3.18.95 \(\int (A+B x) \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1795]

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

[Out]

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

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Rubi [A]
time = 0.06, 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, 78} \begin {gather*} -\frac {2 b^3 (d+e x)^{11/2} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac {4 b^2 (d+e x)^{9/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac {4 b (d+e x)^{7/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6}+\frac {2 (d+e x)^{5/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{5 e^6}-\frac {2 (d+e x)^{3/2} (b d-a e)^4 (B d-A e)}{3 e^6}+\frac {2 b^4 B (d+e x)^{13/2}}{13 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*Sqrt[d + e*x]*(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)^(3/2))/(3*e^6) + (2*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x
)^(5/2))/(5*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^(7/2))/(7*e^6) + (4*b^2*(b*d - a
*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^(9/2))/(9*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(11/
2))/(11*e^6) + (2*b^4*B*(d + e*x)^(13/2))/(13*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 78

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

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Mathematica [A]
time = 0.29, size = 339, normalized size = 1.56 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (3003 a^4 e^4 (-2 B d+5 A e+3 B e x)+1716 a^3 b e^3 \left (7 A e (-2 d+3 e x)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-858 a^2 b^2 e^2 \left (-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )\right )+52 a b^3 e \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+b^4 \left (13 A e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )-5 B \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(3003*a^4*e^4*(-2*B*d + 5*A*e + 3*B*e*x) + 1716*a^3*b*e^3*(7*A*e*(-2*d + 3*e*x) + B*(8*d^2
- 12*d*e*x + 15*e^2*x^2)) - 858*a^2*b^2*e^2*(-3*A*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + B*(16*d^3 - 24*d^2*e*x +
30*d*e^2*x^2 - 35*e^3*x^3)) + 52*a*b^3*e*(11*A*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + B*(128*
d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) + b^4*(13*A*e*(128*d^4 - 192*d^3*e*x + 240
*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) - 5*B*(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 +
630*d*e^4*x^4 - 693*e^5*x^5))))/(45045*e^6)

