3.1793 \(\int (A+B x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=218 \[ -\frac{2 b^3 (d+e x)^{15/2} (-4 a B e-A b e+5 b B d)}{15 e^6}+\frac{4 b^2 (d+e x)^{13/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{13 e^6}-\frac{4 b (d+e x)^{11/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{11 e^6}+\frac{2 (d+e x)^{9/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{9 e^6}-\frac{2 (d+e x)^{7/2} (b d-a e)^4 (B d-A e)}{7 e^6}+\frac{2 b^4 B (d+e x)^{17/2}}{17 e^6} \]

[Out]

(-2*(b*d - a*e)^4*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^6) + (2*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x
)^(9/2))/(9*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^6) + (4*b^2*(b*d -
a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^
(15/2))/(15*e^6) + (2*b^4*B*(d + e*x)^(17/2))/(17*e^6)

________________________________________________________________________________________

Rubi [A]  time = 0.0968407, 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} \[ -\frac{2 b^3 (d+e x)^{15/2} (-4 a B e-A b e+5 b B d)}{15 e^6}+\frac{4 b^2 (d+e x)^{13/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{13 e^6}-\frac{4 b (d+e x)^{11/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{11 e^6}+\frac{2 (d+e x)^{9/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{9 e^6}-\frac{2 (d+e x)^{7/2} (b d-a e)^4 (B d-A e)}{7 e^6}+\frac{2 b^4 B (d+e x)^{17/2}}{17 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(5/2)*(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)^(7/2))/(7*e^6) + (2*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x
)^(9/2))/(9*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^6) + (4*b^2*(b*d -
a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^(13/2))/(13*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^
(15/2))/(15*e^6) + (2*b^4*B*(d + e*x)^(17/2))/(17*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)^{5/2} \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)^{5/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (-B d+A e) (d+e x)^{5/2}}{e^5}+\frac{(-b d+a e)^3 (-5 b B d+4 A b e+a B e) (d+e x)^{7/2}}{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)^{9/2}}{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)^{11/2}}{e^5}+\frac{b^3 (-5 b B d+A b e+4 a B e) (d+e x)^{13/2}}{e^5}+\frac{b^4 B (d+e x)^{15/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (b d-a e)^4 (B d-A e) (d+e x)^{7/2}}{7 e^6}+\frac{2 (b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{9/2}}{9 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)^{11/2}}{11 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)^{13/2}}{13 e^6}-\frac{2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{15/2}}{15 e^6}+\frac{2 b^4 B (d+e x)^{17/2}}{17 e^6}\\ \end{align*}

Mathematica [A]  time = 0.186569, size = 183, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (-51051 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+117810 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-139230 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+85085 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)-109395 (b d-a e)^4 (B d-A e)+45045 b^4 B (d+e x)^5\right )}{765765 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(7/2)*(-109395*(b*d - a*e)^4*(B*d - A*e) + 85085*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e
*x) - 139230*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^2 + 117810*b^2*(b*d - a*e)*(5*b*B*d - 2*A
*b*e - 3*a*B*e)*(d + e*x)^3 - 51051*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^4 + 45045*b^4*B*(d + e*x)^5))/(7
65765*e^6)

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Maple [B]  time = 0.011, size = 469, normalized size = 2.2 \begin{align*}{\frac{90090\,{b}^{4}B{x}^{5}{e}^{5}+102102\,A{b}^{4}{e}^{5}{x}^{4}+408408\,Ba{b}^{3}{e}^{5}{x}^{4}-60060\,B{b}^{4}d{e}^{4}{x}^{4}+471240\,Aa{b}^{3}{e}^{5}{x}^{3}-62832\,A{b}^{4}d{e}^{4}{x}^{3}+706860\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-251328\,Ba{b}^{3}d{e}^{4}{x}^{3}+36960\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+835380\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-257040\,Aa{b}^{3}d{e}^{4}{x}^{2}+34272\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+556920\,B{a}^{3}b{e}^{5}{x}^{2}-385560\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+137088\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-20160\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+680680\,A{a}^{3}b{e}^{5}x-371280\,A{a}^{2}{b}^{2}d{e}^{4}x+114240\,Aa{b}^{3}{d}^{2}{e}^{3}x-15232\,A{b}^{4}{d}^{3}{e}^{2}x+170170\,B{a}^{4}{e}^{5}x-247520\,B{a}^{3}bd{e}^{4}x+171360\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-60928\,Ba{b}^{3}{d}^{3}{e}^{2}x+8960\,B{b}^{4}{d}^{4}ex+218790\,A{a}^{4}{e}^{5}-194480\,Ad{a}^{3}b{e}^{4}+106080\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-32640\,Aa{b}^{3}{d}^{3}{e}^{2}+4352\,A{d}^{4}{b}^{4}e-48620\,B{a}^{4}d{e}^{4}+70720\,B{d}^{2}{a}^{3}b{e}^{3}-48960\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+17408\,Ba{b}^{3}{d}^{4}e-2560\,{b}^{4}B{d}^{5}}{765765\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

