3.11.77 \(\int (A+B x) (d+e x)^{5/2} (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=267 \[ -\frac {2 (d+e x)^{13/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{13 e^6}+\frac {2 (d+e x)^{11/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{11 e^6}-\frac {2 d^2 (d+e x)^{7/2} (B d-A e) (c d-b e)^2}{7 e^6}-\frac {2 c (d+e x)^{15/2} (-A c e-2 b B e+5 B c d)}{15 e^6}+\frac {2 d (d+e x)^{9/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{9 e^6}+\frac {2 B c^2 (d+e x)^{17/2}}{17 e^6} \]

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Rubi [A]  time = 0.15, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{13/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{13 e^6}+\frac {2 (d+e x)^{11/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{11 e^6}-\frac {2 d^2 (d+e x)^{7/2} (B d-A e) (c d-b e)^2}{7 e^6}-\frac {2 c (d+e x)^{15/2} (-A c e-2 b B e+5 B c d)}{15 e^6}+\frac {2 d (d+e x)^{9/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{9 e^6}+\frac {2 B c^2 (d+e x)^{17/2}}{17 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(5/2)*(b*x + c*x^2)^2,x]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(7/2))/(7*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(9/2))/(9*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*
e + 3*b^2*e^2))*(d + e*x)^(11/2))/(11*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))
*(d + e*x)^(13/2))/(13*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(15/2))/(15*e^6) + (2*B*c^2*(d + e*x)
^(17/2))/(17*e^6)

Rule 771

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

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2 (d+e x)^{5/2}}{e^5}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{7/2}}{e^5}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{9/2}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{11/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{13/2}}{e^5}+\frac {B c^2 (d+e x)^{15/2}}{e^5}\right ) \, dx\\ &=-\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{7/2}}{7 e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{9/2}}{9 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{11/2}}{11 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{15/2}}{15 e^6}+\frac {2 B c^2 (d+e x)^{17/2}}{17 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 273, normalized size = 1.02 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (17 A e \left (65 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 b c e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+B \left (255 b^2 e^2 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+34 b c e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )-5 c^2 \left (256 d^5-896 d^4 e x+2016 d^3 e^2 x^2-3696 d^2 e^3 x^3+6006 d e^4 x^4-9009 e^5 x^5\right )\right )\right )}{765765 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(17*A*e*(65*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 30*b*c*e*(-16*d^3 + 56*d^2*e*x - 126*
d*e^2*x^2 + 231*e^3*x^3) + c^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)) + B
*(255*b^2*e^2*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 34*b*c*e*(128*d^4 - 448*d^3*e*x + 1008*d^
2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) - 5*c^2*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2 - 3696*d^2*e^3*x^
3 + 6006*d*e^4*x^4 - 9009*e^5*x^5))))/(765765*e^6)

