3.1843 \(\int (A+B x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=308 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^5 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.18531, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {770, 77} \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

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

Rule 770

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

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

Mathematica [A]  time = 0.271471, size = 163, normalized size = 0.53 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{9/2} \left (-7293 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+25245 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-9945 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+12155 (b d-a e)^3 (B d-A e)+6435 b^3 B (d+e x)^4\right )}{109395 e^5 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*((a + b*x)^2)^(3/2)*(d + e*x)^(9/2)*(12155*(b*d - a*e)^3*(B*d - A*e) - 9945*(b*d - a*e)^2*(4*b*B*d - 3*A*b*
e - a*B*e)*(d + e*x) + 25245*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 7293*b^2*(4*b*B*d - A*b*e -
3*a*B*e)*(d + e*x)^3 + 6435*b^3*B*(d + e*x)^4))/(109395*e^5*(a + b*x)^3)

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Maple [A]  time = 0.009, size = 317, normalized size = 1. \begin{align*}{\frac{12870\,B{x}^{4}{b}^{3}{e}^{4}+14586\,A{x}^{3}{b}^{3}{e}^{4}+43758\,B{x}^{3}a{b}^{2}{e}^{4}-6864\,B{x}^{3}{b}^{3}d{e}^{3}+50490\,A{x}^{2}a{b}^{2}{e}^{4}-6732\,A{x}^{2}{b}^{3}d{e}^{3}+50490\,B{x}^{2}{a}^{2}b{e}^{4}-20196\,B{x}^{2}a{b}^{2}d{e}^{3}+3168\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+59670\,Ax{a}^{2}b{e}^{4}-18360\,Axa{b}^{2}d{e}^{3}+2448\,Ax{b}^{3}{d}^{2}{e}^{2}+19890\,Bx{a}^{3}{e}^{4}-18360\,Bx{a}^{2}bd{e}^{3}+7344\,Bxa{b}^{2}{d}^{2}{e}^{2}-1152\,Bx{b}^{3}{d}^{3}e+24310\,A{a}^{3}{e}^{4}-13260\,Ad{e}^{3}{a}^{2}b+4080\,Aa{b}^{2}{d}^{2}{e}^{2}-544\,A{b}^{3}{d}^{3}e-4420\,Bd{e}^{3}{a}^{3}+4080\,B{a}^{2}b{d}^{2}{e}^{2}-1632\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{109395\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{9}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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)^(3/2),x)

[Out]

