3.1223 \(\int (A+B x) (d+e x)^{3/2} (b x+c x^2)^2 \, dx\)
Optimal. Leaf size=267 \[ -\frac {2 (d+e x)^{11/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{11 e^6}+\frac {2 (d+e x)^{9/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{9 e^6}-\frac {2 d^2 (d+e x)^{5/2} (B d-A e) (c d-b e)^2}{5 e^6}-\frac {2 c (d+e x)^{13/2} (-A c e-2 b B e+5 B c d)}{13 e^6}+\frac {2 d (d+e x)^{7/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{7 e^6}+\frac {2 B c^2 (d+e x)^{15/2}}{15 e^6} \]
[Out]
-2/5*d^2*(-A*e+B*d)*(-b*e+c*d)^2*(e*x+d)^(5/2)/e^6+2/7*d*(-b*e+c*d)*(B*d*(-3*b*e+5*c*d)-2*A*e*(-b*e+2*c*d))*(e
*x+d)^(7/2)/e^6+2/9*(A*e*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))*(e*x+d)^(9/2)/e^
6-2/11*(2*A*c*e*(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))*(e*x+d)^(11/2)/e^6-2/13*c*(-A*c*e-2*B*b*e+5*B*c
*d)*(e*x+d)^(13/2)/e^6+2/15*B*c^2*(e*x+d)^(15/2)/e^6
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Rubi [A] time = 0.16, 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} \[ -\frac {2 (d+e x)^{11/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{11 e^6}+\frac {2 (d+e x)^{9/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{9 e^6}-\frac {2 d^2 (d+e x)^{5/2} (B d-A e) (c d-b e)^2}{5 e^6}-\frac {2 c (d+e x)^{13/2} (-A c e-2 b B e+5 B c d)}{13 e^6}+\frac {2 d (d+e x)^{7/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{7 e^6}+\frac {2 B c^2 (d+e x)^{15/2}}{15 e^6} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]
[Out]
(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(5/2))/(5*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)^(7/2))/(7*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)^(9/2))/(9*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)^(11/2))/(11*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(13/2))/(13*e^6) + (2*B*c^2*(d + e*x)^(
15/2))/(15*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)^{3/2} \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)^{3/2}}{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)^{5/2}}{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)^{7/2}}{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)^{9/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{11/2}}{e^5}+\frac {B c^2 (d+e x)^{13/2}}{e^5}\right ) \, dx\\ &=-\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{5/2}}{5 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)^{7/2}}{7 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)^{9/2}}{9 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)^{11/2}}{11 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{13/2}}{13 e^6}+\frac {2 B c^2 (d+e x)^{15/2}}{15 e^6}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 272, normalized size = 1.02 \[ \frac {2 (d+e x)^{5/2} \left (A e \left (143 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+78 b c e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+B \left (39 b^2 e^2 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+6 b c e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+c^2 \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )\right )\right )}{45045 e^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^(3/2)*(b*x + c*x^2)^2,x]
[Out]
(2*(d + e*x)^(5/2)*(A*e*(143*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 78*b*c*e*(-16*d^3 + 40*d^2*e*x - 70*d*e
^2*x^2 + 105*e^3*x^3) + 3*c^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)) + B*(3
9*b^2*e^2*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 6*b*c*e*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x
^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + c^2*(-256*d^5 + 640*d^4*e*x - 1120*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*
d*e^4*x^4 + 3003*e^5*x^5))))/(45045*e^6)
