3*(c*d - b*e)^3 - 3968055*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 3357585*d*
(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 969969*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^
2*e^2)*(d + e*x)^3 + 2567565*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 2297295*c^2*(2*c*d - b*e)*(d +
e*x)^5 + 692835*c^3*(d + e*x)^6))/(14549535*e^7)

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fricas [B]  time = 0.91, size = 478, normalized size = 1.93 \[ \frac {2 \, {\left (692835 \, c^{3} e^{10} x^{10} + 5120 \, c^{3} d^{10} - 26880 \, b c^{2} d^{9} e + 51072 \, b^{2} c d^{8} e^{2} - 36176 \, b^{3} d^{7} e^{3} + 36465 \, {\left (64 \, c^{3} d e^{9} + 63 \, b c^{2} e^{10}\right )} x^{9} + 19305 \, {\left (138 \, c^{3} d^{2} e^{8} + 406 \, b c^{2} d e^{9} + 133 \, b^{2} c e^{10}\right )} x^{8} + 429 \, {\left (2420 \, c^{3} d^{3} e^{7} + 21210 \, b c^{2} d^{2} e^{8} + 20748 \, b^{2} c d e^{9} + 2261 \, b^{3} e^{10}\right )} x^{7} + 231 \, {\left (5 \, c^{3} d^{4} e^{6} + 15720 \, b c^{2} d^{3} e^{7} + 45714 \, b^{2} c d^{2} e^{8} + 14858 \, b^{3} d e^{9}\right )} x^{6} - 63 \, {\left (20 \, c^{3} d^{5} e^{5} - 105 \, b c^{2} d^{4} e^{6} - 69084 \, b^{2} c d^{3} e^{7} - 66538 \, b^{3} d^{2} e^{8}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{6} e^{4} - 210 \, b c^{2} d^{5} e^{5} + 399 \, b^{2} c d^{4} e^{6} + 51680 \, b^{3} d^{3} e^{7}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{7} e^{3} - 1680 \, b c^{2} d^{6} e^{4} + 3192 \, b^{2} c d^{5} e^{5} - 2261 \, b^{3} d^{4} e^{6}\right )} x^{3} + 6 \, {\left (320 \, c^{3} d^{8} e^{2} - 1680 \, b c^{2} d^{7} e^{3} + 3192 \, b^{2} c d^{6} e^{4} - 2261 \, b^{3} d^{5} e^{5}\right )} x^{2} - 8 \, {\left (320 \, c^{3} d^{9} e - 1680 \, b c^{2} d^{8} e^{2} + 3192 \, b^{2} c d^{7} e^{3} - 2261 \, b^{3} d^{6} e^{4}\right )} x\right )} \sqrt {e x + d}}{14549535 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

2/14549535*(692835*c^3*e^10*x^10 + 5120*c^3*d^10 - 26880*b*c^2*d^9*e + 51072*b^2*c*d^8*e^2 - 36176*b^3*d^7*e^3
+ 36465*(64*c^3*d*e^9 + 63*b*c^2*e^10)*x^9 + 19305*(138*c^3*d^2*e^8 + 406*b*c^2*d*e^9 + 133*b^2*c*e^10)*x^8 +
429*(2420*c^3*d^3*e^7 + 21210*b*c^2*d^2*e^8 + 20748*b^2*c*d*e^9 + 2261*b^3*e^10)*x^7 + 231*(5*c^3*d^4*e^6 + 1
5720*b*c^2*d^3*e^7 + 45714*b^2*c*d^2*e^8 + 14858*b^3*d*e^9)*x^6 - 63*(20*c^3*d^5*e^5 - 105*b*c^2*d^4*e^6 - 690
84*b^2*c*d^3*e^7 - 66538*b^3*d^2*e^8)*x^5 + 35*(40*c^3*d^6*e^4 - 210*b*c^2*d^5*e^5 + 399*b^2*c*d^4*e^6 + 51680
*b^3*d^3*e^7)*x^4 - 5*(320*c^3*d^7*e^3 - 1680*b*c^2*d^6*e^4 + 3192*b^2*c*d^5*e^5 - 2261*b^3*d^4*e^6)*x^3 + 6*(
320*c^3*d^8*e^2 - 1680*b*c^2*d^7*e^3 + 3192*b^2*c*d^6*e^4 - 2261*b^3*d^5*e^5)*x^2 - 8*(320*c^3*d^9*e - 1680*b*
c^2*d^8*e^2 + 3192*b^2*c*d^7*e^3 - 2261*b^3*d^6*e^4)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.31, size = 2065, normalized size = 8.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