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Maple [A]
time = 0.91, size = 352, normalized size = 1.61 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (207) = 414\).
time = 0.28, size = 428, normalized size = 1.96 \begin {gather*} \frac {2}{45045} \, {\left (3465 \, {\left (x e + d\right )}^{\frac {13}{2}} B b^{4} - 4095 \, {\left (5 \, B b^{4} d - 4 \, B a b^{3} e - A b^{4} e\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 10010 \, {\left (5 \, B b^{4} d^{2} + 3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2} - 2 \, {\left (4 \, B a b^{3} e + A b^{4} e\right )} d\right )} {\left (x e + d\right )}^{\frac {9}{2}} - 12870 \, {\left (5 \, B b^{4} d^{3} - 2 \, B a^{3} b e^{3} - 3 \, A a^{2} b^{2} e^{3} - 3 \, {\left (4 \, B a b^{3} e + A b^{4} e\right )} d^{2} + 3 \, {\left (3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, B b^{4} d^{4} + B a^{4} e^{4} + 4 \, A a^{3} b e^{4} - 4 \, {\left (4 \, B a b^{3} e + A b^{4} e\right )} d^{3} + 6 \, {\left (3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} d^{2} - 4 \, {\left (2 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} d\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 15015 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} e + A b^{4} e\right )} d^{4} + 2 \, {\left (3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} d^{3} - 2 \, {\left (2 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} d^{2} + {\left (B a^{4} e^{4} + 4 \, A a^{3} b e^{4}\right )} d\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3465*(x*e + d)^(13/2)*B*b^4 - 4095*(5*B*b^4*d - 4*B*a*b^3*e - A*b^4*e)*(x*e + d)^(11/2) + 10010*(5*B*
b^4*d^2 + 3*B*a^2*b^2*e^2 + 2*A*a*b^3*e^2 - 2*(4*B*a*b^3*e + A*b^4*e)*d)*(x*e + d)^(9/2) - 12870*(5*B*b^4*d^3
- 2*B*a^3*b*e^3 - 3*A*a^2*b^2*e^3 - 3*(4*B*a*b^3*e + A*b^4*e)*d^2 + 3*(3*B*a^2*b^2*e^2 + 2*A*a*b^3*e^2)*d)*(x*
e + d)^(7/2) + 9009*(5*B*b^4*d^4 + B*a^4*e^4 + 4*A*a^3*b*e^4 - 4*(4*B*a*b^3*e + A*b^4*e)*d^3 + 6*(3*B*a^2*b^2*
e^2 + 2*A*a*b^3*e^2)*d^2 - 4*(2*B*a^3*b*e^3 + 3*A*a^2*b^2*e^3)*d)*(x*e + d)^(5/2) - 15015*(B*b^4*d^5 - A*a^4*e
^5 - (4*B*a*b^3*e + A*b^4*e)*d^4 + 2*(3*B*a^2*b^2*e^2 + 2*A*a*b^3*e^2)*d^3 - 2*(2*B*a^3*b*e^3 + 3*A*a^2*b^2*e^
3)*d^2 + (B*a^4*e^4 + 4*A*a^3*b*e^4)*d)*(x*e + d)^(3/2))*e^(-6)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (207) = 414\).
time = 1.67, size = 505, normalized size = 2.32 \begin {gather*} -\frac {2}{45045} \, {\left (1280 \, B b^{4} d^{6} - {\left (3465 \, B b^{4} x^{6} + 15015 \, A a^{4} x + 4095 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{5} + 10010 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{4} + 12870 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{3} + 9009 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{2}\right )} e^{6} - {\left (315 \, B b^{4} d x^{5} + 15015 \, A a^{4} d + 455 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d x^{4} + 1430 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d x^{3} + 2574 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d x^{2} + 3003 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d x\right )} e^{5} + 2 \, {\left (175 \, B b^{4} d^{2} x^{4} + 260 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} x^{3} + 858 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} x^{2} + 1716 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} x + 3003 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2}\right )} e^{4} - 16 \, {\left (25 \, B b^{4} d^{3} x^{3} + 39 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} x^{2} + 143 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} x + 429 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3}\right )} e^{3} + 32 \, {\left (15 \, B b^{4} d^{4} x^{2} + 26 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} x + 143 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4}\right )} e^{2} - 128 \, {\left (5 \, B b^{4} d^{5} x + 13 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5}\right )} e\right )} \sqrt {x e + d} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-2/45045*(1280*B*b^4*d^6 - (3465*B*b^4*x^6 + 15015*A*a^4*x + 4095*(4*B*a*b^3 + A*b^4)*x^5 + 10010*(3*B*a^2*b^2
+ 2*A*a*b^3)*x^4 + 12870*(2*B*a^3*b + 3*A*a^2*b^2)*x^3 + 9009*(B*a^4 + 4*A*a^3*b)*x^2)*e^6 - (315*B*b^4*d*x^5
+ 15015*A*a^4*d + 455*(4*B*a*b^3 + A*b^4)*d*x^4 + 1430*(3*B*a^2*b^2 + 2*A*a*b^3)*d*x^3 + 2574*(2*B*a^3*b + 3*
A*a^2*b^2)*d*x^2 + 3003*(B*a^4 + 4*A*a^3*b)*d*x)*e^5 + 2*(175*B*b^4*d^2*x^4 + 260*(4*B*a*b^3 + A*b^4)*d^2*x^3
+ 858*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*x^2 + 1716*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*x + 3003*(B*a^4 + 4*A*a^3*b)*d^2)
*e^4 - 16*(25*B*b^4*d^3*x^3 + 39*(4*B*a*b^3 + A*b^4)*d^3*x^2 + 143*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*x + 429*(2*B*
a^3*b + 3*A*a^2*b^2)*d^3)*e^3 + 32*(15*B*b^4*d^4*x^2 + 26*(4*B*a*b^3 + A*b^4)*d^4*x + 143*(3*B*a^2*b^2 + 2*A*a
*b^3)*d^4)*e^2 - 128*(5*B*b^4*d^5*x + 13*(4*B*a*b^3 + A*b^4)*d^5)*e)*sqrt(x*e + d)*e^(-6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (221) = 442\).
time = 4.13, size = 517, normalized size = 2.37 \begin {gather*} \frac {2 \left (\frac {B b^{4} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A b^{4} e + 4 B a b^{3} e - 5 B b^{4} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (4 A a b^{3} e^{2} - 4 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 16 B a b^{3} d e + 10 B b^{4} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 A a^{2} b^{2} e^{3} - 12 A a b^{3} d e^{2} + 6 A b^{4} d^{2} e + 4 B a^{3} b e^{3} - 18 B a^{2} b^{2} d e^{2} + 24 B a b^{3} d^{2} e - 10 B b^{4} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (4 A a^{3} b e^{4} - 12 A a^{2} b^{2} d e^{3} + 12 A a b^{3} d^{2} e^{2} - 4 A b^{4} d^{3} e + B a^{4} e^{4} - 8 B a^{3} b d e^{3} + 18 B a^{2} b^{2} d^{2} e^{2} - 16 B a b^{3} d^{3} e + 5 B b^{4} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{4} e^{5} - 4 A a^{3} b d e^{4} + 6 A a^{2} b^{2} d^{2} e^{3} - 4 A a b^{3} d^{3} e^{2} + A b^{4} d^{4} e - B a^{4} d e^{4} + 4 B a^{3} b d^{2} e^{3} - 6 B a^{2} b^{2} d^{3} e^{2} + 4 B a b^{3} d^{4} e - B b^{4} d^{5}\right )}{3 e^{5}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2*(e*x+d)**(1/2),x)

