3.1792 \(\int (A+B x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)
Optimal. Leaf size=218 \[ -\frac{2 b^3 (d+e x)^{17/2} (-4 a B e-A b e+5 b B d)}{17 e^6}+\frac{4 b^2 (d+e x)^{15/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{15 e^6}-\frac{4 b (d+e x)^{13/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{13 e^6}+\frac{2 (d+e x)^{11/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{11 e^6}-\frac{2 (d+e x)^{9/2} (b d-a e)^4 (B d-A e)}{9 e^6}+\frac{2 b^4 B (d+e x)^{19/2}}{19 e^6} \]
[Out]
(-2*(b*d - a*e)^4*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^6) + (2*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x
)^(11/2))/(11*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^6) + (4*b^2*(b*d
- a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(15*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x
)^(17/2))/(17*e^6) + (2*b^4*B*(d + e*x)^(19/2))/(19*e^6)
________________________________________________________________________________________
Rubi [A] time = 0.144578, 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} \[ -\frac{2 b^3 (d+e x)^{17/2} (-4 a B e-A b e+5 b B d)}{17 e^6}+\frac{4 b^2 (d+e x)^{15/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{15 e^6}-\frac{4 b (d+e x)^{13/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{13 e^6}+\frac{2 (d+e x)^{11/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{11 e^6}-\frac{2 (d+e x)^{9/2} (b d-a e)^4 (B d-A e)}{9 e^6}+\frac{2 b^4 B (d+e x)^{19/2}}{19 e^6} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^(7/2)*(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)^(9/2))/(9*e^6) + (2*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x
)^(11/2))/(11*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^6) + (4*b^2*(b*d
- a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(15*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x
)^(17/2))/(17*e^6) + (2*b^4*B*(d + e*x)^(19/2))/(19*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)^{7/2} \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)^{7/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (-B d+A e) (d+e x)^{7/2}}{e^5}+\frac{(-b d+a e)^3 (-5 b B d+4 A b e+a B e) (d+e x)^{9/2}}{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)^{11/2}}{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)^{13/2}}{e^5}+\frac{b^3 (-5 b B d+A b e+4 a B e) (d+e x)^{15/2}}{e^5}+\frac{b^4 B (d+e x)^{17/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (b d-a e)^4 (B d-A e) (d+e x)^{9/2}}{9 e^6}+\frac{2 (b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{11/2}}{11 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)^{13/2}}{13 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)^{15/2}}{15 e^6}-\frac{2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{17/2}}{17 e^6}+\frac{2 b^4 B (d+e x)^{19/2}}{19 e^6}\\ \end{align*}
Mathematica [A] time = 0.247509, size = 183, normalized size = 0.84 \[ \frac{2 (d+e x)^{9/2} \left (-122265 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+277134 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-319770 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+188955 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)-230945 (b d-a e)^4 (B d-A e)+109395 b^4 B (d+e x)^5\right )}{2078505 e^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
