3.1442 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=344 \[ -\frac{2 c (d+e x)^{5/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8}+\frac{2 c^2 (d+e x)^{9/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{3 e^8}-\frac{2 c^2 (d+e x)^{7/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{7 e^8}-\frac{2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 \sqrt{d+e x}}-\frac{2 c^3 (d+e x)^{11/2} (7 B d-A e)}{11 e^8}+\frac{2 B c^3 (d+e x)^{13/2}}{13 e^8} \]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(e^8*Sqrt[d + e*x]) + (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)
*Sqrt[d + e*x])/e^8 - (2*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(3/2))/
e^8 - (2*c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^(5/2))/(5*e
^8) - (2*c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(7/2))/(7*e^8) + (2*c^2*(7*B*c*d
^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(9/2))/(3*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(11/2))/(11*e^8) + (2*B*c^
3*(d + e*x)^(13/2))/(13*e^8)

________________________________________________________________________________________

Rubi [A]  time = 0.163118, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {772} \[ -\frac{2 c (d+e x)^{5/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8}+\frac{2 c^2 (d+e x)^{9/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{3 e^8}-\frac{2 c^2 (d+e x)^{7/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{7 e^8}-\frac{2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 \sqrt{d+e x}}-\frac{2 c^3 (d+e x)^{11/2} (7 B d-A e)}{11 e^8}+\frac{2 B c^3 (d+e x)^{13/2}}{13 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(e^8*Sqrt[d + e*x]) + (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)
*Sqrt[d + e*x])/e^8 - (2*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(3/2))/
e^8 - (2*c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^(5/2))/(5*e
^8) - (2*c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(7/2))/(7*e^8) + (2*c^2*(7*B*c*d
^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(9/2))/(3*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(11/2))/(11*e^8) + (2*B*c^
3*(d + e*x)^(13/2))/(13*e^8)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^{3/2}}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^7 \sqrt{d+e x}}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right ) \sqrt{d+e x}}{e^7}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right ) (d+e x)^{3/2}}{e^7}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right ) (d+e x)^{5/2}}{e^7}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^{7/2}}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^{9/2}}{e^7}+\frac{B c^3 (d+e x)^{11/2}}{e^7}\right ) \, dx\\ &=\frac{2 (B d-A e) \left (c d^2+a e^2\right )^3}{e^8 \sqrt{d+e x}}+\frac{2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) \sqrt{d+e x}}{e^8}-\frac{2 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{3/2}}{e^8}-\frac{2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^{5/2}}{5 e^8}-\frac{2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{7/2}}{7 e^8}+\frac{2 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{9/2}}{3 e^8}-\frac{2 c^3 (7 B d-A e) (d+e x)^{11/2}}{11 e^8}+\frac{2 B c^3 (d+e x)^{13/2}}{13 e^8}\\ \end{align*}