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IntegrateAlgebraic [A]  time = 0.19, size = 399, normalized size = 1.49 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (109395 A b^2 d^2 e^3-170170 A b^2 d e^3 (d+e x)+69615 A b^2 e^3 (d+e x)^2-218790 A b c d^3 e^2+510510 A b c d^2 e^2 (d+e x)-417690 A b c d e^2 (d+e x)^2+117810 A b c e^2 (d+e x)^3+109395 A c^2 d^4 e-340340 A c^2 d^3 e (d+e x)+417690 A c^2 d^2 e (d+e x)^2-235620 A c^2 d e (d+e x)^3+51051 A c^2 e (d+e x)^4-109395 b^2 B d^3 e^2+255255 b^2 B d^2 e^2 (d+e x)-208845 b^2 B d e^2 (d+e x)^2+58905 b^2 B e^2 (d+e x)^3+218790 b B c d^4 e-680680 b B c d^3 e (d+e x)+835380 b B c d^2 e (d+e x)^2-471240 b B c d e (d+e x)^3+102102 b B c e (d+e x)^4-109395 B c^2 d^5+425425 B c^2 d^4 (d+e x)-696150 B c^2 d^3 (d+e x)^2+589050 B c^2 d^2 (d+e x)^3-255255 B c^2 d (d+e x)^4+45045 B c^2 (d+e x)^5\right )}{765765 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^(5/2)*(b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(7/2)*(-109395*B*c^2*d^5 + 218790*b*B*c*d^4*e + 109395*A*c^2*d^4*e - 109395*b^2*B*d^3*e^2 - 21879
0*A*b*c*d^3*e^2 + 109395*A*b^2*d^2*e^3 + 425425*B*c^2*d^4*(d + e*x) - 680680*b*B*c*d^3*e*(d + e*x) - 340340*A*
c^2*d^3*e*(d + e*x) + 255255*b^2*B*d^2*e^2*(d + e*x) + 510510*A*b*c*d^2*e^2*(d + e*x) - 170170*A*b^2*d*e^3*(d
+ e*x) - 696150*B*c^2*d^3*(d + e*x)^2 + 835380*b*B*c*d^2*e*(d + e*x)^2 + 417690*A*c^2*d^2*e*(d + e*x)^2 - 2088
45*b^2*B*d*e^2*(d + e*x)^2 - 417690*A*b*c*d*e^2*(d + e*x)^2 + 69615*A*b^2*e^3*(d + e*x)^2 + 589050*B*c^2*d^2*(
d + e*x)^3 - 471240*b*B*c*d*e*(d + e*x)^3 - 235620*A*c^2*d*e*(d + e*x)^3 + 58905*b^2*B*e^2*(d + e*x)^3 + 11781
0*A*b*c*e^2*(d + e*x)^3 - 255255*B*c^2*d*(d + e*x)^4 + 102102*b*B*c*e*(d + e*x)^4 + 51051*A*c^2*e*(d + e*x)^4
+ 45045*B*c^2*(d + e*x)^5))/(765765*e^6)

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fricas [B]  time = 0.42, size = 494, normalized size = 1.85 \begin {gather*} \frac {2 \, {\left (45045 \, B c^{2} e^{8} x^{8} - 1280 \, B c^{2} d^{8} + 8840 \, A b^{2} d^{5} e^{3} + 2176 \, {\left (2 \, B b c + A c^{2}\right )} d^{7} e - 4080 \, {\left (B b^{2} + 2 \, A b c\right )} d^{6} e^{2} + 3003 \, {\left (35 \, B c^{2} d e^{7} + 17 \, {\left (2 \, B b c + A c^{2}\right )} e^{8}\right )} x^{7} + 231 \, {\left (275 \, B c^{2} d^{2} e^{6} + 527 \, {\left (2 \, B b c + A c^{2}\right )} d e^{7} + 255 \, {\left (B b^{2} + 2 \, A b c\right )} e^{8}\right )} x^{6} + 63 \, {\left (5 \, B c^{2} d^{3} e^{5} + 1105 \, A b^{2} e^{8} + 1207 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{6} + 2295 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{7}\right )} x^{5} - 35 \, {\left (10 \, B c^{2} d^{4} e^{4} - 5083 \, A b^{2} d e^{7} - 17 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{5} - 2703 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{6}\right )} x^{4} + 5 \, {\left (80 \, B c^{2} d^{5} e^{3} + 24973 \, A b^{2} d^{2} e^{6} - 136 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e^{4} + 255 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{6} e^{2} - 1105 \, A b^{2} d^{3} e^{5} - 272 \, {\left (2 \, B b c + A c^{2}\right )} d^{5} e^{3} + 510 \, {\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{7} e - 1105 \, A b^{2} d^{4} e^{4} - 272 \, {\left (2 \, B b c + A c^{2}\right )} d^{6} e^{2} + 510 \, {\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{3}\right )} x\right )} \sqrt {e x + d}}{765765 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(45045*B*c^2*e^8*x^8 - 1280*B*c^2*d^8 + 8840*A*b^2*d^5*e^3 + 2176*(2*B*b*c + A*c^2)*d^7*e - 4080*(B*b
^2 + 2*A*b*c)*d^6*e^2 + 3003*(35*B*c^2*d*e^7 + 17*(2*B*b*c + A*c^2)*e^8)*x^7 + 231*(275*B*c^2*d^2*e^6 + 527*(2
*B*b*c + A*c^2)*d*e^7 + 255*(B*b^2 + 2*A*b*c)*e^8)*x^6 + 63*(5*B*c^2*d^3*e^5 + 1105*A*b^2*e^8 + 1207*(2*B*b*c
+ A*c^2)*d^2*e^6 + 2295*(B*b^2 + 2*A*b*c)*d*e^7)*x^5 - 35*(10*B*c^2*d^4*e^4 - 5083*A*b^2*d*e^7 - 17*(2*B*b*c +
A*c^2)*d^3*e^5 - 2703*(B*b^2 + 2*A*b*c)*d^2*e^6)*x^4 + 5*(80*B*c^2*d^5*e^3 + 24973*A*b^2*d^2*e^6 - 136*(2*B*b
*c + A*c^2)*d^4*e^4 + 255*(B*b^2 + 2*A*b*c)*d^3*e^5)*x^3 - 3*(160*B*c^2*d^6*e^2 - 1105*A*b^2*d^3*e^5 - 272*(2*
B*b*c + A*c^2)*d^5*e^3 + 510*(B*b^2 + 2*A*b*c)*d^4*e^4)*x^2 + 4*(160*B*c^2*d^7*e - 1105*A*b^2*d^4*e^4 - 272*(2
*B*b*c + A*c^2)*d^6*e^2 + 510*(B*b^2 + 2*A*b*c)*d^5*e^3)*x)*sqrt(e*x + d)/e^6