(2*(d + e*x)^(9/2)*(-230945*(b*d - a*e)^4*(B*d - A*e) + 188955*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d +
e*x) - 319770*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^2 + 277134*b^2*(b*d - a*e)*(5*b*B*d - 2*
A*b*e - 3*a*B*e)*(d + e*x)^3 - 122265*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^4 + 109395*b^4*B*(d + e*x)^5))
/(2078505*e^6)
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Maple [B] time = 0.008, size = 469, normalized size = 2.2 \begin{align*}{\frac{218790\,{b}^{4}B{x}^{5}{e}^{5}+244530\,A{b}^{4}{e}^{5}{x}^{4}+978120\,Ba{b}^{3}{e}^{5}{x}^{4}-128700\,B{b}^{4}d{e}^{4}{x}^{4}+1108536\,Aa{b}^{3}{e}^{5}{x}^{3}-130416\,A{b}^{4}d{e}^{4}{x}^{3}+1662804\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-521664\,Ba{b}^{3}d{e}^{4}{x}^{3}+68640\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+1918620\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-511632\,Aa{b}^{3}d{e}^{4}{x}^{2}+60192\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+1279080\,B{a}^{3}b{e}^{5}{x}^{2}-767448\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+240768\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-31680\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+1511640\,A{a}^{3}b{e}^{5}x-697680\,A{a}^{2}{b}^{2}d{e}^{4}x+186048\,Aa{b}^{3}{d}^{2}{e}^{3}x-21888\,A{b}^{4}{d}^{3}{e}^{2}x+377910\,B{a}^{4}{e}^{5}x-465120\,B{a}^{3}bd{e}^{4}x+279072\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-87552\,Ba{b}^{3}{d}^{3}{e}^{2}x+11520\,B{b}^{4}{d}^{4}ex+461890\,A{a}^{4}{e}^{5}-335920\,Ad{a}^{3}b{e}^{4}+155040\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-41344\,Aa{b}^{3}{d}^{3}{e}^{2}+4864\,A{d}^{4}{b}^{4}e-83980\,B{a}^{4}d{e}^{4}+103360\,B{d}^{2}{a}^{3}b{e}^{3}-62016\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+19456\,Ba{b}^{3}{d}^{4}e-2560\,{b}^{4}B{d}^{5}}{2078505\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
2/2078505*(e*x+d)^(9/2)*(109395*B*b^4*e^5*x^5+122265*A*b^4*e^5*x^4+489060*B*a*b^3*e^5*x^4-64350*B*b^4*d*e^4*x^
4+554268*A*a*b^3*e^5*x^3-65208*A*b^4*d*e^4*x^3+831402*B*a^2*b^2*e^5*x^3-260832*B*a*b^3*d*e^4*x^3+34320*B*b^4*d
^2*e^3*x^3+959310*A*a^2*b^2*e^5*x^2-255816*A*a*b^3*d*e^4*x^2+30096*A*b^4*d^2*e^3*x^2+639540*B*a^3*b*e^5*x^2-38
3724*B*a^2*b^2*d*e^4*x^2+120384*B*a*b^3*d^2*e^3*x^2-15840*B*b^4*d^3*e^2*x^2+755820*A*a^3*b*e^5*x-348840*A*a^2*
b^2*d*e^4*x+93024*A*a*b^3*d^2*e^3*x-10944*A*b^4*d^3*e^2*x+188955*B*a^4*e^5*x-232560*B*a^3*b*d*e^4*x+139536*B*a
^2*b^2*d^2*e^3*x-43776*B*a*b^3*d^3*e^2*x+5760*B*b^4*d^4*e*x+230945*A*a^4*e^5-167960*A*a^3*b*d*e^4+77520*A*a^2*
b^2*d^2*e^3-20672*A*a*b^3*d^3*e^2+2432*A*b^4*d^4*e-41990*B*a^4*d*e^4+51680*B*a^3*b*d^2*e^3-31008*B*a^2*b^2*d^3
*e^2+9728*B*a*b^3*d^4*e-1280*B*b^4*d^5)/e^6
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Maxima [B] time = 1.00282, size = 552, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (109395 \,{\left (e x + d\right )}^{\frac{19}{2}} B b^{4} - 122265 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 277134 \,{\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{15}{2}} - 319770 \,{\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{13}{2}} + 188955 \,{\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{11}{2}} - 230945 \,{\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{9}{2}}\right )}}{2078505 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