Mathematica [A]  time = 0.30186, size = 373, normalized size = 1.08 \[ \frac{2 B \left (9009 a^2 c e^4 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )+15015 a^3 e^6 (2 d+e x)+715 a c^2 e^2 \left (-32 d^3 e^2 x^2+16 d^2 e^3 x^3+128 d^4 e x+256 d^5-10 d e^4 x^4+7 e^5 x^5\right )+35 c^3 \left (-256 d^5 e^2 x^2+128 d^4 e^3 x^3-80 d^3 e^4 x^4+56 d^2 e^5 x^5+1024 d^6 e x+2048 d^7-42 d e^6 x^6+33 e^7 x^7\right )\right )-26 A e \left (1155 a^2 c e^4 \left (8 d^2+4 d e x-e^2 x^2\right )+1155 a^3 e^6+99 a c^2 e^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )+5 c^3 \left (-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+512 d^5 e x+1024 d^6+28 d e^5 x^5-21 e^6 x^6\right )\right )}{15015 e^8 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-26*A*e*(1155*a^3*e^6 + 1155*a^2*c*e^4*(8*d^2 + 4*d*e*x - e^2*x^2) + 99*a*c^2*e^2*(128*d^4 + 64*d^3*e*x - 16*
d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) + 5*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40
*d^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6)) + 2*B*(15015*a^3*e^6*(2*d + e*x) + 9009*a^2*c*e^4*(16*d^3 + 8*d^2*e
*x - 2*d*e^2*x^2 + e^3*x^3) + 715*a*c^2*e^2*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^
4*x^4 + 7*e^5*x^5) + 35*c^3*(2048*d^7 + 1024*d^6*e*x - 256*d^5*e^2*x^2 + 128*d^4*e^3*x^3 - 80*d^3*e^4*x^4 + 56
*d^2*e^5*x^5 - 42*d*e^6*x^6 + 33*e^7*x^7)))/(15015*e^8*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 489, normalized size = 1.4 \begin{align*} -{\frac{-2310\,B{c}^{3}{x}^{7}{e}^{7}-2730\,A{c}^{3}{e}^{7}{x}^{6}+2940\,B{c}^{3}d{e}^{6}{x}^{6}+3640\,A{c}^{3}d{e}^{6}{x}^{5}-10010\,Ba{c}^{2}{e}^{7}{x}^{5}-3920\,B{c}^{3}{d}^{2}{e}^{5}{x}^{5}-12870\,Aa{c}^{2}{e}^{7}{x}^{4}-5200\,A{c}^{3}{d}^{2}{e}^{5}{x}^{4}+14300\,Ba{c}^{2}d{e}^{6}{x}^{4}+5600\,B{c}^{3}{d}^{3}{e}^{4}{x}^{4}+20592\,Aa{c}^{2}d{e}^{6}{x}^{3}+8320\,A{c}^{3}{d}^{3}{e}^{4}{x}^{3}-18018\,B{a}^{2}c{e}^{7}{x}^{3}-22880\,Ba{c}^{2}{d}^{2}{e}^{5}{x}^{3}-8960\,B{c}^{3}{d}^{4}{e}^{3}{x}^{3}-30030\,A{a}^{2}c{e}^{7}{x}^{2}-41184\,Aa{c}^{2}{d}^{2}{e}^{5}{x}^{2}-16640\,A{c}^{3}{d}^{4}{e}^{3}{x}^{2}+36036\,B{a}^{2}cd{e}^{6}{x}^{2}+45760\,Ba{c}^{2}{d}^{3}{e}^{4}{x}^{2}+17920\,B{c}^{3}{d}^{5}{e}^{2}{x}^{2}+120120\,A{a}^{2}cd{e}^{6}x+164736\,Aa{c}^{2}{d}^{3}{e}^{4}x+66560\,A{c}^{3}{d}^{5}{e}^{2}x-30030\,B{a}^{3}{e}^{7}x-144144\,B{a}^{2}c{d}^{2}{e}^{5}x-183040\,Ba{c}^{2}{d}^{4}{e}^{3}x-71680\,B{c}^{3}{d}^{6}ex+30030\,A{a}^{3}{e}^{7}+240240\,A{a}^{2}c{d}^{2}{e}^{5}+329472\,Aa{c}^{2}{d}^{4}{e}^{3}+133120\,A{c}^{3}{d}^{6}e-60060\,B{a}^{3}d{e}^{6}-288288\,B{a}^{2}c{d}^{3}{e}^{4}-366080\,Ba{c}^{2}{d}^{5}{e}^{2}-143360\,B{c}^{3}{d}^{7}}{15015\,{e}^{8}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(3/2),x)

[Out]

