3.16.74 \(\int (A+B x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=218 \[ -\frac {2 b^3 (d+e x)^{13/2} (-4 a B e-A b e+5 b B d)}{13 e^6}+\frac {4 b^2 (d+e x)^{11/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{11 e^6}-\frac {4 b (d+e x)^{9/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6}+\frac {2 (d+e x)^{7/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac {2 (d+e x)^{5/2} (b d-a e)^4 (B d-A e)}{5 e^6}+\frac {2 b^4 B (d+e x)^{15/2}}{15 e^6} \]

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Rubi [A]  time = 0.10, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 77} \begin {gather*} -\frac {2 b^3 (d+e x)^{13/2} (-4 a B e-A b e+5 b B d)}{13 e^6}+\frac {4 b^2 (d+e x)^{11/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{11 e^6}-\frac {4 b (d+e x)^{9/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6}+\frac {2 (d+e x)^{7/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac {2 (d+e x)^{5/2} (b d-a e)^4 (B d-A e)}{5 e^6}+\frac {2 b^4 B (d+e x)^{15/2}}{15 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 183, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (-3465 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+8190 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-10010 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+6435 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)-9009 (b d-a e)^4 (B d-A e)+3003 b^4 B (d+e x)^5\right )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(-9009*(b*d - a*e)^4*(B*d - A*e) + 6435*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x)
- 10010*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^2 + 8190*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e -
3*a*B*e)*(d + e*x)^3 - 3465*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^4 + 3003*b^4*B*(d + e*x)^5))/(45045*e^6
)

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IntegrateAlgebraic [B]  time = 0.25, size = 543, normalized size = 2.49 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9009 a^4 A e^5+6435 a^4 B e^4 (d+e x)-9009 a^4 B d e^4+25740 a^3 A b e^4 (d+e x)-36036 a^3 A b d e^4+36036 a^3 b B d^2 e^3-51480 a^3 b B d e^3 (d+e x)+20020 a^3 b B e^3 (d+e x)^2+54054 a^2 A b^2 d^2 e^3-77220 a^2 A b^2 d e^3 (d+e x)+30030 a^2 A b^2 e^3 (d+e x)^2-54054 a^2 b^2 B d^3 e^2+115830 a^2 b^2 B d^2 e^2 (d+e x)-90090 a^2 b^2 B d e^2 (d+e x)^2+24570 a^2 b^2 B e^2 (d+e x)^3-36036 a A b^3 d^3 e^2+77220 a A b^3 d^2 e^2 (d+e x)-60060 a A b^3 d e^2 (d+e x)^2+16380 a A b^3 e^2 (d+e x)^3+36036 a b^3 B d^4 e-102960 a b^3 B d^3 e (d+e x)+120120 a b^3 B d^2 e (d+e x)^2-65520 a b^3 B d e (d+e x)^3+13860 a b^3 B e (d+e x)^4+9009 A b^4 d^4 e-25740 A b^4 d^3 e (d+e x)+30030 A b^4 d^2 e (d+e x)^2-16380 A b^4 d e (d+e x)^3+3465 A b^4 e (d+e x)^4-9009 b^4 B d^5+32175 b^4 B d^4 (d+e x)-50050 b^4 B d^3 (d+e x)^2+40950 b^4 B d^2 (d+e x)^3-17325 b^4 B d (d+e x)^4+3003 b^4 B (d+e x)^5\right )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(-9009*b^4*B*d^5 + 9009*A*b^4*d^4*e + 36036*a*b^3*B*d^4*e - 36036*a*A*b^3*d^3*e^2 - 54054*a
^2*b^2*B*d^3*e^2 + 54054*a^2*A*b^2*d^2*e^3 + 36036*a^3*b*B*d^2*e^3 - 36036*a^3*A*b*d*e^4 - 9009*a^4*B*d*e^4 +
9009*a^4*A*e^5 + 32175*b^4*B*d^4*(d + e*x) - 25740*A*b^4*d^3*e*(d + e*x) - 102960*a*b^3*B*d^3*e*(d + e*x) + 77
220*a*A*b^3*d^2*e^2*(d + e*x) + 115830*a^2*b^2*B*d^2*e^2*(d + e*x) - 77220*a^2*A*b^2*d*e^3*(d + e*x) - 51480*a
^3*b*B*d*e^3*(d + e*x) + 25740*a^3*A*b*e^4*(d + e*x) + 6435*a^4*B*e^4*(d + e*x) - 50050*b^4*B*d^3*(d + e*x)^2
+ 30030*A*b^4*d^2*e*(d + e*x)^2 + 120120*a*b^3*B*d^2*e*(d + e*x)^2 - 60060*a*A*b^3*d*e^2*(d + e*x)^2 - 90090*a
^2*b^2*B*d*e^2*(d + e*x)^2 + 30030*a^2*A*b^2*e^3*(d + e*x)^2 + 20020*a^3*b*B*e^3*(d + e*x)^2 + 40950*b^4*B*d^2
*(d + e*x)^3 - 16380*A*b^4*d*e*(d + e*x)^3 - 65520*a*b^3*B*d*e*(d + e*x)^3 + 16380*a*A*b^3*e^2*(d + e*x)^3 + 2
4570*a^2*b^2*B*e^2*(d + e*x)^3 - 17325*b^4*B*d*(d + e*x)^4 + 3465*A*b^4*e*(d + e*x)^4 + 13860*a*b^3*B*e*(d + e
*x)^4 + 3003*b^4*B*(d + e*x)^5))/(45045*e^6)