2/109395*(e*x+d)^(9/2)*(6435*B*b^3*e^4*x^4+7293*A*b^3*e^4*x^3+21879*B*a*b^2*e^4*x^3-3432*B*b^3*d*e^3*x^3+25245
*A*a*b^2*e^4*x^2-3366*A*b^3*d*e^3*x^2+25245*B*a^2*b*e^4*x^2-10098*B*a*b^2*d*e^3*x^2+1584*B*b^3*d^2*e^2*x^2+298
35*A*a^2*b*e^4*x-9180*A*a*b^2*d*e^3*x+1224*A*b^3*d^2*e^2*x+9945*B*a^3*e^4*x-9180*B*a^2*b*d*e^3*x+3672*B*a*b^2*
d^2*e^2*x-576*B*b^3*d^3*e*x+12155*A*a^3*e^4-6630*A*a^2*b*d*e^3+2040*A*a*b^2*d^2*e^2-272*A*b^3*d^3*e-2210*B*a^3
*d*e^3+2040*B*a^2*b*d^2*e^2-816*B*a*b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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Maxima [B]  time = 1.05221, size = 941, normalized size = 3.06 \begin{align*} \frac{2 \,{\left (429 \, b^{3} e^{7} x^{7} - 16 \, b^{3} d^{7} + 120 \, a b^{2} d^{6} e - 390 \, a^{2} b d^{5} e^{2} + 715 \, a^{3} d^{4} e^{3} + 33 \,{\left (46 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 9 \,{\left (206 \, b^{3} d^{2} e^{5} + 600 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 5 \,{\left (160 \, b^{3} d^{3} e^{4} + 1374 \, a b^{2} d^{2} e^{5} + 1326 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} + 5 \,{\left (b^{3} d^{4} e^{3} + 636 \, a b^{2} d^{3} e^{4} + 1794 \, a^{2} b d^{2} e^{5} + 572 \, a^{3} d e^{6}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{5} e^{2} - 15 \, a b^{2} d^{4} e^{3} - 1560 \, a^{2} b d^{3} e^{4} - 1430 \, a^{3} d^{2} e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{6} e - 60 \, a b^{2} d^{5} e^{2} + 195 \, a^{2} b d^{4} e^{3} + 2860 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d} A}{6435 \, e^{4}} + \frac{2 \,{\left (6435 \, b^{3} e^{8} x^{8} + 128 \, b^{3} d^{8} - 816 \, a b^{2} d^{7} e + 2040 \, a^{2} b d^{6} e^{2} - 2210 \, a^{3} d^{5} e^{3} + 429 \,{\left (52 \, b^{3} d e^{7} + 51 \, a b^{2} e^{8}\right )} x^{7} + 33 \,{\left (802 \, b^{3} d^{2} e^{6} + 2346 \, a b^{2} d e^{7} + 765 \, a^{2} b e^{8}\right )} x^{6} + 9 \,{\left (1212 \, b^{3} d^{3} e^{5} + 10506 \, a b^{2} d^{2} e^{6} + 10200 \, a^{2} b d e^{7} + 1105 \, a^{3} e^{8}\right )} x^{5} + 5 \,{\left (7 \, b^{3} d^{4} e^{4} + 8160 \, a b^{2} d^{3} e^{5} + 23358 \, a^{2} b d^{2} e^{6} + 7514 \, a^{3} d e^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{5} e^{3} - 51 \, a b^{2} d^{4} e^{4} - 10812 \, a^{2} b d^{3} e^{5} - 10166 \, a^{3} d^{2} e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{6} e^{2} - 102 \, a b^{2} d^{5} e^{3} + 255 \, a^{2} b d^{4} e^{4} + 8840 \, a^{3} d^{3} e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{7} e - 408 \, a b^{2} d^{6} e^{2} + 1020 \, a^{2} b d^{5} e^{3} - 1105 \, a^{3} d^{4} e^{4}\right )} x\right )} \sqrt{e x + d} B}{109395 \, e^{5}} \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)^(3/2),x, algorithm="maxima")

[Out]

2/6435*(429*b^3*e^7*x^7 - 16*b^3*d^7 + 120*a*b^2*d^6*e - 390*a^2*b*d^5*e^2 + 715*a^3*d^4*e^3 + 33*(46*b^3*d*e^
6 + 45*a*b^2*e^7)*x^6 + 9*(206*b^3*d^2*e^5 + 600*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 5*(160*b^3*d^3*e^4 + 1374*
a*b^2*d^2*e^5 + 1326*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 + 5*(b^3*d^4*e^3 + 636*a*b^2*d^3*e^4 + 1794*a^2*b*d^2*e^5
+ 572*a^3*d*e^6)*x^3 - 3*(2*b^3*d^5*e^2 - 15*a*b^2*d^4*e^3 - 1560*a^2*b*d^3*e^4 - 1430*a^3*d^2*e^5)*x^2 + (8*b
^3*d^6*e - 60*a*b^2*d^5*e^2 + 195*a^2*b*d^4*e^3 + 2860*a^3*d^3*e^4)*x)*sqrt(e*x + d)*A/e^4 + 2/109395*(6435*b^
3*e^8*x^8 + 128*b^3*d^8 - 816*a*b^2*d^7*e + 2040*a^2*b*d^6*e^2 - 2210*a^3*d^5*e^3 + 429*(52*b^3*d*e^7 + 51*a*b
^2*e^8)*x^7 + 33*(802*b^3*d^2*e^6 + 2346*a*b^2*d*e^7 + 765*a^2*b*e^8)*x^6 + 9*(1212*b^3*d^3*e^5 + 10506*a*b^2*
d^2*e^6 + 10200*a^2*b*d*e^7 + 1105*a^3*e^8)*x^5 + 5*(7*b^3*d^4*e^4 + 8160*a*b^2*d^3*e^5 + 23358*a^2*b*d^2*e^6
+ 7514*a^3*d*e^7)*x^4 - 5*(8*b^3*d^5*e^3 - 51*a*b^2*d^4*e^4 - 10812*a^2*b*d^3*e^5 - 10166*a^3*d^2*e^6)*x^3 + 3
*(16*b^3*d^6*e^2 - 102*a*b^2*d^5*e^3 + 255*a^2*b*d^4*e^4 + 8840*a^3*d^3*e^5)*x^2 - (64*b^3*d^7*e - 408*a*b^2*d
^6*e^2 + 1020*a^2*b*d^5*e^3 - 1105*a^3*d^4*e^4)*x)*sqrt(e*x + d)*B/e^5