-2/15015/(e*x+d)^(1/2)*(-1155*B*c^3*e^7*x^7-1365*A*c^3*e^7*x^6+1470*B*c^3*d*e^6*x^6+1820*A*c^3*d*e^6*x^5-5005*
B*a*c^2*e^7*x^5-1960*B*c^3*d^2*e^5*x^5-6435*A*a*c^2*e^7*x^4-2600*A*c^3*d^2*e^5*x^4+7150*B*a*c^2*d*e^6*x^4+2800
*B*c^3*d^3*e^4*x^4+10296*A*a*c^2*d*e^6*x^3+4160*A*c^3*d^3*e^4*x^3-9009*B*a^2*c*e^7*x^3-11440*B*a*c^2*d^2*e^5*x
^3-4480*B*c^3*d^4*e^3*x^3-15015*A*a^2*c*e^7*x^2-20592*A*a*c^2*d^2*e^5*x^2-8320*A*c^3*d^4*e^3*x^2+18018*B*a^2*c
*d*e^6*x^2+22880*B*a*c^2*d^3*e^4*x^2+8960*B*c^3*d^5*e^2*x^2+60060*A*a^2*c*d*e^6*x+82368*A*a*c^2*d^3*e^4*x+3328
0*A*c^3*d^5*e^2*x-15015*B*a^3*e^7*x-72072*B*a^2*c*d^2*e^5*x-91520*B*a*c^2*d^4*e^3*x-35840*B*c^3*d^6*e*x+15015*
A*a^3*e^7+120120*A*a^2*c*d^2*e^5+164736*A*a*c^2*d^4*e^3+66560*A*c^3*d^6*e-30030*B*a^3*d*e^6-144144*B*a^2*c*d^3
*e^4-183040*B*a*c^2*d^5*e^2-71680*B*c^3*d^7)/e^8

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Maxima [A]  time = 1.03071, size = 622, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (\frac{1155 \,{\left (e x + d\right )}^{\frac{13}{2}} B c^{3} - 1365 \,{\left (7 \, B c^{3} d - A c^{3} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 5005 \,{\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 2145 \,{\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 3003 \,{\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 15015 \,{\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15015 \,{\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} \sqrt{e x + d}}{e^{7}} + \frac{15015 \,{\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )}}{\sqrt{e x + d} e^{7}}\right )}}{15015 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/15015*((1155*(e*x + d)^(13/2)*B*c^3 - 1365*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(11/2) + 5005*(7*B*c^3*d^2 - 2*A*
c^3*d*e + B*a*c^2*e^2)*(e*x + d)^(9/2) - 2145*(35*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e^
3)*(e*x + d)^(7/2) + 3003*(35*B*c^3*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e^3 + 3*B*a^2*c*e
^4)*(e*x + d)^(5/2) - 15015*(7*B*c^3*d^5 - 5*A*c^3*d^4*e + 10*B*a*c^2*d^3*e^2 - 6*A*a*c^2*d^2*e^3 + 3*B*a^2*c*
d*e^4 - A*a^2*c*e^5)*(e*x + d)^(3/2) + 15015*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 15*B*a*c^2*d^4*e^2 - 12*A*a*c^2*d^
3*e^3 + 9*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^5 + B*a^3*e^6)*sqrt(e*x + d))/e^7 + 15015*(B*c^3*d^7 - A*c^3*d^6*e +
3*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 - A*a^3*e^7)/(sqr
t(e*x + d)*e^7))/e