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fricas [B]  time = 0.42, size = 649, normalized size = 2.98 \begin {gather*} \frac {2 \, {\left (3003 \, B b^{4} e^{7} x^{7} - 256 \, B b^{4} d^{7} + 9009 \, A a^{4} d^{2} e^{5} + 384 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{6} e - 1248 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{5} e^{2} + 2288 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} e^{3} - 2574 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} e^{4} + 231 \, {\left (16 \, B b^{4} d e^{6} + 15 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{7}\right )} x^{6} + 63 \, {\left (B b^{4} d^{2} e^{5} + 70 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{6} + 130 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{7}\right )} x^{5} - 35 \, {\left (2 \, B b^{4} d^{3} e^{4} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{5} - 312 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{6} - 286 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{7}\right )} x^{4} + 5 \, {\left (16 \, B b^{4} d^{4} e^{3} - 24 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{4} + 78 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{5} + 2860 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{6} + 1287 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{7}\right )} x^{3} - 3 \, {\left (32 \, B b^{4} d^{5} e^{2} - 3003 \, A a^{4} e^{7} - 48 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{3} + 156 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{4} - 286 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{5} - 3432 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{6}\right )} x^{2} + {\left (128 \, B b^{4} d^{6} e + 18018 \, A a^{4} d e^{6} - 192 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e^{2} + 624 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{3} - 1144 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{4} + 1287 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^4*e^7*x^7 - 256*B*b^4*d^7 + 9009*A*a^4*d^2*e^5 + 384*(4*B*a*b^3 + A*b^4)*d^6*e - 1248*(3*B*a
^2*b^2 + 2*A*a*b^3)*d^5*e^2 + 2288*(2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^3 - 2574*(B*a^4 + 4*A*a^3*b)*d^3*e^4 + 231*
(16*B*b^4*d*e^6 + 15*(4*B*a*b^3 + A*b^4)*e^7)*x^6 + 63*(B*b^4*d^2*e^5 + 70*(4*B*a*b^3 + A*b^4)*d*e^6 + 130*(3*
B*a^2*b^2 + 2*A*a*b^3)*e^7)*x^5 - 35*(2*B*b^4*d^3*e^4 - 3*(4*B*a*b^3 + A*b^4)*d^2*e^5 - 312*(3*B*a^2*b^2 + 2*A
*a*b^3)*d*e^6 - 286*(2*B*a^3*b + 3*A*a^2*b^2)*e^7)*x^4 + 5*(16*B*b^4*d^4*e^3 - 24*(4*B*a*b^3 + A*b^4)*d^3*e^4
+ 78*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^5 + 2860*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^6 + 1287*(B*a^4 + 4*A*a^3*b)*e^7)*
x^3 - 3*(32*B*b^4*d^5*e^2 - 3003*A*a^4*e^7 - 48*(4*B*a*b^3 + A*b^4)*d^4*e^3 + 156*(3*B*a^2*b^2 + 2*A*a*b^3)*d^
3*e^4 - 286*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^5 - 3432*(B*a^4 + 4*A*a^3*b)*d*e^6)*x^2 + (128*B*b^4*d^6*e + 18018
*A*a^4*d*e^6 - 192*(4*B*a*b^3 + A*b^4)*d^5*e^2 + 624*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^3 - 1144*(2*B*a^3*b + 3*A
*a^2*b^2)*d^3*e^4 + 1287*(B*a^4 + 4*A*a^3*b)*d^2*e^5)*x)*sqrt(e*x + d)/e^6