2/765765*(e*x+d)^(7/2)*(45045*B*b^4*e^5*x^5+51051*A*b^4*e^5*x^4+204204*B*a*b^3*e^5*x^4-30030*B*b^4*d*e^4*x^4+2
35620*A*a*b^3*e^5*x^3-31416*A*b^4*d*e^4*x^3+353430*B*a^2*b^2*e^5*x^3-125664*B*a*b^3*d*e^4*x^3+18480*B*b^4*d^2*
e^3*x^3+417690*A*a^2*b^2*e^5*x^2-128520*A*a*b^3*d*e^4*x^2+17136*A*b^4*d^2*e^3*x^2+278460*B*a^3*b*e^5*x^2-19278
0*B*a^2*b^2*d*e^4*x^2+68544*B*a*b^3*d^2*e^3*x^2-10080*B*b^4*d^3*e^2*x^2+340340*A*a^3*b*e^5*x-185640*A*a^2*b^2*
d*e^4*x+57120*A*a*b^3*d^2*e^3*x-7616*A*b^4*d^3*e^2*x+85085*B*a^4*e^5*x-123760*B*a^3*b*d*e^4*x+85680*B*a^2*b^2*
d^2*e^3*x-30464*B*a*b^3*d^3*e^2*x+4480*B*b^4*d^4*e*x+109395*A*a^4*e^5-97240*A*a^3*b*d*e^4+53040*A*a^2*b^2*d^2*
e^3-16320*A*a*b^3*d^3*e^2+2176*A*b^4*d^4*e-24310*B*a^4*d*e^4+35360*B*a^3*b*d^2*e^3-24480*B*a^2*b^2*d^3*e^2+870
4*B*a*b^3*d^4*e-1280*B*b^4*d^5)/e^6

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Maxima [B]  time = 0.975611, size = 552, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (45045 \,{\left (e x + d\right )}^{\frac{17}{2}} B b^{4} - 51051 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 117810 \,{\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{13}{2}} - 139230 \,{\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{11}{2}} + 85085 \,{\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{9}{2}} - 109395 \,{\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{7}{2}}\right )}}{765765 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

2/765765*(45045*(e*x + d)^(17/2)*B*b^4 - 51051*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x + d)^(15/2) + 117810*(
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)^(13/2) - 139230*(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)
^(11/2) + 85085*(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)^(9/2) - 109395*(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)^(7/2))/e^6

