3.1795 \(\int (A+B x) \sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

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

[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)

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Rubi [A]  time = 0.0979345, antiderivative size = 218, normalized size of antiderivative = 1., 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} \[ -\frac{2 b^3 (d+e x)^{11/2} (-4 a B e-A b e+5 b B d)}{11 e^6}+\frac{4 b^2 (d+e x)^{9/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac{4 b (d+e x)^{7/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6}+\frac{2 (d+e x)^{5/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{5 e^6}-\frac{2 (d+e x)^{3/2} (b d-a e)^4 (B d-A e)}{3 e^6}+\frac{2 b^4 B (d+e x)^{13/2}}{13 e^6} \]

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 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) \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*}

Mathematica [A]  time = 0.159967, size = 183, normalized size = 0.84 \[ \frac{2 (d+e x)^{3/2} \left (-4095 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+10010 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-12870 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+9009 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)-15015 (b d-a e)^4 (B d-A e)+3465 b^4 B (d+e x)^5\right )}{45045 e^6} \]

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

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Maple [B]  time = 0.009, size = 469, normalized size = 2.2 \begin{align*}{\frac{6930\,{b}^{4}B{x}^{5}{e}^{5}+8190\,A{b}^{4}{e}^{5}{x}^{4}+32760\,Ba{b}^{3}{e}^{5}{x}^{4}-6300\,B{b}^{4}d{e}^{4}{x}^{4}+40040\,Aa{b}^{3}{e}^{5}{x}^{3}-7280\,A{b}^{4}d{e}^{4}{x}^{3}+60060\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-29120\,Ba{b}^{3}d{e}^{4}{x}^{3}+5600\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+77220\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-34320\,Aa{b}^{3}d{e}^{4}{x}^{2}+6240\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+51480\,B{a}^{3}b{e}^{5}{x}^{2}-51480\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+24960\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-4800\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+72072\,A{a}^{3}b{e}^{5}x-61776\,A{a}^{2}{b}^{2}d{e}^{4}x+27456\,Aa{b}^{3}{d}^{2}{e}^{3}x-4992\,A{b}^{4}{d}^{3}{e}^{2}x+18018\,B{a}^{4}{e}^{5}x-41184\,B{a}^{3}bd{e}^{4}x+41184\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-19968\,Ba{b}^{3}{d}^{3}{e}^{2}x+3840\,B{b}^{4}{d}^{4}ex+30030\,A{a}^{4}{e}^{5}-48048\,Ad{a}^{3}b{e}^{4}+41184\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-18304\,Aa{b}^{3}{d}^{3}{e}^{2}+3328\,A{b}^{4}{d}^{4}e-12012\,B{a}^{4}d{e}^{4}+27456\,B{d}^{2}{a}^{3}b{e}^{3}-27456\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+13312\,Ba{b}^{3}{d}^{4}e-2560\,{b}^{4}B{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

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)

[Out]

2/45045*(e*x+d)^(3/2)*(3465*B*b^4*e^5*x^5+4095*A*b^4*e^5*x^4+16380*B*a*b^3*e^5*x^4-3150*B*b^4*d*e^4*x^4+20020*
A*a*b^3*e^5*x^3-3640*A*b^4*d*e^4*x^3+30030*B*a^2*b^2*e^5*x^3-14560*B*a*b^3*d*e^4*x^3+2800*B*b^4*d^2*e^3*x^3+38
610*A*a^2*b^2*e^5*x^2-17160*A*a*b^3*d*e^4*x^2+3120*A*b^4*d^2*e^3*x^2+25740*B*a^3*b*e^5*x^2-25740*B*a^2*b^2*d*e
^4*x^2+12480*B*a*b^3*d^2*e^3*x^2-2400*B*b^4*d^3*e^2*x^2+36036*A*a^3*b*e^5*x-30888*A*a^2*b^2*d*e^4*x+13728*A*a*
b^3*d^2*e^3*x-2496*A*b^4*d^3*e^2*x+9009*B*a^4*e^5*x-20592*B*a^3*b*d*e^4*x+20592*B*a^2*b^2*d^2*e^3*x-9984*B*a*b
^3*d^3*e^2*x+1920*B*b^4*d^4*e*x+15015*A*a^4*e^5-24024*A*a^3*b*d*e^4+20592*A*a^2*b^2*d^2*e^3-9152*A*a*b^3*d^3*e
^2+1664*A*b^4*d^4*e-6006*B*a^4*d*e^4+13728*B*a^3*b*d^2*e^3-13728*B*a^2*b^2*d^3*e^2+6656*B*a*b^3*d^4*e-1280*B*b
^4*d^5)/e^6