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giac [B]  time = 0.29, size = 2007, normalized size = 7.52

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

2/765765*(51051*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*d^3*e^(-2) + 21879*(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*b^2*d^3*e^(-3) + 437
58*(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)*A*b*c*d^3*e^(-3)
+ 4862*(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*b*c*d^3*e^(-4) + 2431*(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)*A*c^2*d^3*e^(-4) + 1105*(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^3*e^(-5) + 65637*(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)*A*b^2*d^2*e^(-2) + 7293*(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*b^2*d^2*e^(-3) + 14586*(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)*A*b*c*d^2*e^(-3) + 6630*(
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*b*c*d^2*e^(-4) + 3315*(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)*A*c^2*d
^2*e^(-4) + 765*(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^2*e^(-5) + 729
3*(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)*A*b^2*d*e^(-2) + 3315*(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*b^2*d*e^(-3) + 6630*(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)*A*b*c*d*e^(-3) + 1530*(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*b*c*d*e^(-4) + 765*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8
580*(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)*A*c^2*
d*e^(-4) + 357*(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*e^(-5) + 1105*(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)*A*b^2*e^(-2) + 255*(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*b^2*e^(-3) + 510*(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)*A*b*c*e^(-3) + 238*(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 + 1501
5*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*b*c*e^(-4) + 119*(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)*A*c^2*e^(-4) + 7*(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*e^(-5))*e^(-1)

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maple [A]  time = 0.05, size = 341, normalized size = 1.28 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (45045 B \,c^{2} x^{5} e^{5}+51051 A \,c^{2} e^{5} x^{4}+102102 B b c \,e^{5} x^{4}-30030 B \,c^{2} d \,e^{4} x^{4}+117810 A b c \,e^{5} x^{3}-31416 A \,c^{2} d \,e^{4} x^{3}+58905 B \,b^{2} e^{5} x^{3}-62832 B b c d \,e^{4} x^{3}+18480 B \,c^{2} d^{2} e^{3} x^{3}+69615 A \,b^{2} e^{5} x^{2}-64260 A b c d \,e^{4} x^{2}+17136 A \,c^{2} d^{2} e^{3} x^{2}-32130 B \,b^{2} d \,e^{4} x^{2}+34272 B b c \,d^{2} e^{3} x^{2}-10080 B \,c^{2} d^{3} e^{2} x^{2}-30940 A \,b^{2} d \,e^{4} x +28560 A b c \,d^{2} e^{3} x -7616 A \,c^{2} d^{3} e^{2} x +14280 B \,b^{2} d^{2} e^{3} x -15232 B b c \,d^{3} e^{2} x +4480 B \,c^{2} d^{4} e x +8840 A \,b^{2} d^{2} e^{3}-8160 A b c \,d^{3} e^{2}+2176 A \,c^{2} d^{4} e -4080 B \,b^{2} d^{3} e^{2}+4352 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{765765 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)*(c*x^2+b*x)^2,x)