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Fricas [B]  time = 1.32161, size = 1758, normalized size = 8.06 \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)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^4*e^8*x^8 - 1280*B*b^4*d^8 + 109395*A*a^4*d^3*e^5 + 2176*(4*B*a*b^3 + A*b^4)*d^7*e - 8160*
(3*B*a^2*b^2 + 2*A*a*b^3)*d^6*e^2 + 17680*(2*B*a^3*b + 3*A*a^2*b^2)*d^5*e^3 - 24310*(B*a^4 + 4*A*a^3*b)*d^4*e^
4 + 3003*(35*B*b^4*d*e^7 + 17*(4*B*a*b^3 + A*b^4)*e^8)*x^7 + 231*(275*B*b^4*d^2*e^6 + 527*(4*B*a*b^3 + A*b^4)*
d*e^7 + 510*(3*B*a^2*b^2 + 2*A*a*b^3)*e^8)*x^6 + 63*(5*B*b^4*d^3*e^5 + 1207*(4*B*a*b^3 + A*b^4)*d^2*e^6 + 4590
*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^7 + 2210*(2*B*a^3*b + 3*A*a^2*b^2)*e^8)*x^5 - 35*(10*B*b^4*d^4*e^4 - 17*(4*B*a*
b^3 + A*b^4)*d^3*e^5 - 5406*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^6 - 10166*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^7 - 2431*(
B*a^4 + 4*A*a^3*b)*e^8)*x^4 + 5*(80*B*b^4*d^5*e^3 + 21879*A*a^4*e^8 - 136*(4*B*a*b^3 + A*b^4)*d^4*e^4 + 510*(3
*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^5 + 49946*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^6 + 46189*(B*a^4 + 4*A*a^3*b)*d*e^7)*x
^3 - 3*(160*B*b^4*d^6*e^2 - 109395*A*a^4*d*e^7 - 272*(4*B*a*b^3 + A*b^4)*d^5*e^3 + 1020*(3*B*a^2*b^2 + 2*A*a*b
^3)*d^4*e^4 - 2210*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^5 - 60775*(B*a^4 + 4*A*a^3*b)*d^2*e^6)*x^2 + (640*B*b^4*d^7
*e + 328185*A*a^4*d^2*e^6 - 1088*(4*B*a*b^3 + A*b^4)*d^6*e^2 + 4080*(3*B*a^2*b^2 + 2*A*a*b^3)*d^5*e^3 - 8840*(
2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^4 + 12155*(B*a^4 + 4*A*a^3*b)*d^3*e^5)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 66.8528, size = 2193, normalized size = 10.06 \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)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*A*a**4*d*(-d*(d + e*x)**(3/
2)/3 + (d + e*x)**(5/2)/5)/e + 2*A*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 + 8*A*a**3*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 16*A*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**2 + 8*A*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**2 + 12*A*a**2*b**2*d**2*(d**2*(d + e*x)**(3/2)/3
- 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 24*A*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**3 + 12*A*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**3
+ 8*A*a*b**3*d**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**4 + 16*A*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**4 + 8*A*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**4 + 2*A*b**4*d**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**5 + 4*A*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**5 + 2*A*b**4*(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 - 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**5 +
2*B*a**4*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*B*a**4*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(
d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*B*a**4*(-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**2 + 8*B*a**3*b*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)
**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 16*B*a**3*b*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**3 + 8*B*a**3*b*(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**3 + 12*B*a**2*b**2*d**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**4 + 24*
B*a**2*b**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**4 + 12*B*a**2*b**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**4
+ 8*B*a*b**3*d**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**5 + 16*B*a*b**3*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*
*5 + 8*B*a*b**3*(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 - 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**5 + 2*B*
b**4*d**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**6 + 4*B*b**4*d*(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 - 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 + 2*B*b**4*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)
**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d +
e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**6

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Giac [B]  time = 1.28114, size = 2623, normalized size = 12.03 \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)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2/765765*(51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^4*d^2*e^(-1) + 204204*(3*(x*e + d)^(5/2) - 5*(x*
e + d)^(3/2)*d)*A*a^3*b*d^2*e^(-1) + 29172*(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^2*e^(-2) + 43758*(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^
2*e^(-2) + 14586*(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^2*e^(-3) + 9724*(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^2*e^(-3) + 884*(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^2*e^(-4) + 221*(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^2*e^(-4) + 85*(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^2*e^(-5) + 255255*(x*e + d)^(3/2)*
A*a^4*d^2 + 14586*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^4*d*e^(-1) + 58344*
(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*d*e^(-1) + 19448*(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*d*e^(-2) + 29172*(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*d*e^(-2
) + 2652*(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*d*e^(-3) + 1768*(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*d*e^(-3) + 680*(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*d*e^(-4) + 170*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10
010*(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*d*e^(-4) + 34*(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*d*e^
(-5) + 102102*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^4*d + 2431*(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^4*e^(-1) + 9724*(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^3*b*e^(-1) + 884*(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^3
*b*e^(-2) + 1326*(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^2*b^2*e^(-2) + 510*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 1
0010*(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^2*b^2*e^(-3) + 340*(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)*A*a*b^3*e^(-3) + 68*(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*a*b^3*e^(-4) + 17*(3003*(x*e + d)^(15/2) - 20
790*(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)*A*b^4*e^(-4) + 7*(6435*(x*e + d)^(17/2) - 51051*(x*e +
d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185
*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*B*b^4*e^(-5) + 7293*(15*(x*e +
d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^4)*e^(-1)