[Out]

2*(B*b**4*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(A*b**4*e + 4*B*a*b**3*e - 5*B*b**4*d)/(11*e**5) + (
d + e*x)**(9/2)*(4*A*a*b**3*e**2 - 4*A*b**4*d*e + 6*B*a**2*b**2*e**2 - 16*B*a*b**3*d*e + 10*B*b**4*d**2)/(9*e*
*5) + (d + e*x)**(7/2)*(6*A*a**2*b**2*e**3 - 12*A*a*b**3*d*e**2 + 6*A*b**4*d**2*e + 4*B*a**3*b*e**3 - 18*B*a**
2*b**2*d*e**2 + 24*B*a*b**3*d**2*e - 10*B*b**4*d**3)/(7*e**5) + (d + e*x)**(5/2)*(4*A*a**3*b*e**4 - 12*A*a**2*
b**2*d*e**3 + 12*A*a*b**3*d**2*e**2 - 4*A*b**4*d**3*e + B*a**4*e**4 - 8*B*a**3*b*d*e**3 + 18*B*a**2*b**2*d**2*
e**2 - 16*B*a*b**3*d**3*e + 5*B*b**4*d**4)/(5*e**5) + (d + e*x)**(3/2)*(A*a**4*e**5 - 4*A*a**3*b*d*e**4 + 6*A*
a**2*b**2*d**2*e**3 - 4*A*a*b**3*d**3*e**2 + A*b**4*d**4*e - B*a**4*d*e**4 + 4*B*a**3*b*d**2*e**3 - 6*B*a**2*b
**2*d**3*e**2 + 4*B*a*b**3*d**4*e - B*b**4*d**5)/(3*e**5))/e

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1144 vs. \(2 (207) = 414\).
time = 1.46, size = 1144, normalized size = 5.25 \begin {gather*} \frac {2}{45045} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{4} d e^{\left (-1\right )} + 60060 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{3} b d e^{\left (-1\right )} + 12012 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{3} b d e^{\left (-2\right )} + 18018 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a^{2} b^{2} d e^{\left (-2\right )} + 7722 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a^{2} b^{2} d e^{\left (-3\right )} + 5148 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A a b^{3} d e^{\left (-3\right )} + 572 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B a b^{3} d e^{\left (-4\right )} + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} A b^{4} d e^{\left (-4\right )} + 65 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} B b^{4} d e^{\left (-5\right )} + 3003 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{4} e^{\left (-1\right )} + 12012 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a^{3} b e^{\left (-1\right )} + 5148 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a^{3} b e^{\left (-2\right )} + 7722 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A a^{2} b^{2} e^{\left (-2\right )} + 858 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B a^{2} b^{2} e^{\left (-3\right )} + 572 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} A a b^{3} e^{\left (-3\right )} + 260 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} B a b^{3} e^{\left (-4\right )} + 65 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} A b^{4} e^{\left (-4\right )} + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} B b^{4} e^{\left (-5\right )} + 45045 \, \sqrt {x e + d} A a^{4} d + 15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{4}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^4*d*e^(-1) + 60060*((x*e + d)^(3/2) - 3*sqrt(x*e + d)
*d)*A*a^3*b*d*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*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*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*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*e^(-3) + 5
72*(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*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*b^4*d*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*e^(-5) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^4*e^(-1) + 12012*(3*
(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^3*b*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)*B*a^3*b*e^(-2) + 7722*(5*(x*e + d)^(7/2) - 2
1*(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*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)*B*a^2*b
^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)*A*a*b^3*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*a*b^3*e^(-4) + 65*(6
3*(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^4*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*s
qrt(x*e + d)*d^6)*B*b^4*e^(-5) + 45045*sqrt(x*e + d)*A*a^4*d + 15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a
^4)*e^(-1)

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Mupad [B]
time = 1.93, size = 197, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{11\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}\,\left (4\,A\,b\,e+B\,a\,e-5\,B\,b\,d\right )}{5\,e^6}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{7\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{9\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

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

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