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Fricas [B]  time = 1.36451, size = 1431, normalized size = 4.65 \begin{align*} \frac{2 \,{\left (6435 \, B b^{3} e^{8} x^{8} + 128 \, B b^{3} d^{8} + 12155 \, A a^{3} d^{4} e^{4} - 272 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{7} e + 2040 \,{\left (B a^{2} b + A a b^{2}\right )} d^{6} e^{2} - 2210 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5} e^{3} + 429 \,{\left (52 \, B b^{3} d e^{7} + 17 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{8}\right )} x^{7} + 33 \,{\left (802 \, B b^{3} d^{2} e^{6} + 782 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{7} + 765 \,{\left (B a^{2} b + A a b^{2}\right )} e^{8}\right )} x^{6} + 9 \,{\left (1212 \, B b^{3} d^{3} e^{5} + 3502 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{6} + 10200 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{7} + 1105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{8}\right )} x^{5} + 5 \,{\left (7 \, B b^{3} d^{4} e^{4} + 2431 \, A a^{3} e^{8} + 2720 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{5} + 23358 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{6} + 7514 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{7}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{5} e^{3} - 9724 \, A a^{3} d e^{7} - 17 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{4} - 10812 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{5} - 10166 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{6} e^{2} + 24310 \, A a^{3} d^{2} e^{6} - 34 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{3} + 255 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{4} + 8840 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{7} e - 48620 \, A a^{3} d^{3} e^{5} - 136 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e^{2} + 1020 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{3} - 1105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}{109395 \, e^{5}} \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)^(3/2),x, algorithm="fricas")

[Out]

2/109395*(6435*B*b^3*e^8*x^8 + 128*B*b^3*d^8 + 12155*A*a^3*d^4*e^4 - 272*(3*B*a*b^2 + A*b^3)*d^7*e + 2040*(B*a
^2*b + A*a*b^2)*d^6*e^2 - 2210*(B*a^3 + 3*A*a^2*b)*d^5*e^3 + 429*(52*B*b^3*d*e^7 + 17*(3*B*a*b^2 + A*b^3)*e^8)
*x^7 + 33*(802*B*b^3*d^2*e^6 + 782*(3*B*a*b^2 + A*b^3)*d*e^7 + 765*(B*a^2*b + A*a*b^2)*e^8)*x^6 + 9*(1212*B*b^
3*d^3*e^5 + 3502*(3*B*a*b^2 + A*b^3)*d^2*e^6 + 10200*(B*a^2*b + A*a*b^2)*d*e^7 + 1105*(B*a^3 + 3*A*a^2*b)*e^8)
*x^5 + 5*(7*B*b^3*d^4*e^4 + 2431*A*a^3*e^8 + 2720*(3*B*a*b^2 + A*b^3)*d^3*e^5 + 23358*(B*a^2*b + A*a*b^2)*d^2*
e^6 + 7514*(B*a^3 + 3*A*a^2*b)*d*e^7)*x^4 - 5*(8*B*b^3*d^5*e^3 - 9724*A*a^3*d*e^7 - 17*(3*B*a*b^2 + A*b^3)*d^4
*e^4 - 10812*(B*a^2*b + A*a*b^2)*d^3*e^5 - 10166*(B*a^3 + 3*A*a^2*b)*d^2*e^6)*x^3 + 3*(16*B*b^3*d^6*e^2 + 2431
0*A*a^3*d^2*e^6 - 34*(3*B*a*b^2 + A*b^3)*d^5*e^3 + 255*(B*a^2*b + A*a*b^2)*d^4*e^4 + 8840*(B*a^3 + 3*A*a^2*b)*
d^3*e^5)*x^2 - (64*B*b^3*d^7*e - 48620*A*a^3*d^3*e^5 - 136*(3*B*a*b^2 + A*b^3)*d^6*e^2 + 1020*(B*a^2*b + A*a*b
^2)*d^5*e^3 - 1105*(B*a^3 + 3*A*a^2*b)*d^4*e^4)*x)*sqrt(e*x + d)/e^5