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giac [B]  time = 0.31, size = 1937, normalized size = 8.89

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)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^4*d^2*e^(-1) + 60060*((x*e + d)^(3/2) - 3*sqrt(x*e +
d)*d)*A*a^3*b*d^2*e^(-1) + 12012*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^3*b*d^2
*e^(-2) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*b^2*d^2*e^(-2) + 7722*
(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*a^2*b^2*d^2*e^(-3
) + 5148*(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*a*b^3*d^
2*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)*B*a*b^3*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 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*b^4*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*b^4*d^2*e^(-5) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^4*
d*e^(-1) + 24024*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^3*b*d*e^(-1) + 10296*(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*a^3*b*d*e^(-2) + 15
444*(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*a^2*b^2*d*e^(
-2) + 1716*(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 + 3
15*sqrt(x*e + d)*d^4)*B*a^2*b^2*d*e^(-3) + 1144*(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*a*b^3*d*e^(-3) + 520*(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*a*b^3*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 - 13
86*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*b^4*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*b^4*d*e^(-5) + 45045*sqrt(x*e + d)*A*a^4*d^2 + 3003
0*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^4*d + 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*a^4*e^(-1) + 5148*(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*a^3*b*e^(-1) + 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)*B*a^3*b*e^(-2) + 858*(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*a^2*b^2
*e^(-2) + 390*(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*a^2*b^2*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)*A*a*b^3*e^(-3) + 60*(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*a*b^3*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*b^4*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*b^4*e^(-5) + 30
03*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^4)*e^(-1)

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maple [B]  time = 0.05, size = 469, normalized size = 2.15 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 b^{4} B \,x^{5} e^{5}+3465 A \,b^{4} e^{5} x^{4}+13860 B a \,b^{3} e^{5} x^{4}-2310 B \,b^{4} d \,e^{4} x^{4}+16380 A a \,b^{3} e^{5} x^{3}-2520 A \,b^{4} d \,e^{4} x^{3}+24570 B \,a^{2} b^{2} e^{5} x^{3}-10080 B a \,b^{3} d \,e^{4} x^{3}+1680 B \,b^{4} d^{2} e^{3} x^{3}+30030 A \,a^{2} b^{2} e^{5} x^{2}-10920 A a \,b^{3} d \,e^{4} x^{2}+1680 A \,b^{4} d^{2} e^{3} x^{2}+20020 B \,a^{3} b \,e^{5} x^{2}-16380 B \,a^{2} b^{2} d \,e^{4} x^{2}+6720 B a \,b^{3} d^{2} e^{3} x^{2}-1120 B \,b^{4} d^{3} e^{2} x^{2}+25740 A \,a^{3} b \,e^{5} x -17160 A \,a^{2} b^{2} d \,e^{4} x +6240 A a \,b^{3} d^{2} e^{3} x -960 A \,b^{4} d^{3} e^{2} x +6435 B \,a^{4} e^{5} x -11440 B \,a^{3} b d \,e^{4} x +9360 B \,a^{2} b^{2} d^{2} e^{3} x -3840 B a \,b^{3} d^{3} e^{2} x +640 B \,b^{4} d^{4} e x +9009 A \,a^{4} e^{5}-10296 A \,a^{3} b d \,e^{4}+6864 A \,a^{2} b^{2} d^{2} e^{3}-2496 A a \,b^{3} d^{3} e^{2}+384 A \,b^{4} d^{4} e -2574 B \,a^{4} d \,e^{4}+4576 B \,d^{2} a^{3} b \,e^{3}-3744 B \,d^{3} a^{2} b^{2} e^{2}+1536 B a \,b^{3} d^{4} e -256 B \,b^{4} d^{5}\right )}{45045 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2/45045*(e*x+d)^(5/2)*(3003*B*b^4*e^5*x^5+3465*A*b^4*e^5*x^4+13860*B*a*b^3*e^5*x^4-2310*B*b^4*d*e^4*x^4+16380*
A*a*b^3*e^5*x^3-2520*A*b^4*d*e^4*x^3+24570*B*a^2*b^2*e^5*x^3-10080*B*a*b^3*d*e^4*x^3+1680*B*b^4*d^2*e^3*x^3+30
030*A*a^2*b^2*e^5*x^2-10920*A*a*b^3*d*e^4*x^2+1680*A*b^4*d^2*e^3*x^2+20020*B*a^3*b*e^5*x^2-16380*B*a^2*b^2*d*e
^4*x^2+6720*B*a*b^3*d^2*e^3*x^2-1120*B*b^4*d^3*e^2*x^2+25740*A*a^3*b*e^5*x-17160*A*a^2*b^2*d*e^4*x+6240*A*a*b^
3*d^2*e^3*x-960*A*b^4*d^3*e^2*x+6435*B*a^4*e^5*x-11440*B*a^3*b*d*e^4*x+9360*B*a^2*b^2*d^2*e^3*x-3840*B*a*b^3*d
^3*e^2*x+640*B*b^4*d^4*e*x+9009*A*a^4*e^5-10296*A*a^3*b*d*e^4+6864*A*a^2*b^2*d^2*e^3-2496*A*a*b^3*d^3*e^2+384*
A*b^4*d^4*e-2574*B*a^4*d*e^4+4576*B*a^3*b*d^2*e^3-3744*B*a^2*b^2*d^3*e^2+1536*B*a*b^3*d^4*e-256*B*b^4*d^5)/e^6