[Out]
2/2078505*(109395*(e*x + d)^(19/2)*B*b^4 - 122265*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x + d)^(17/2) + 27713
4*(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)^(15/2) - 319770*(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)^(13/2) + 188955*(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)^(11/2) - 230945*(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)^(9/2))/e^6
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Fricas [B] time = 1.38598, size = 2052, normalized size = 9.41 \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)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
[Out]
2/2078505*(109395*B*b^4*e^9*x^9 - 1280*B*b^4*d^9 + 230945*A*a^4*d^4*e^5 + 2432*(4*B*a*b^3 + A*b^4)*d^8*e - 103
36*(3*B*a^2*b^2 + 2*A*a*b^3)*d^7*e^2 + 25840*(2*B*a^3*b + 3*A*a^2*b^2)*d^6*e^3 - 41990*(B*a^4 + 4*A*a^3*b)*d^5
*e^4 + 6435*(58*B*b^4*d*e^8 + 19*(4*B*a*b^3 + A*b^4)*e^9)*x^8 + 858*(505*B*b^4*d^2*e^7 + 494*(4*B*a*b^3 + A*b^
4)*d*e^8 + 323*(3*B*a^2*b^2 + 2*A*a*b^3)*e^9)*x^7 + 66*(2620*B*b^4*d^3*e^6 + 7619*(4*B*a*b^3 + A*b^4)*d^2*e^7
+ 14858*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^8 + 4845*(2*B*a^3*b + 3*A*a^2*b^2)*e^9)*x^6 + 9*(35*B*b^4*d^4*e^5 + 2302
8*(4*B*a*b^3 + A*b^4)*d^3*e^6 + 133076*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^7 + 129200*(2*B*a^3*b + 3*A*a^2*b^2)*d*
e^8 + 20995*(B*a^4 + 4*A*a^3*b)*e^9)*x^5 - 5*(70*B*b^4*d^5*e^4 - 46189*A*a^4*e^9 - 133*(4*B*a*b^3 + A*b^4)*d^4
*e^5 - 103360*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^6 - 295868*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^7 - 142766*(B*a^4 + 4
*A*a^3*b)*d*e^8)*x^4 + 10*(40*B*b^4*d^6*e^3 + 92378*A*a^4*d*e^8 - 76*(4*B*a*b^3 + A*b^4)*d^5*e^4 + 323*(3*B*a^
2*b^2 + 2*A*a*b^3)*d^4*e^5 + 68476*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^6 + 96577*(B*a^4 + 4*A*a^3*b)*d^2*e^7)*x^3
- 6*(80*B*b^4*d^7*e^2 - 230945*A*a^4*d^2*e^7 - 152*(4*B*a*b^3 + A*b^4)*d^6*e^3 + 646*(3*B*a^2*b^2 + 2*A*a*b^3)
*d^5*e^4 - 1615*(2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^5 - 83980*(B*a^4 + 4*A*a^3*b)*d^3*e^6)*x^2 + (640*B*b^4*d^8*e
+ 923780*A*a^4*d^3*e^6 - 1216*(4*B*a*b^3 + A*b^4)*d^7*e^2 + 5168*(3*B*a^2*b^2 + 2*A*a*b^3)*d^6*e^3 - 12920*(2*
B*a^3*b + 3*A*a^2*b^2)*d^5*e^4 + 20995*(B*a^4 + 4*A*a^3*b)*d^4*e^5)*x)*sqrt(e*x + d)/e^6
________________________________________________________________________________________
Sympy [A] time = 22.5503, size = 2091, normalized size = 9.59 \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)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
Piecewise((2*A*a**4*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**4*d**3*x*sqrt(d + e*x)/9 + 4*A*a**4*d**2*e*x**2*sqrt(d +
e*x)/3 + 8*A*a**4*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a**4*e**3*x**4*sqrt(d + e*x)/9 - 16*A*a**3*b*d**5*sqrt(d
+ e*x)/(99*e**2) + 8*A*a**3*b*d**4*x*sqrt(d + e*x)/(99*e) + 64*A*a**3*b*d**3*x**2*sqrt(d + e*x)/33 + 368*A*a**
3*b*d**2*e*x**3*sqrt(d + e*x)/99 + 272*A*a**3*b*d*e**2*x**4*sqrt(d + e*x)/99 + 8*A*a**3*b*e**3*x**5*sqrt(d + e
*x)/11 + 32*A*a**2*b**2*d**6*sqrt(d + e*x)/(429*e**3) - 16*A*a**2*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 4*A*a