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fricas [A] time = 0.63, size = 425, normalized size = 1.59 \[ \frac {2 \, {\left (3003 \, B c^{2} e^{7} x^{7} - 256 \, B c^{2} d^{7} + 1144 \, A b^{2} d^{4} e^{3} + 384 \, {\left (2 \, B b c + A c^{2}\right )} d^{6} e - 624 \, {\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{2} + 231 \, {\left (16 \, B c^{2} d e^{6} + 15 \, {\left (2 \, B b c + A c^{2}\right )} e^{7}\right )} x^{6} + 63 \, {\left (B c^{2} d^{2} e^{5} + 70 \, {\left (2 \, B b c + A c^{2}\right )} d e^{6} + 65 \, {\left (B b^{2} + 2 \, A b c\right )} e^{7}\right )} x^{5} - 35 \, {\left (2 \, B c^{2} d^{3} e^{4} - 143 \, A b^{2} e^{7} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{5} - 156 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{6}\right )} x^{4} + 5 \, {\left (16 \, B c^{2} d^{4} e^{3} + 1430 \, A b^{2} d e^{6} - 24 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{4} + 39 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{5}\right )} x^{3} - 3 \, {\left (32 \, B c^{2} d^{5} e^{2} - 143 \, A b^{2} d^{2} e^{5} - 48 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e^{3} + 78 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{4}\right )} x^{2} + 4 \, {\left (32 \, B c^{2} d^{6} e - 143 \, A b^{2} d^{3} e^{4} - 48 \, {\left (2 \, B b c + A c^{2}\right )} d^{5} e^{2} + 78 \, {\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(3/2)*(c*x^2+b*x)^2,x, algorithm="fricas")
[Out]
2/45045*(3003*B*c^2*e^7*x^7 - 256*B*c^2*d^7 + 1144*A*b^2*d^4*e^3 + 384*(2*B*b*c + A*c^2)*d^6*e - 624*(B*b^2 +
2*A*b*c)*d^5*e^2 + 231*(16*B*c^2*d*e^6 + 15*(2*B*b*c + A*c^2)*e^7)*x^6 + 63*(B*c^2*d^2*e^5 + 70*(2*B*b*c + A*c
^2)*d*e^6 + 65*(B*b^2 + 2*A*b*c)*e^7)*x^5 - 35*(2*B*c^2*d^3*e^4 - 143*A*b^2*e^7 - 3*(2*B*b*c + A*c^2)*d^2*e^5
- 156*(B*b^2 + 2*A*b*c)*d*e^6)*x^4 + 5*(16*B*c^2*d^4*e^3 + 1430*A*b^2*d*e^6 - 24*(2*B*b*c + A*c^2)*d^3*e^4 + 3
9*(B*b^2 + 2*A*b*c)*d^2*e^5)*x^3 - 3*(32*B*c^2*d^5*e^2 - 143*A*b^2*d^2*e^5 - 48*(2*B*b*c + A*c^2)*d^4*e^3 + 78
*(B*b^2 + 2*A*b*c)*d^3*e^4)*x^2 + 4*(32*B*c^2*d^6*e - 143*A*b^2*d^3*e^4 - 48*(2*B*b*c + A*c^2)*d^5*e^2 + 78*(B
*b^2 + 2*A*b*c)*d^4*e^3)*x)*sqrt(e*x + d)/e^6
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giac [B] time = 0.24, size = 1382, normalized size = 5.18 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(3/2)*(c*x^2+b*x)^2,x, algorithm="giac")
[Out]
2/45045*(3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*d^2*e^(-2) + 1287*(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^2*e^(-3) + 2574*(
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^2*e^(-3) + 2
86*(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^2*e^(-4) + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 4
20*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*c^2*d^2*e^(-4) + 65*(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^2*e^(-5) + 2574*(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*e^(-2) + 286*(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*e^(-3) + 572*(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*e^(-3) + 260*(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*e^(-4) + 130*(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*e^(-4) + 30*(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*e^(-5) + 143*(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*e^(-2) +
65*(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^2*e^(-3) + 130*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 9
90*(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*e^
(-3) + 30*(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*e^(-4) + 15*(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*c^2*e^(-4) + 7*(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*e^(-5))*e^(-1)
