3.1794 \(\int (A+B x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0985036, 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} \[ -\frac{2 b^3 (d+e x)^{13/2} (-4 a B e-A b e+5 b B d)}{13 e^6}+\frac{4 b^2 (d+e x)^{11/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{11 e^6}-\frac{4 b (d+e x)^{9/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6}+\frac{2 (d+e x)^{7/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac{2 (d+e x)^{5/2} (b d-a e)^4 (B d-A e)}{5 e^6}+\frac{2 b^4 B (d+e x)^{15/2}}{15 e^6} \]

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*}

Mathematica [A]  time = 0.164181, size = 183, normalized size = 0.84 \[ \frac{2 (d+e x)^{5/2} \left (-3465 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+8190 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-10010 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+6435 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)-9009 (b d-a e)^4 (B d-A e)+3003 b^4 B (d+e x)^5\right )}{45045 e^6} \]

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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Maple [B]  time = 0.009, size = 469, normalized size = 2.2 \begin{align*}{\frac{6006\,{b}^{4}B{x}^{5}{e}^{5}+6930\,A{b}^{4}{e}^{5}{x}^{4}+27720\,Ba{b}^{3}{e}^{5}{x}^{4}-4620\,B{b}^{4}d{e}^{4}{x}^{4}+32760\,Aa{b}^{3}{e}^{5}{x}^{3}-5040\,A{b}^{4}d{e}^{4}{x}^{3}+49140\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-20160\,Ba{b}^{3}d{e}^{4}{x}^{3}+3360\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+60060\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-21840\,Aa{b}^{3}d{e}^{4}{x}^{2}+3360\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+40040\,B{a}^{3}b{e}^{5}{x}^{2}-32760\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+13440\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-2240\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+51480\,A{a}^{3}b{e}^{5}x-34320\,A{a}^{2}{b}^{2}d{e}^{4}x+12480\,Aa{b}^{3}{d}^{2}{e}^{3}x-1920\,A{b}^{4}{d}^{3}{e}^{2}x+12870\,B{a}^{4}{e}^{5}x-22880\,B{a}^{3}bd{e}^{4}x+18720\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-7680\,Ba{b}^{3}{d}^{3}{e}^{2}x+1280\,B{b}^{4}{d}^{4}ex+18018\,A{a}^{4}{e}^{5}-20592\,Ad{a}^{3}b{e}^{4}+13728\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-4992\,Aa{b}^{3}{d}^{3}{e}^{2}+768\,A{d}^{4}{b}^{4}e-5148\,B{a}^{4}d{e}^{4}+9152\,B{d}^{2}{a}^{3}b{e}^{3}-7488\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+3072\,Ba{b}^{3}{d}^{4}e-512\,{b}^{4}B{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

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 = 0.98393, size = 552, normalized size = 2.53 \begin{align*} \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{align*}

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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Fricas [B]  time = 1.3608, size = 1439, normalized size = 6.6 \begin{align*} \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{align*}

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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Sympy [A]  time = 41.3024, size = 1297, normalized size = 5.95 \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)*(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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Giac [B]  time = 1.20947, size = 1551, normalized size = 7.11 \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)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^4*d*e^(-1) + 12012*(3*(x*e + d)^(5/2) - 5*(x*e + d
)^(3/2)*d)*A*a^3*b*d*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*d*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*d*e^(-2) + 85
8*(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*d
*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*d*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*d*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*d*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 - 3003*(x*e + d)^(3/2)*d^5)*B*b^4*d*e^(-5) + 15015*(x*e + d)^(3/2)*A*a^4*d + 429*(15*(x*e + d)^(7
/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^4*e^(-1) + 1716*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5
/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^3*b*e^(-1) + 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)*B*a^3*b*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)*A*a^2*b^2*e^(-2) + 78*(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^2*b^2*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)*A*a*b^3*e^(-3) + 20*(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 - 3003*(x*e + d)^(3/2)*d^5)*B*a*b^3*e^(-4) + 5*(6
93*(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 - 3003*(x*e + d)^(3/2)*d^5)*A*b^4*e^(-4) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*
d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2
)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*B*b^4*e^(-5) + 3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^4)*e^(-1)