3.1221 \(\int (A+B x) (d+e x)^{7/2} (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=267 \[ -\frac{2 (d+e x)^{15/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{15 e^6}+\frac{2 (d+e x)^{13/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{13 e^6}-\frac{2 d^2 (d+e x)^{9/2} (B d-A e) (c d-b e)^2}{9 e^6}-\frac{2 c (d+e x)^{17/2} (-A c e-2 b B e+5 B c d)}{17 e^6}+\frac{2 d (d+e x)^{11/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{11 e^6}+\frac{2 B c^2 (d+e x)^{19/2}}{19 e^6} \]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(9/2))/(9*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(11/2))/(11*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*
d*e + 3*b^2*e^2))*(d + e*x)^(13/2))/(13*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2
))*(d + e*x)^(15/2))/(15*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(17/2))/(17*e^6) + (2*B*c^2*(d + e*
x)^(19/2))/(19*e^6)

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Rubi [A]  time = 0.213962, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ -\frac{2 (d+e x)^{15/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{15 e^6}+\frac{2 (d+e x)^{13/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{13 e^6}-\frac{2 d^2 (d+e x)^{9/2} (B d-A e) (c d-b e)^2}{9 e^6}-\frac{2 c (d+e x)^{17/2} (-A c e-2 b B e+5 B c d)}{17 e^6}+\frac{2 d (d+e x)^{11/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{11 e^6}+\frac{2 B c^2 (d+e x)^{19/2}}{19 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(7/2)*(b*x + c*x^2)^2,x]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(9/2))/(9*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(11/2))/(11*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*
d*e + 3*b^2*e^2))*(d + e*x)^(13/2))/(13*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2
))*(d + e*x)^(15/2))/(15*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(17/2))/(17*e^6) + (2*B*c^2*(d + e*
x)^(19/2))/(19*e^6)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac{d^2 (B d-A e) (c d-b e)^2 (d+e x)^{7/2}}{e^5}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{9/2}}{e^5}+\frac{\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{11/2}}{e^5}+\frac{\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{13/2}}{e^5}+\frac{c (-5 B c d+2 b B e+A c e) (d+e x)^{15/2}}{e^5}+\frac{B c^2 (d+e x)^{17/2}}{e^5}\right ) \, dx\\ &=-\frac{2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{9/2}}{9 e^6}+\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{11/2}}{11 e^6}+\frac{2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^6}-\frac{2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{15/2}}{15 e^6}-\frac{2 c (5 B c d-2 b B e-A c e) (d+e x)^{17/2}}{17 e^6}+\frac{2 B c^2 (d+e x)^{19/2}}{19 e^6}\\ \end{align*}

Mathematica [A]  time = 0.243568, size = 273, normalized size = 1.02 \[ \frac{2 (d+e x)^{9/2} \left (19 A e \left (85 b^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+34 b c e \left (72 d^2 e x-16 d^3-198 d e^2 x^2+429 e^3 x^3\right )+c^2 \left (1584 d^2 e^2 x^2-576 d^3 e x+128 d^4-3432 d e^3 x^3+6435 e^4 x^4\right )\right )+B \left (323 b^2 e^2 \left (72 d^2 e x-16 d^3-198 d e^2 x^2+429 e^3 x^3\right )+38 b c e \left (1584 d^2 e^2 x^2-576 d^3 e x+128 d^4-3432 d e^3 x^3+6435 e^4 x^4\right )-5 c^2 \left (3168 d^3 e^2 x^2-6864 d^2 e^3 x^3-1152 d^4 e x+256 d^5+12870 d e^4 x^4-21879 e^5 x^5\right )\right )\right )}{2078505 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(7/2)*(b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(9/2)*(19*A*e*(85*b^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + 34*b*c*e*(-16*d^3 + 72*d^2*e*x - 198*
d*e^2*x^2 + 429*e^3*x^3) + c^2*(128*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4)) + B
*(323*b^2*e^2*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + 38*b*c*e*(128*d^4 - 576*d^3*e*x + 1584*d^
2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4) - 5*c^2*(256*d^5 - 1152*d^4*e*x + 3168*d^3*e^2*x^2 - 6864*d^2*e^3*x
^3 + 12870*d*e^4*x^4 - 21879*e^5*x^5))))/(2078505*e^6)