2/14549535*(415701*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*
b^3*d^4*e^(-3) + 138567*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^
(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^2*c*d^4*e^(-4) + 62985*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990
*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b*c^2*d^4*
e^(-5) + 4845*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2
)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*c^3*d^4*e^(-6) + 184756*
(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e
+ d)*d^4)*b^3*d^3*e^(-3) + 251940*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 13
86*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^2*c*d^3*e^(-4) + 58140*(231*(x*e
+ d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(
5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b*c^2*d^3*e^(-5) + 9044*(429*(x*e + d)^(15/2) -
3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 -
27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^3*d^3*e^(-6) + 125970*(63*(
x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^
(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^3*d^2*e^(-3) + 87210*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 50
05*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003
*sqrt(x*e + d)*d^6)*b^2*c*d^2*e^(-4) + 40698*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)
^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e +
d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b*c^2*d^2*e^(-5) + 798*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)
*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)
^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*c^3*d^2*e^(-6
) + 19380*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^
3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^3*d*e^(-3) + 27132*(429*(x
*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e
+ d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^2*c*d*e^(-
4) + 1596*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(
11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e
+ d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*b*c^2*d*e^(-5) + 252*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(1
7/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(
x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 -
230945*sqrt(x*e + d)*d^9)*c^3*d*e^(-6) + 2261*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d
)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e
+ d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^3*e^(-3) + 399*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d +
235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2
)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*b^2*c*e^(-4) + 189
*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*
d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e
+ d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9)*b*c^2*e^(-5) + 15*(46189*(x*e + d)^(21
/2) - 510510*(x*e + d)^(19/2)*d + 2567565*(x*e + d)^(17/2)*d^2 - 7759752*(x*e + d)^(15/2)*d^3 + 15668730*(x*e
+ d)^(13/2)*d^4 - 22221108*(x*e + d)^(11/2)*d^5 + 22632610*(x*e + d)^(9/2)*d^6 - 16628040*(x*e + d)^(7/2)*d^7
+ 8729721*(x*e + d)^(5/2)*d^8 - 3233230*(x*e + d)^(3/2)*d^9 + 969969*sqrt(x*e + d)*d^10)*c^3*e^(-6))*e^(-1)

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maple [A]  time = 0.12, size = 286, normalized size = 1.15 \[ -\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (-692835 c^{3} x^{6} e^{6}-2297295 b \,c^{2} e^{6} x^{5}+437580 c^{3} d \,e^{5} x^{5}-2567565 b^{2} c \,e^{6} x^{4}+1351350 b \,c^{2} d \,e^{5} x^{4}-257400 c^{3} d^{2} e^{4} x^{4}-969969 b^{3} e^{6} x^{3}+1369368 b^{2} c d \,e^{5} x^{3}-720720 b \,c^{2} d^{2} e^{4} x^{3}+137280 c^{3} d^{3} e^{3} x^{3}+447678 b^{3} d \,e^{5} x^{2}-632016 b^{2} c \,d^{2} e^{4} x^{2}+332640 b \,c^{2} d^{3} e^{3} x^{2}-63360 c^{3} d^{4} e^{2} x^{2}-162792 b^{3} d^{2} e^{4} x +229824 b^{2} c \,d^{3} e^{3} x -120960 b \,c^{2} d^{4} e^{2} x +23040 c^{3} d^{5} e x +36176 b^{3} d^{3} e^{3}-51072 b^{2} c \,d^{4} e^{2}+26880 b \,c^{2} d^{5} e -5120 c^{3} d^{6}\right )}{14549535 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)*(c*x^2+b*x)^3,x)

[Out]

-2/14549535*(e*x+d)^(9/2)*(-692835*c^3*e^6*x^6-2297295*b*c^2*e^6*x^5+437580*c^3*d*e^5*x^5-2567565*b^2*c*e^6*x^
4+1351350*b*c^2*d*e^5*x^4-257400*c^3*d^2*e^4*x^4-969969*b^3*e^6*x^3+1369368*b^2*c*d*e^5*x^3-720720*b*c^2*d^2*e
^4*x^3+137280*c^3*d^3*e^3*x^3+447678*b^3*d*e^5*x^2-632016*b^2*c*d^2*e^4*x^2+332640*b*c^2*d^3*e^3*x^2-63360*c^3
*d^4*e^2*x^2-162792*b^3*d^2*e^4*x+229824*b^2*c*d^3*e^3*x-120960*b*c^2*d^4*e^2*x+23040*c^3*d^5*e*x+36176*b^3*d^
3*e^3-51072*b^2*c*d^4*e^2+26880*b*c^2*d^5*e-5120*c^3*d^6)/e^7