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maple [A] time = 0.05, size = 341, normalized size = 1.28 \[ \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 B \,c^{2} x^{5} e^{5}+3465 A \,c^{2} e^{5} x^{4}+6930 B b c \,e^{5} x^{4}-2310 B \,c^{2} d \,e^{4} x^{4}+8190 A b c \,e^{5} x^{3}-2520 A \,c^{2} d \,e^{4} x^{3}+4095 B \,b^{2} e^{5} x^{3}-5040 B b c d \,e^{4} x^{3}+1680 B \,c^{2} d^{2} e^{3} x^{3}+5005 A \,b^{2} e^{5} x^{2}-5460 A b c d \,e^{4} x^{2}+1680 A \,c^{2} d^{2} e^{3} x^{2}-2730 B \,b^{2} d \,e^{4} x^{2}+3360 B b c \,d^{2} e^{3} x^{2}-1120 B \,c^{2} d^{3} e^{2} x^{2}-2860 A \,b^{2} d \,e^{4} x +3120 A b c \,d^{2} e^{3} x -960 A \,c^{2} d^{3} e^{2} x +1560 B \,b^{2} d^{2} e^{3} x -1920 B b c \,d^{3} e^{2} x +640 B \,c^{2} d^{4} e x +1144 A \,b^{2} d^{2} e^{3}-1248 A b c \,d^{3} e^{2}+384 A \,c^{2} d^{4} e -624 B \,b^{2} d^{3} e^{2}+768 B b c \,d^{4} e -256 B \,c^{2} d^{5}\right )}{45045 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(3/2)*(c*x^2+b*x)^2,x)
[Out]
2/45045*(e*x+d)^(5/2)*(3003*B*c^2*e^5*x^5+3465*A*c^2*e^5*x^4+6930*B*b*c*e^5*x^4-2310*B*c^2*d*e^4*x^4+8190*A*b*
c*e^5*x^3-2520*A*c^2*d*e^4*x^3+4095*B*b^2*e^5*x^3-5040*B*b*c*d*e^4*x^3+1680*B*c^2*d^2*e^3*x^3+5005*A*b^2*e^5*x
^2-5460*A*b*c*d*e^4*x^2+1680*A*c^2*d^2*e^3*x^2-2730*B*b^2*d*e^4*x^2+3360*B*b*c*d^2*e^3*x^2-1120*B*c^2*d^3*e^2*
x^2-2860*A*b^2*d*e^4*x+3120*A*b*c*d^2*e^3*x-960*A*c^2*d^3*e^2*x+1560*B*b^2*d^2*e^3*x-1920*B*b*c*d^3*e^2*x+640*
B*c^2*d^4*e*x+1144*A*b^2*d^2*e^3-1248*A*b*c*d^3*e^2+384*A*c^2*d^4*e-624*B*b^2*d^3*e^2+768*B*b*c*d^4*e-256*B*c^
2*d^5)/e^6
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maxima [A] time = 0.54, size = 291, normalized size = 1.09 \[ \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} B c^{2} - 3465 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 4095 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 9009 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(3/2)*(c*x^2+b*x)^2,x, algorithm="maxima")
[Out]
2/45045*(3003*(e*x + d)^(15/2)*B*c^2 - 3465*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(13/2) + 4095*(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)^(11/2) - 5005*(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)^(9/2) + 6435*(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)^(7/2) - 9009*(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)^(5/2))/e^6
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mupad [B] time = 0.06, size = 254, normalized size = 0.95 \[ \frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{13\,e^6}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{9\,e^6}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{11\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{7\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x + c*x^2)^2*(A + B*x)*(d + e*x)^(3/2),x)
[Out]
((d + e*x)^(13/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(13*e^6) + ((d + e*x)^(9/2)*(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))/(9*e^6) + ((d + e*x)^(11/2)*(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))/(11*e^6) + (2*B*c^2*(d + e*x)^(15/2))/(15*e^6) -
(2*d*(b*e - c*d)*(d + e*x)^(7/2)*(2*A*b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/(7*e^6) + (2*d^2*(A*e - B*d
)*(b*e - c*d)^2*(d + e*x)^(5/2))/(5*e^6)
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sympy [B] time = 33.46, size = 937, normalized size = 3.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**(3/2)*(c*x**2+b*x)**2,x)
[Out]
2*A*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 2*A*b**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**3 + 4*A*b*c*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**4 + 4*A*b
*c*(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*A*c**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**5 + 2*A*c**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 + 2*B*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**4 + 2*B*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**4 + 4*B*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**5 + 4*B*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**5 + 2*B*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)**(1
3/2)/13)/e**6 + 2*B*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 - 2
0*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
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