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Maple [A]  time = 0.009, size = 341, normalized size = 1.3 \begin{align*}{\frac{218790\,B{c}^{2}{x}^{5}{e}^{5}+244530\,A{c}^{2}{e}^{5}{x}^{4}+489060\,Bbc{e}^{5}{x}^{4}-128700\,B{c}^{2}d{e}^{4}{x}^{4}+554268\,Abc{e}^{5}{x}^{3}-130416\,A{c}^{2}d{e}^{4}{x}^{3}+277134\,B{b}^{2}{e}^{5}{x}^{3}-260832\,Bbcd{e}^{4}{x}^{3}+68640\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+319770\,A{b}^{2}{e}^{5}{x}^{2}-255816\,Abcd{e}^{4}{x}^{2}+60192\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-127908\,B{b}^{2}d{e}^{4}{x}^{2}+120384\,Bbc{d}^{2}{e}^{3}{x}^{2}-31680\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-116280\,A{b}^{2}d{e}^{4}x+93024\,Abc{d}^{2}{e}^{3}x-21888\,A{c}^{2}{d}^{3}{e}^{2}x+46512\,B{b}^{2}{d}^{2}{e}^{3}x-43776\,Bbc{d}^{3}{e}^{2}x+11520\,B{c}^{2}{d}^{4}ex+25840\,A{b}^{2}{d}^{2}{e}^{3}-20672\,Abc{d}^{3}{e}^{2}+4864\,A{c}^{2}{d}^{4}e-10336\,B{b}^{2}{d}^{3}{e}^{2}+9728\,Bbc{d}^{4}e-2560\,B{c}^{2}{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)*(c*x^2+b*x)^2,x)

[Out]

2/2078505*(e*x+d)^(9/2)*(109395*B*c^2*e^5*x^5+122265*A*c^2*e^5*x^4+244530*B*b*c*e^5*x^4-64350*B*c^2*d*e^4*x^4+
277134*A*b*c*e^5*x^3-65208*A*c^2*d*e^4*x^3+138567*B*b^2*e^5*x^3-130416*B*b*c*d*e^4*x^3+34320*B*c^2*d^2*e^3*x^3
+159885*A*b^2*e^5*x^2-127908*A*b*c*d*e^4*x^2+30096*A*c^2*d^2*e^3*x^2-63954*B*b^2*d*e^4*x^2+60192*B*b*c*d^2*e^3
*x^2-15840*B*c^2*d^3*e^2*x^2-58140*A*b^2*d*e^4*x+46512*A*b*c*d^2*e^3*x-10944*A*c^2*d^3*e^2*x+23256*B*b^2*d^2*e
^3*x-21888*B*b*c*d^3*e^2*x+5760*B*c^2*d^4*e*x+12920*A*b^2*d^2*e^3-10336*A*b*c*d^3*e^2+2432*A*c^2*d^4*e-5168*B*
b^2*d^3*e^2+4864*B*b*c*d^4*e-1280*B*c^2*d^5)/e^6

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Maxima [A]  time = 1.04615, size = 393, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (109395 \,{\left (e x + d\right )}^{\frac{19}{2}} B c^{2} - 122265 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 138567 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 159885 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 188955 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 230945 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\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)*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

2/2078505*(109395*(e*x + d)^(19/2)*B*c^2 - 122265*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(17/2) + 138567*
(10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(15/2) - 159885*(10*B*c^2*d^3 - A*b
^2*e^3 - 6*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(13/2) + 188955*(5*B*c^2*d^4 - 2*A*b
^2*d*e^3 - 4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*(e*x + d)^(11/2) - 230945*(B*c^2*d^5 - A*b
^2*d^2*e^3 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)*(e*x + d)^(9/2))/e^6