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Fricas [A]  time = 1.51395, size = 1131, normalized size = 3.29 \begin{align*} \frac{2 \,{\left (1155 \, B c^{3} e^{7} x^{7} + 71680 \, B c^{3} d^{7} - 66560 \, A c^{3} d^{6} e + 183040 \, B a c^{2} d^{5} e^{2} - 164736 \, A a c^{2} d^{4} e^{3} + 144144 \, B a^{2} c d^{3} e^{4} - 120120 \, A a^{2} c d^{2} e^{5} + 30030 \, B a^{3} d e^{6} - 15015 \, A a^{3} e^{7} - 105 \,{\left (14 \, B c^{3} d e^{6} - 13 \, A c^{3} e^{7}\right )} x^{6} + 35 \,{\left (56 \, B c^{3} d^{2} e^{5} - 52 \, A c^{3} d e^{6} + 143 \, B a c^{2} e^{7}\right )} x^{5} - 5 \,{\left (560 \, B c^{3} d^{3} e^{4} - 520 \, A c^{3} d^{2} e^{5} + 1430 \, B a c^{2} d e^{6} - 1287 \, A a c^{2} e^{7}\right )} x^{4} +{\left (4480 \, B c^{3} d^{4} e^{3} - 4160 \, A c^{3} d^{3} e^{4} + 11440 \, B a c^{2} d^{2} e^{5} - 10296 \, A a c^{2} d e^{6} + 9009 \, B a^{2} c e^{7}\right )} x^{3} -{\left (8960 \, B c^{3} d^{5} e^{2} - 8320 \, A c^{3} d^{4} e^{3} + 22880 \, B a c^{2} d^{3} e^{4} - 20592 \, A a c^{2} d^{2} e^{5} + 18018 \, B a^{2} c d e^{6} - 15015 \, A a^{2} c e^{7}\right )} x^{2} +{\left (35840 \, B c^{3} d^{6} e - 33280 \, A c^{3} d^{5} e^{2} + 91520 \, B a c^{2} d^{4} e^{3} - 82368 \, A a c^{2} d^{3} e^{4} + 72072 \, B a^{2} c d^{2} e^{5} - 60060 \, A a^{2} c d e^{6} + 15015 \, B a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{15015 \,{\left (e^{9} x + d e^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/15015*(1155*B*c^3*e^7*x^7 + 71680*B*c^3*d^7 - 66560*A*c^3*d^6*e + 183040*B*a*c^2*d^5*e^2 - 164736*A*a*c^2*d^
4*e^3 + 144144*B*a^2*c*d^3*e^4 - 120120*A*a^2*c*d^2*e^5 + 30030*B*a^3*d*e^6 - 15015*A*a^3*e^7 - 105*(14*B*c^3*
d*e^6 - 13*A*c^3*e^7)*x^6 + 35*(56*B*c^3*d^2*e^5 - 52*A*c^3*d*e^6 + 143*B*a*c^2*e^7)*x^5 - 5*(560*B*c^3*d^3*e^
4 - 520*A*c^3*d^2*e^5 + 1430*B*a*c^2*d*e^6 - 1287*A*a*c^2*e^7)*x^4 + (4480*B*c^3*d^4*e^3 - 4160*A*c^3*d^3*e^4
+ 11440*B*a*c^2*d^2*e^5 - 10296*A*a*c^2*d*e^6 + 9009*B*a^2*c*e^7)*x^3 - (8960*B*c^3*d^5*e^2 - 8320*A*c^3*d^4*e
^3 + 22880*B*a*c^2*d^3*e^4 - 20592*A*a*c^2*d^2*e^5 + 18018*B*a^2*c*d*e^6 - 15015*A*a^2*c*e^7)*x^2 + (35840*B*c
^3*d^6*e - 33280*A*c^3*d^5*e^2 + 91520*B*a*c^2*d^4*e^3 - 82368*A*a*c^2*d^3*e^4 + 72072*B*a^2*c*d^2*e^5 - 60060
*A*a^2*c*d*e^6 + 15015*B*a^3*e^7)*x)*sqrt(e*x + d)/(e^9*x + d*e^8)

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Sympy [A]  time = 78.034, size = 461, normalized size = 1.34 \begin{align*} \frac{2 B c^{3} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{8}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (2 A c^{3} e - 14 B c^{3} d\right )}{11 e^{8}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (- 12 A c^{3} d e + 6 B a c^{2} e^{2} + 42 B c^{3} d^{2}\right )}{9 e^{8}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (6 A a c^{2} e^{3} + 30 A c^{3} d^{2} e - 30 B a c^{2} d e^{2} - 70 B c^{3} d^{3}\right )}{7 e^{8}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 24 A a c^{2} d e^{3} - 40 A c^{3} d^{3} e + 6 B a^{2} c e^{4} + 60 B a c^{2} d^{2} e^{2} + 70 B c^{3} d^{4}\right )}{5 e^{8}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (6 A a^{2} c e^{5} + 36 A a c^{2} d^{2} e^{3} + 30 A c^{3} d^{4} e - 18 B a^{2} c d e^{4} - 60 B a c^{2} d^{3} e^{2} - 42 B c^{3} d^{5}\right )}{3 e^{8}} + \frac{\sqrt{d + e x} \left (- 12 A a^{2} c d e^{5} - 24 A a c^{2} d^{3} e^{3} - 12 A c^{3} d^{5} e + 2 B a^{3} e^{6} + 18 B a^{2} c d^{2} e^{4} + 30 B a c^{2} d^{4} e^{2} + 14 B c^{3} d^{6}\right )}{e^{8}} + \frac{2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{e^{8} \sqrt{d + e x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(3/2),x)

[Out]