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maxima [A]  time = 1.41, size = 271, normalized size = 1.09 \[ \frac {2 \, {\left (692835 \, {\left (e x + d\right )}^{\frac {21}{2}} c^{3} - 2297295 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {19}{2}} + 2567565 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )} {\left (e x + d\right )}^{\frac {17}{2}} - 969969 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 3357585 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 3968055 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1616615 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{14549535 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

2/14549535*(692835*(e*x + d)^(21/2)*c^3 - 2297295*(2*c^3*d - b*c^2*e)*(e*x + d)^(19/2) + 2567565*(5*c^3*d^2 -
5*b*c^2*d*e + b^2*c*e^2)*(e*x + d)^(17/2) - 969969*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e
*x + d)^(15/2) + 3357585*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(13/2) - 3968055
*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d)^(11/2) + 1616615*(c^3*d^6 - 3*b*c^2*d^5
*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*(e*x + d)^(9/2))/e^7

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mupad [B]  time = 0.23, size = 239, normalized size = 0.96 \[ \frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,b^3\,e^3-24\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e-40\,c^3\,d^3\right )}{15\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{21/2}}{21\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{19/2}}{19\,e^7}+\frac {{\left (d+e\,x\right )}^{17/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2\right )}{17\,e^7}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{13\,e^7}-\frac {2\,d^3\,{\left (b\,e-c\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3*(d + e*x)^(7/2),x)

[Out]

((d + e*x)^(15/2)*(2*b^3*e^3 - 40*c^3*d^3 + 60*b*c^2*d^2*e - 24*b^2*c*d*e^2))/(15*e^7) + (2*c^3*(d + e*x)^(21/
2))/(21*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e*x)^(19/2))/(19*e^7) + ((d + e*x)^(17/2)*(30*c^3*d^2 + 6*b^2*c*e^
2 - 30*b*c^2*d*e))/(17*e^7) + ((d + e*x)^(13/2)*(30*c^3*d^4 - 6*b^3*d*e^3 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e)
)/(13*e^7) - (2*d^3*(b*e - c*d)^3*(d + e*x)^(9/2))/(9*e^7) + (6*d^2*(b*e - c*d)^2*(b*e - 2*c*d)*(d + e*x)^(11/
2))/(11*e^7)

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sympy [B]  time = 61.79, size = 1741, normalized size = 7.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)*(c*x**2+b*x)**3,x)

[Out]

2*b**3*d**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/
9)/e**4 + 6*b**3*d**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(
d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 6*b**3*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) -
10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**
4 + 2*b**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*
x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**4 + 6*b**2*c*
d**3*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9
+ (d + e*x)**(11/2)/11)/e**5 + 18*b**2*c*d**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d
+ e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 18*b**2
*c*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9
/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5 + 6*b**2*c*(-d**7
*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d*
*3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**5
+ 6*b*c**2*d**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d +
e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 18*b*c**2*d**2*(d**6*(d + e*x)**(3/2)
/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**
(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 18*b*c**2*d*(-d**7*(d + e*x)**(3/2)/3 + 7*
d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11
+ 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**6 + 6*b*c**2*(d**8*(d + e
*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d +
e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d
+ e*x)**(19/2)/19)/e**6 + 2*c**3*d**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x
)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)*
*(15/2)/15)/e**7 + 6*c**3*d**2*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2)
+ 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(
15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 6*c**3*d*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6
*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 +
28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**7 + 2*c**3*(-d**9*(d + e*x
)**(3/2)/3 + 9*d**8*(d + e*x)**(5/2)/5 - 36*d**7*(d + e*x)**(7/2)/7 + 28*d**6*(d + e*x)**(9/2)/3 - 126*d**5*(d
+ e*x)**(11/2)/11 + 126*d**4*(d + e*x)**(13/2)/13 - 28*d**3*(d + e*x)**(15/2)/5 + 36*d**2*(d + e*x)**(17/2)/1
7 - 9*d*(d + e*x)**(19/2)/19 + (d + e*x)**(21/2)/21)/e**7

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