[Out]

2/765765*(e*x+d)^(7/2)*(45045*B*c^2*e^5*x^5+51051*A*c^2*e^5*x^4+102102*B*b*c*e^5*x^4-30030*B*c^2*d*e^4*x^4+117
810*A*b*c*e^5*x^3-31416*A*c^2*d*e^4*x^3+58905*B*b^2*e^5*x^3-62832*B*b*c*d*e^4*x^3+18480*B*c^2*d^2*e^3*x^3+6961
5*A*b^2*e^5*x^2-64260*A*b*c*d*e^4*x^2+17136*A*c^2*d^2*e^3*x^2-32130*B*b^2*d*e^4*x^2+34272*B*b*c*d^2*e^3*x^2-10
080*B*c^2*d^3*e^2*x^2-30940*A*b^2*d*e^4*x+28560*A*b*c*d^2*e^3*x-7616*A*c^2*d^3*e^2*x+14280*B*b^2*d^2*e^3*x-152
32*B*b*c*d^3*e^2*x+4480*B*c^2*d^4*e*x+8840*A*b^2*d^2*e^3-8160*A*b*c*d^3*e^2+2176*A*c^2*d^4*e-4080*B*b^2*d^3*e^
2+4352*B*b*c*d^4*e-1280*B*c^2*d^5)/e^6

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maxima [A]  time = 0.61, size = 291, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (45045 \, {\left (e x + d\right )}^{\frac {17}{2}} B c^{2} - 51051 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 58905 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 69615 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 109395 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{765765 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/765765*(45045*(e*x + d)^(17/2)*B*c^2 - 51051*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(15/2) + 58905*(10*
B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(13/2) - 69615*(10*B*c^2*d^3 - A*b^2*e^
3 - 6*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(11/2) + 85085*(5*B*c^2*d^4 - 2*A*b^2*d*e
^3 - 4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*(e*x + d)^(9/2) - 109395*(B*c^2*d^5 - A*b^2*d^2*
e^3 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)*(e*x + d)^(7/2))/e^6

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mupad [B]  time = 1.62, size = 254, normalized size = 0.95 \begin {gather*} \frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{15\,e^6}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{11\,e^6}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{13\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{9\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2*(A + B*x)*(d + e*x)^(5/2),x)

[Out]

((d + e*x)^(15/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(15*e^6) + ((d + e*x)^(11/2)*(2*A*b^2*e^3 - 20*B*c^2*d
^3 + 12*A*c^2*d^2*e - 6*B*b^2*d*e^2 - 12*A*b*c*d*e^2 + 24*B*b*c*d^2*e))/(11*e^6) + ((d + e*x)^(13/2)*(2*B*b^2*
e^2 + 20*B*c^2*d^2 + 4*A*b*c*e^2 - 8*A*c^2*d*e - 16*B*b*c*d*e))/(13*e^6) + (2*B*c^2*(d + e*x)^(17/2))/(17*e^6)
- (2*d*(b*e - c*d)*(d + e*x)^(9/2)*(2*A*b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/(9*e^6) + (2*d^2*(A*e - B
*d)*(b*e - c*d)^2*(d + e*x)^(7/2))/(7*e^6)

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sympy [B]  time = 52.79, size = 1556, normalized size = 5.83

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(5/2)*(c*x**2+b*x)**2,x)

[Out]

2*A*b**2*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 4*A*b**2*d*(-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**3 + 2*A*b**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**3 + 4*A*b*c*d**2*(-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 + 8*A*b*c*d*(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 + 4*A*b*c*(-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*A*c**2*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**5 + 4*A*c**2*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**5 + 2*A*c**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**5 + 2*B*b**2*d**2*(-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 + 4*B*b**2*d*(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 + 2*B*b**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**4 + 4*B*b*c*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**5 + 8*B*b*c*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**5 + 4*B*b*c*(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 + 2*B*c**2*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**6 + 4*B*c**2*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**6 + 2*B*c**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**6

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