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Maxima [B]  time = 0.995635, size = 552, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} B b^{4} - 4095 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 10010 \,{\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{9}{2}} - 12870 \,{\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{7}{2}} + 9009 \,{\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{5}{2}} - 15015 \,{\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{3}{2}}\right )}}{45045 \, e^{6}} \end{align*}

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

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Fricas [B]  time = 1.34668, size = 1187, normalized size = 5.44 \begin{align*} \frac{2 \,{\left (3465 \, B b^{4} e^{6} x^{6} - 1280 \, B b^{4} d^{6} + 15015 \, A a^{4} d e^{5} + 1664 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e - 4576 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{2} + 6864 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{3} - 6006 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{4} + 315 \,{\left (B b^{4} d e^{5} + 13 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{6}\right )} x^{5} - 35 \,{\left (10 \, B b^{4} d^{2} e^{4} - 13 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{5} - 286 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{6}\right )} x^{4} + 10 \,{\left (40 \, B b^{4} d^{3} e^{3} - 52 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{4} + 143 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{5} + 1287 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{6}\right )} x^{3} - 3 \,{\left (160 \, B b^{4} d^{4} e^{2} - 208 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{3} + 572 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{4} - 858 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{5} - 3003 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{6}\right )} x^{2} +{\left (640 \, B b^{4} d^{5} e + 15015 \, A a^{4} e^{6} - 832 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{2} + 2288 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{3} - 3432 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{4} + 3003 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \end{align*}

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*(3465*B*b^4*e^6*x^6 - 1280*B*b^4*d^6 + 15015*A*a^4*d*e^5 + 1664*(4*B*a*b^3 + A*b^4)*d^5*e - 4576*(3*B*
a^2*b^2 + 2*A*a*b^3)*d^4*e^2 + 6864*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^3 - 6006*(B*a^4 + 4*A*a^3*b)*d^2*e^4 + 315
*(B*b^4*d*e^5 + 13*(4*B*a*b^3 + A*b^4)*e^6)*x^5 - 35*(10*B*b^4*d^2*e^4 - 13*(4*B*a*b^3 + A*b^4)*d*e^5 - 286*(3
*B*a^2*b^2 + 2*A*a*b^3)*e^6)*x^4 + 10*(40*B*b^4*d^3*e^3 - 52*(4*B*a*b^3 + A*b^4)*d^2*e^4 + 143*(3*B*a^2*b^2 +
2*A*a*b^3)*d*e^5 + 1287*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*x^3 - 3*(160*B*b^4*d^4*e^2 - 208*(4*B*a*b^3 + A*b^4)*d^
3*e^3 + 572*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^4 - 858*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 - 3003*(B*a^4 + 4*A*a^3*b)
*e^6)*x^2 + (640*B*b^4*d^5*e + 15015*A*a^4*e^6 - 832*(4*B*a*b^3 + A*b^4)*d^4*e^2 + 2288*(3*B*a^2*b^2 + 2*A*a*b
^3)*d^3*e^3 - 3432*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 + 3003*(B*a^4 + 4*A*a^3*b)*d*e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [B]  time = 8.17922, size = 517, normalized size = 2.37 \begin{align*} \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}} \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}} \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}} \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{align*}

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]  time = 1.18279, size = 684, normalized size = 3.14 \begin{align*} \text{result too large to display} \end{align*}

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*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^4*e^(-1) + 12012*(3*(x*e + d)^(5/2) - 5*(x*e + d)^
(3/2)*d)*A*a^3*b*e^(-1) + 1716*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^3*b*e^
(-2) + 2574*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^2*b^2*e^(-2) + 858*(35*(x
*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a^2*b^2*e^(-3) +
572*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a*b^3*e
^(-3) + 52*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^
3 + 1155*(x*e + d)^(3/2)*d^4)*B*a*b^3*e^(-4) + 13*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e +
d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*b^4*e^(-4) + 5*(693*(x*e + d)^(13/2) -
4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3
003*(x*e + d)^(3/2)*d^5)*B*b^4*e^(-5) + 15015*(x*e + d)^(3/2)*A*a^4)*e^(-1)