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.39498, size = 3051, normalized size = 9.91 \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)^(3/2),x, algorithm="giac")

[Out]

2/765765*(51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^3*d^3*e^(-1)*sgn(b*x + a) + 153153*(3*(x*e + d)^
(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^2*b*d^3*e^(-1)*sgn(b*x + a) + 21879*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*
d + 35*(x*e + d)^(3/2)*d^2)*B*a^2*b*d^3*e^(-2)*sgn(b*x + a) + 21879*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d
+ 35*(x*e + d)^(3/2)*d^2)*A*a*b^2*d^3*e^(-2)*sgn(b*x + a) + 7293*(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*b^2*d^3*e^(-3)*sgn(b*x + a) + 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)*A*b^3*d^3*e^(-3)*sgn(b*x + a) +
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 + 11
55*(x*e + d)^(3/2)*d^4)*B*b^3*d^3*e^(-4)*sgn(b*x + a) + 255255*(x*e + d)^(3/2)*A*a^3*d^3*sgn(b*x + a) + 21879*
(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^3*d^2*e^(-1)*sgn(b*x + a) + 65637*(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*d^2*e^(-1)*sgn(b*x + a) + 21879*(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*d^2*e^(-2
)*sgn(b*x + a) + 21879*(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^2*d^2*e^(-2)*sgn(b*x + a) + 1989*(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^2*d^2*e^(-3)*sgn(b*x + a) + 663*(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^3*d^2*e^(-3)*sgn(b*x + a) + 255*(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^3*d^2*
e^(-4)*sgn(b*x + a) + 153153*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^3*d^2*sgn(b*x + a) + 7293*(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*d*e^(-1)*sgn(b*x
+ a) + 21879*(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*d*e^(-1)*sgn(b*x + a) + 1989*(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*d*e^(-2)*sgn(b*x + a) + 1989*(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^2*d*e^(-2)*sgn(b*x + a) + 765*(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^2*d*e^(-3)*sgn(b
*x + a) + 255*(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^3*d*e^(-3)*sgn(b*x + a) + 51*(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^3*d*e^(-4)*sgn(b*x + a) + 21879*(15*(
x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^3*d*sgn(b*x + a) + 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)*
B*a^3*e^(-1)*sgn(b*x + a) + 663*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 27
72*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*a^2*b*e^(-1)*sgn(b*x + a) + 255*(693*(x*e + d)^(13/2) - 4
095*(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 - 30
03*(x*e + d)^(3/2)*d^5)*B*a^2*b*e^(-2)*sgn(b*x + a) + 255*(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*a
*b^2*e^(-2)*sgn(b*x + a) + 51*(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^2*e^(-3)*sgn(b*x + a) + 17*(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^3*e^(-3)*sgn(b*x + a) + 7*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(1
3/2)*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^3*e^(-4)*sgn(b*x + a) + 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)*A*a^3*sgn(b*x + a))*e^(-1)