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Fricas [B]  time = 1.8073, size = 1320, normalized size = 4.94 \begin{align*} \frac{2 \,{\left (109395 \, B c^{2} e^{9} x^{9} - 1280 \, B c^{2} d^{9} + 12920 \, A b^{2} d^{6} e^{3} + 2432 \,{\left (2 \, B b c + A c^{2}\right )} d^{8} e - 5168 \,{\left (B b^{2} + 2 \, A b c\right )} d^{7} e^{2} + 6435 \,{\left (58 \, B c^{2} d e^{8} + 19 \,{\left (2 \, B b c + A c^{2}\right )} e^{9}\right )} x^{8} + 429 \,{\left (1010 \, B c^{2} d^{2} e^{7} + 988 \,{\left (2 \, B b c + A c^{2}\right )} d e^{8} + 323 \,{\left (B b^{2} + 2 \, A b c\right )} e^{9}\right )} x^{7} + 33 \,{\left (5240 \, B c^{2} d^{3} e^{6} + 4845 \, A b^{2} e^{9} + 15238 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{7} + 14858 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{8}\right )} x^{6} + 9 \,{\left (35 \, B c^{2} d^{4} e^{5} + 64600 \, A b^{2} d e^{8} + 23028 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{6} + 66538 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{7}\right )} x^{5} - 5 \,{\left (70 \, B c^{2} d^{5} e^{4} - 147934 \, A b^{2} d^{2} e^{7} - 133 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e^{5} - 51680 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{6}\right )} x^{4} + 5 \,{\left (80 \, B c^{2} d^{6} e^{3} + 68476 \, A b^{2} d^{3} e^{6} - 152 \,{\left (2 \, B b c + A c^{2}\right )} d^{5} e^{4} + 323 \,{\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{5}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{7} e^{2} - 1615 \, A b^{2} d^{4} e^{5} - 304 \,{\left (2 \, B b c + A c^{2}\right )} d^{6} e^{3} + 646 \,{\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{4}\right )} x^{2} + 4 \,{\left (160 \, B c^{2} d^{8} e - 1615 \, A b^{2} d^{5} e^{4} - 304 \,{\left (2 \, B b c + A c^{2}\right )} d^{7} e^{2} + 646 \,{\left (B b^{2} + 2 \, A b c\right )} d^{6} e^{3}\right )} x\right )} \sqrt{e x + d}}{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)*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

2/2078505*(109395*B*c^2*e^9*x^9 - 1280*B*c^2*d^9 + 12920*A*b^2*d^6*e^3 + 2432*(2*B*b*c + A*c^2)*d^8*e - 5168*(
B*b^2 + 2*A*b*c)*d^7*e^2 + 6435*(58*B*c^2*d*e^8 + 19*(2*B*b*c + A*c^2)*e^9)*x^8 + 429*(1010*B*c^2*d^2*e^7 + 98
8*(2*B*b*c + A*c^2)*d*e^8 + 323*(B*b^2 + 2*A*b*c)*e^9)*x^7 + 33*(5240*B*c^2*d^3*e^6 + 4845*A*b^2*e^9 + 15238*(
2*B*b*c + A*c^2)*d^2*e^7 + 14858*(B*b^2 + 2*A*b*c)*d*e^8)*x^6 + 9*(35*B*c^2*d^4*e^5 + 64600*A*b^2*d*e^8 + 2302
8*(2*B*b*c + A*c^2)*d^3*e^6 + 66538*(B*b^2 + 2*A*b*c)*d^2*e^7)*x^5 - 5*(70*B*c^2*d^5*e^4 - 147934*A*b^2*d^2*e^
7 - 133*(2*B*b*c + A*c^2)*d^4*e^5 - 51680*(B*b^2 + 2*A*b*c)*d^3*e^6)*x^4 + 5*(80*B*c^2*d^6*e^3 + 68476*A*b^2*d
^3*e^6 - 152*(2*B*b*c + A*c^2)*d^5*e^4 + 323*(B*b^2 + 2*A*b*c)*d^4*e^5)*x^3 - 3*(160*B*c^2*d^7*e^2 - 1615*A*b^
2*d^4*e^5 - 304*(2*B*b*c + A*c^2)*d^6*e^3 + 646*(B*b^2 + 2*A*b*c)*d^5*e^4)*x^2 + 4*(160*B*c^2*d^8*e - 1615*A*b
^2*d^5*e^4 - 304*(2*B*b*c + A*c^2)*d^7*e^2 + 646*(B*b^2 + 2*A*b*c)*d^6*e^3)*x)*sqrt(e*x + d)/e^6

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Sympy [A]  time = 23.4658, size = 1352, normalized size = 5.06 \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)**(7/2)*(c*x**2+b*x)**2,x)

[Out]