**2*b**2*d**4*x**2*sqrt(d + e*x)/(143*e) + 848*A*a**2*b**2*d**3*x**3*sqrt(d + e*x)/429 + 1832*A*a**2*b**2*d**2
*e*x**4*sqrt(d + e*x)/429 + 480*A*a**2*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 12*A*a**2*b**2*e**3*x**6*sqrt(d +
e*x)/13 - 128*A*a*b**3*d**7*sqrt(d + e*x)/(6435*e**4) + 64*A*a*b**3*d**6*x*sqrt(d + e*x)/(6435*e**3) - 16*A*a*
b**3*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 8*A*a*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 1280*A*a*b**3*d**3*x*
*4*sqrt(d + e*x)/1287 + 1648*A*a*b**3*d**2*e*x**5*sqrt(d + e*x)/715 + 368*A*a*b**3*d*e**2*x**6*sqrt(d + e*x)/1
95 + 8*A*a*b**3*e**3*x**7*sqrt(d + e*x)/15 + 256*A*b**4*d**8*sqrt(d + e*x)/(109395*e**5) - 128*A*b**4*d**7*x*s
qrt(d + e*x)/(109395*e**4) + 32*A*b**4*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 16*A*b**4*d**5*x**3*sqrt(d + e*x
)/(21879*e**2) + 14*A*b**4*d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*A*b**4*d**3*x**5*sqrt(d + e*x)/12155 + 160
4*A*b**4*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*A*b**4*d*e**2*x**7*sqrt(d + e*x)/255 + 2*A*b**4*e**3*x**8*sqrt(d
+ e*x)/17 - 4*B*a**4*d**5*sqrt(d + e*x)/(99*e**2) + 2*B*a**4*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a**4*d**3*x**
2*sqrt(d + e*x)/33 + 92*B*a**4*d**2*e*x**3*sqrt(d + e*x)/99 + 68*B*a**4*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a**
4*e**3*x**5*sqrt(d + e*x)/11 + 64*B*a**3*b*d**6*sqrt(d + e*x)/(1287*e**3) - 32*B*a**3*b*d**5*x*sqrt(d + e*x)/(
1287*e**2) + 8*B*a**3*b*d**4*x**2*sqrt(d + e*x)/(429*e) + 1696*B*a**3*b*d**3*x**3*sqrt(d + e*x)/1287 + 3664*B*
a**3*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 320*B*a**3*b*d*e**2*x**5*sqrt(d + e*x)/143 + 8*B*a**3*b*e**3*x**6*sqrt
(d + e*x)/13 - 64*B*a**2*b**2*d**7*sqrt(d + e*x)/(2145*e**4) + 32*B*a**2*b**2*d**6*x*sqrt(d + e*x)/(2145*e**3)
- 8*B*a**2*b**2*d**5*x**2*sqrt(d + e*x)/(715*e**2) + 4*B*a**2*b**2*d**4*x**3*sqrt(d + e*x)/(429*e) + 640*B*a*
*2*b**2*d**3*x**4*sqrt(d + e*x)/429 + 2472*B*a**2*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 184*B*a**2*b**2*d*e**2*
x**6*sqrt(d + e*x)/65 + 4*B*a**2*b**2*e**3*x**7*sqrt(d + e*x)/5 + 1024*B*a*b**3*d**8*sqrt(d + e*x)/(109395*e**
5) - 512*B*a*b**3*d**7*x*sqrt(d + e*x)/(109395*e**4) + 128*B*a*b**3*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 64*
B*a*b**3*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 56*B*a*b**3*d**4*x**4*sqrt(d + e*x)/(21879*e) + 9696*B*a*b**3*
d**3*x**5*sqrt(d + e*x)/12155 + 6416*B*a*b**3*d**2*e*x**6*sqrt(d + e*x)/3315 + 416*B*a*b**3*d*e**2*x**7*sqrt(d
+ e*x)/255 + 8*B*a*b**3*e**3*x**8*sqrt(d + e*x)/17 - 512*B*b**4*d**9*sqrt(d + e*x)/(415701*e**6) + 256*B*b**4
*d**8*x*sqrt(d + e*x)/(415701*e**5) - 64*B*b**4*d**7*x**2*sqrt(d + e*x)/(138567*e**4) + 160*B*b**4*d**6*x**3*s
qrt(d + e*x)/(415701*e**3) - 140*B*b**4*d**5*x**4*sqrt(d + e*x)/(415701*e**2) + 14*B*b**4*d**4*x**5*sqrt(d + e
*x)/(46189*e) + 2096*B*b**4*d**3*x**6*sqrt(d + e*x)/12597 + 404*B*b**4*d**2*e*x**7*sqrt(d + e*x)/969 + 116*B*b
**4*d*e**2*x**8*sqrt(d + e*x)/323 + 2*B*b**4*e**3*x**9*sqrt(d + e*x)/19, Ne(e, 0)), (d**(7/2)*(A*a**4*x + 2*A*
a**3*b*x**2 + 2*A*a**2*b**2*x**3 + A*a*b**3*x**4 + A*b**4*x**5/5 + B*a**4*x**2/2 + 4*B*a**3*b*x**3/3 + 3*B*a**
2*b**2*x**4/2 + 4*B*a*b**3*x**5/5 + B*b**4*x**6/6), True))
________________________________________________________________________________________
Giac [B] time = 1.37168, size = 3869, normalized size = 17.75 \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)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