2*B*c**3*(d + e*x)**(13/2)/(13*e**8) + (d + e*x)**(11/2)*(2*A*c**3*e - 14*B*c**3*d)/(11*e**8) + (d + e*x)**(9/
2)*(-12*A*c**3*d*e + 6*B*a*c**2*e**2 + 42*B*c**3*d**2)/(9*e**8) + (d + e*x)**(7/2)*(6*A*a*c**2*e**3 + 30*A*c**
3*d**2*e - 30*B*a*c**2*d*e**2 - 70*B*c**3*d**3)/(7*e**8) + (d + e*x)**(5/2)*(-24*A*a*c**2*d*e**3 - 40*A*c**3*d
**3*e + 6*B*a**2*c*e**4 + 60*B*a*c**2*d**2*e**2 + 70*B*c**3*d**4)/(5*e**8) + (d + e*x)**(3/2)*(6*A*a**2*c*e**5
+ 36*A*a*c**2*d**2*e**3 + 30*A*c**3*d**4*e - 18*B*a**2*c*d*e**4 - 60*B*a*c**2*d**3*e**2 - 42*B*c**3*d**5)/(3*
e**8) + sqrt(d + e*x)*(-12*A*a**2*c*d*e**5 - 24*A*a*c**2*d**3*e**3 - 12*A*c**3*d**5*e + 2*B*a**3*e**6 + 18*B*a
**2*c*d**2*e**4 + 30*B*a*c**2*d**4*e**2 + 14*B*c**3*d**6)/e**8 + 2*(-A*e + B*d)*(a*e**2 + c*d**2)**3/(e**8*sqr
t(d + e*x))

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Giac [A]  time = 1.20201, size = 830, normalized size = 2.41 \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)*(c*x^2+a)^3/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/15015*(1155*(x*e + d)^(13/2)*B*c^3*e^96 - 9555*(x*e + d)^(11/2)*B*c^3*d*e^96 + 35035*(x*e + d)^(9/2)*B*c^3*d
^2*e^96 - 75075*(x*e + d)^(7/2)*B*c^3*d^3*e^96 + 105105*(x*e + d)^(5/2)*B*c^3*d^4*e^96 - 105105*(x*e + d)^(3/2
)*B*c^3*d^5*e^96 + 105105*sqrt(x*e + d)*B*c^3*d^6*e^96 + 1365*(x*e + d)^(11/2)*A*c^3*e^97 - 10010*(x*e + d)^(9
/2)*A*c^3*d*e^97 + 32175*(x*e + d)^(7/2)*A*c^3*d^2*e^97 - 60060*(x*e + d)^(5/2)*A*c^3*d^3*e^97 + 75075*(x*e +
d)^(3/2)*A*c^3*d^4*e^97 - 90090*sqrt(x*e + d)*A*c^3*d^5*e^97 + 5005*(x*e + d)^(9/2)*B*a*c^2*e^98 - 32175*(x*e
+ d)^(7/2)*B*a*c^2*d*e^98 + 90090*(x*e + d)^(5/2)*B*a*c^2*d^2*e^98 - 150150*(x*e + d)^(3/2)*B*a*c^2*d^3*e^98 +
225225*sqrt(x*e + d)*B*a*c^2*d^4*e^98 + 6435*(x*e + d)^(7/2)*A*a*c^2*e^99 - 36036*(x*e + d)^(5/2)*A*a*c^2*d*e
^99 + 90090*(x*e + d)^(3/2)*A*a*c^2*d^2*e^99 - 180180*sqrt(x*e + d)*A*a*c^2*d^3*e^99 + 9009*(x*e + d)^(5/2)*B*
a^2*c*e^100 - 45045*(x*e + d)^(3/2)*B*a^2*c*d*e^100 + 135135*sqrt(x*e + d)*B*a^2*c*d^2*e^100 + 15015*(x*e + d)
^(3/2)*A*a^2*c*e^101 - 90090*sqrt(x*e + d)*A*a^2*c*d*e^101 + 15015*sqrt(x*e + d)*B*a^3*e^102)*e^(-104) + 2*(B*
c^3*d^7 - A*c^3*d^6*e + 3*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a^3*
d*e^6 - A*a^3*e^7)*e^(-8)/sqrt(x*e + d)