Piecewise((16*A*b**2*d**6*sqrt(d + e*x)/(1287*e**3) - 8*A*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*A*b**2*d**
4*x**2*sqrt(d + e*x)/(429*e) + 424*A*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 916*A*b**2*d**2*e*x**4*sqrt(d + e*x)/
1287 + 80*A*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 2*A*b**2*e**3*x**6*sqrt(d + e*x)/13 - 64*A*b*c*d**7*sqrt(d +
e*x)/(6435*e**4) + 32*A*b*c*d**6*x*sqrt(d + e*x)/(6435*e**3) - 8*A*b*c*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 4
*A*b*c*d**4*x**3*sqrt(d + e*x)/(1287*e) + 640*A*b*c*d**3*x**4*sqrt(d + e*x)/1287 + 824*A*b*c*d**2*e*x**5*sqrt(
d + e*x)/715 + 184*A*b*c*d*e**2*x**6*sqrt(d + e*x)/195 + 4*A*b*c*e**3*x**7*sqrt(d + e*x)/15 + 256*A*c**2*d**8*
sqrt(d + e*x)/(109395*e**5) - 128*A*c**2*d**7*x*sqrt(d + e*x)/(109395*e**4) + 32*A*c**2*d**6*x**2*sqrt(d + e*x
)/(36465*e**3) - 16*A*c**2*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*A*c**2*d**4*x**4*sqrt(d + e*x)/(21879*e)
+ 2424*A*c**2*d**3*x**5*sqrt(d + e*x)/12155 + 1604*A*c**2*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*A*c**2*d*e**2*x
**7*sqrt(d + e*x)/255 + 2*A*c**2*e**3*x**8*sqrt(d + e*x)/17 - 32*B*b**2*d**7*sqrt(d + e*x)/(6435*e**4) + 16*B*
b**2*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*B*b**2*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 2*B*b**2*d**4*x**3*sqrt
(d + e*x)/(1287*e) + 320*B*b**2*d**3*x**4*sqrt(d + e*x)/1287 + 412*B*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B
*b**2*d*e**2*x**6*sqrt(d + e*x)/195 + 2*B*b**2*e**3*x**7*sqrt(d + e*x)/15 + 512*B*b*c*d**8*sqrt(d + e*x)/(1093
95*e**5) - 256*B*b*c*d**7*x*sqrt(d + e*x)/(109395*e**4) + 64*B*b*c*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 32*B
*b*c*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 28*B*b*c*d**4*x**4*sqrt(d + e*x)/(21879*e) + 4848*B*b*c*d**3*x**5*
sqrt(d + e*x)/12155 + 3208*B*b*c*d**2*e*x**6*sqrt(d + e*x)/3315 + 208*B*b*c*d*e**2*x**7*sqrt(d + e*x)/255 + 4*
B*b*c*e**3*x**8*sqrt(d + e*x)/17 - 512*B*c**2*d**9*sqrt(d + e*x)/(415701*e**6) + 256*B*c**2*d**8*x*sqrt(d + e*
x)/(415701*e**5) - 64*B*c**2*d**7*x**2*sqrt(d + e*x)/(138567*e**4) + 160*B*c**2*d**6*x**3*sqrt(d + e*x)/(41570
1*e**3) - 140*B*c**2*d**5*x**4*sqrt(d + e*x)/(415701*e**2) + 14*B*c**2*d**4*x**5*sqrt(d + e*x)/(46189*e) + 209
6*B*c**2*d**3*x**6*sqrt(d + e*x)/12597 + 404*B*c**2*d**2*e*x**7*sqrt(d + e*x)/969 + 116*B*c**2*d*e**2*x**8*sqr
t(d + e*x)/323 + 2*B*c**2*e**3*x**9*sqrt(d + e*x)/19, Ne(e, 0)), (d**(7/2)*(A*b**2*x**3/3 + A*b*c*x**4/2 + A*c
**2*x**5/5 + B*b**2*x**4/4 + 2*B*b*c*x**5/5 + B*c**2*x**6/6), True))

________________________________________________________________________________________

Giac [B]  time = 1.42348, size = 2709, normalized size = 10.15 \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)*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

2/14549535*(138567*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*b^2*d^3*e^(-2) + 461
89*(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*b^2*d^3*
e^(-3) + 92378*(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*b*c*d^3*e^(-3) + 8398*(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*b*c*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*c^2*d^3*e^(-4) + 161
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 + 90
09*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*c^2*d^3*e^(-5) + 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)*A*b^2*d^2*e^(-2) + 12597*(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*
b^2*d^2*e^(-3) + 25194*(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*c*d^2*e^(-3) + 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)*B*b*c*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 - 128
70*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*A*c^2*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*c^2*d^2*e^(-5) + 12597*(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^2*d*e^(-2) + 4845*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12
870*(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^2*d*e^(-3) + 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*b*c*d*e^(-3) + 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 + 15015*(x*e + d)^(3/2)*d^6)*B*b*c*d*e^(-4) + 969*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61
425*(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*c^2*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*c^2*d*e^(-5) + 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)*A*b^2*e^(-2) + 323*(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^2*e^(-3) + 646*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 1
00100*(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*c*e^(-3) + 266*(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 - 3
6465*(x*e + d)^(3/2)*d^7)*B*b*c*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 + 15315
3*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*A*c^2*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 - 129
32920*(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*c^2*e^(-5))*e^(-1)