[Out]
2/14549535*(969969*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^4*d^3*e^(-1) + 3879876*(3*(x*e + d)^(5/2) - 5
*(x*e + d)^(3/2)*d)*A*a^3*b*d^3*e^(-1) + 554268*(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^3*e^(-2) + 831402*(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^3*e^(-2) + 277134*(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^3*e^(-3) + 184756*(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^3*e^(-3) + 16796*(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^3*e^(-4) + 4199*(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^3*e^(-4) + 1615*(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^3*e^(-5) + 4849845*(
x*e + d)^(3/2)*A*a^4*d^3 + 415701*(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
^2*e^(-1) + 1662804*(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^2*e^(-1) +
554268*(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^2*e^(-2) + 831402*(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^2*e^(-2) + 75582*(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^2*e^(-3) + 50388*(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^2*e^(-3) + 19380*(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^2*e^(-4) + 4845*(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*b^4*d^2*e^(-4) + 969*(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^2*e^(-5) + 2909907*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^4*d^2
+ 138567*(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*d*e^(-1) + 554268*(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*d*e^(-1) + 50388*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2
772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a^3*b*d*e^(-2) + 75582*(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*d*e^
(-2) + 29070*(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^2*b^2*d*e^(-3) + 19380*(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*d*e^(-3) + 3876*(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 + 15
015*(x*e + d)^(3/2)*d^6)*B*a*b^3*d*e^(-4) + 969*(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)*A*b^4*d*e^(-4) + 399*(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*d*e^(-5) + 415701*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/
2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^4*d + 4199*(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^4*e^(-1) + 16796*(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^3
*b*e^(-1) + 6460*(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^3*b*e^(-2) + 9690*(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^2*b^2*e^(-2) + 1938*(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 + 150
15*(x*e + d)^(3/2)*d^6)*B*a^2*b^2*e^(-3) + 1292*(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)*A*a*b^3*e^(-3) + 532*(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*a*b^3*e^(-4) + 133*(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)*A*b^4*e^(-4) + 7*(109395*(x*e + d)^
(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x
*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7
+ 692835*(x*e + d)^(3/2)*d^8)*B*b^4*e^(-5) + 46189*(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^4)*e^(-1)