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maxima [B]  time = 0.50, size = 409, normalized size = 1.88 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} B b^{4} - 3465 \, {\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 8190 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 10010 \, {\left (5 \, B b^{4} d^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 9009 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{45045 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*B*b^4 - 3465*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x + d)^(13/2) + 8190*(5*B*b
^4*d^2 - 2*(4*B*a*b^3 + A*b^4)*d*e + (3*B*a^2*b^2 + 2*A*a*b^3)*e^2)*(e*x + d)^(11/2) - 10010*(5*B*b^4*d^3 - 3*
(4*B*a*b^3 + A*b^4)*d^2*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 - (2*B*a^3*b + 3*A*a^2*b^2)*e^3)*(e*x + d)^(9/2)
+ 6435*(5*B*b^4*d^4 - 4*(4*B*a*b^3 + A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^2 - 4*(2*B*a^3*b + 3*A*
a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^3*b)*e^4)*(e*x + d)^(7/2) - 9009*(B*b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)*
d^4*e + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4)
*(e*x + d)^(5/2))/e^6

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mupad [B]  time = 0.06, size = 197, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{13\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}\,\left (4\,A\,b\,e+B\,a\,e-5\,B\,b\,d\right )}{7\,e^6}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{9\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{11\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

((d + e*x)^(13/2)*(2*A*b^4*e - 10*B*b^4*d + 8*B*a*b^3*e))/(13*e^6) + (2*(a*e - b*d)^3*(d + e*x)^(7/2)*(4*A*b*e
+ B*a*e - 5*B*b*d))/(7*e^6) + (2*B*b^4*(d + e*x)^(15/2))/(15*e^6) + (2*(A*e - B*d)*(a*e - b*d)^4*(d + e*x)^(5
/2))/(5*e^6) + (4*b*(a*e - b*d)^2*(d + e*x)^(9/2)*(3*A*b*e + 2*B*a*e - 5*B*b*d))/(9*e^6) + (4*b^2*(a*e - b*d)*
(d + e*x)^(11/2)*(2*A*b*e + 3*B*a*e - 5*B*b*d))/(11*e^6)

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sympy [A]  time = 45.22, size = 1297, normalized size = 5.95

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)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a**4*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 8*A*a**3*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*A*a**3*b*(d**2*(d
+ e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 12*A*a**2*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 + 12*A*a**2*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 + 8*A*a*b**3*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 + 8*A*a*b**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**4 + 2*A*b**4*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*b**4*(-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*a**4*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 2*B*a**4*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d +
e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 8*B*a**3*b*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 + 8*B*a**3*b*(-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 + 12*B*a**2*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 + 12*B*a**2*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 + 8*B*a*b**3*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 + 8*B*a*b**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**5 + 2*B*b**4*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**6 + 2*B*b**4*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 1
5*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

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