3.1444 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=346 \[ -\frac{2 c \sqrt{d+e x} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8}+\frac{6 c^2 (d+e x)^{5/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8}-\frac{2 c^2 (d+e x)^{3/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{3 e^8}+\frac{6 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^{5/2}}-\frac{2 c^3 (d+e x)^{7/2} (7 B d-A e)}{7 e^8}+\frac{2 B c^3 (d+e x)^{9/2}}{9 e^8} \]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(5*e^8*(d + e*x)^(5/2)) - (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*
e^2))/(3*e^8*(d + e*x)^(3/2)) + (6*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))/(e^8*S
qrt[d + e*x]) - (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))*Sqrt[d + e*
x])/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)^(3/2))/(3*e^8) + (6*c^2*(7*B
*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(5/2))/(5*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(7/2))/(7*e^8) + (2*B*
c^3*(d + e*x)^(9/2))/(9*e^8)

________________________________________________________________________________________

Rubi [A]  time = 0.161639, antiderivative size = 346, 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} \[ -\frac{2 c \sqrt{d+e x} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8}+\frac{6 c^2 (d+e x)^{5/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8}-\frac{2 c^2 (d+e x)^{3/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{3 e^8}+\frac{6 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^{5/2}}-\frac{2 c^3 (d+e x)^{7/2} (7 B d-A e)}{7 e^8}+\frac{2 B c^3 (d+e x)^{9/2}}{9 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(5*e^8*(d + e*x)^(5/2)) - (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*
e^2))/(3*e^8*(d + e*x)^(3/2)) + (6*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))/(e^8*S
qrt[d + e*x]) - (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))*Sqrt[d + e*
x])/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)^(3/2))/(3*e^8) + (6*c^2*(7*B
*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(5/2))/(5*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(7/2))/(7*e^8) + (2*B*
c^3*(d + e*x)^(9/2))/(9*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)^{7/2}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^{7/2}}+\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 (d+e x)^{5/2}}+\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 )}{e^7 (d+e x)^{3/2}}-\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 )}{e^7 \sqrt{d+e x}}+\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 ) \sqrt{d+e x}}{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)^{3/2}}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^{5/2}}{e^7}+\frac{B c^3 (d+e x)^{7/2}}{e^7}\right ) \, dx\\ &=\frac{2 (B d-A e) \left (c d^2+a e^2\right )^3}{5 e^8 (d+e x)^{5/2}}-\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 )}{3 e^8 (d+e x)^{3/2}}+\frac{6 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 )}{e^8 \sqrt{d+e x}}-\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 ) \sqrt{d+e x}}{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)^{3/2}}{3 e^8}+\frac{6 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{5/2}}{5 e^8}-\frac{2 c^3 (7 B d-A e) (d+e x)^{7/2}}{7 e^8}+\frac{2 B c^3 (d+e x)^{9/2}}{9 e^8}\\ \end{align*}

Mathematica [A]  time = 0.288793, size = 373, normalized size = 1.08 \[ \frac{2 \left (7 B \left (27 a^2 c e^4 \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )-3 a^3 e^6 (2 d+5 e x)+9 a c^2 e^2 \left (480 d^3 e^2 x^2+80 d^2 e^3 x^3+640 d^4 e x+256 d^5-10 d e^4 x^4+3 e^5 x^5\right )+c^3 \left (3840 d^5 e^2 x^2+640 d^4 e^3 x^3-80 d^3 e^4 x^4+24 d^2 e^5 x^5+5120 d^6 e x+2048 d^7-10 d e^6 x^6+5 e^7 x^7\right )\right )-9 A e \left (7 a^2 c e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+7 a^3 e^6+7 a c^2 e^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )+c^3 \left (1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+2560 d^5 e x+1024 d^6+12 d e^5 x^5-5 e^6 x^6\right )\right )\right )}{315 e^8 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-9*A*e*(7*a^3*e^6 + 7*a^2*c*e^4*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 7*a*c^2*e^2*(128*d^4 + 320*d^3*e*x + 240
*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) + c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 -
40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6)) + 7*B*(-3*a^3*e^6*(2*d + 5*e*x) + 27*a^2*c*e^4*(16*d^3 + 40*d^2*e
*x + 30*d*e^2*x^2 + 5*e^3*x^3) + 9*a*c^2*e^2*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*
e^4*x^4 + 3*e^5*x^5) + c^3*(2048*d^7 + 5120*d^6*e*x + 3840*d^5*e^2*x^2 + 640*d^4*e^3*x^3 - 80*d^3*e^4*x^4 + 24
*d^2*e^5*x^5 - 10*d*e^6*x^6 + 5*e^7*x^7))))/(315*e^8*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 489, normalized size = 1.4 \begin{align*} -{\frac{-70\,B{c}^{3}{x}^{7}{e}^{7}-90\,A{c}^{3}{e}^{7}{x}^{6}+140\,B{c}^{3}d{e}^{6}{x}^{6}+216\,A{c}^{3}d{e}^{6}{x}^{5}-378\,Ba{c}^{2}{e}^{7}{x}^{5}-336\,B{c}^{3}{d}^{2}{e}^{5}{x}^{5}-630\,Aa{c}^{2}{e}^{7}{x}^{4}-720\,A{c}^{3}{d}^{2}{e}^{5}{x}^{4}+1260\,Ba{c}^{2}d{e}^{6}{x}^{4}+1120\,B{c}^{3}{d}^{3}{e}^{4}{x}^{4}+5040\,Aa{c}^{2}d{e}^{6}{x}^{3}+5760\,A{c}^{3}{d}^{3}{e}^{4}{x}^{3}-1890\,B{a}^{2}c{e}^{7}{x}^{3}-10080\,Ba{c}^{2}{d}^{2}{e}^{5}{x}^{3}-8960\,B{c}^{3}{d}^{4}{e}^{3}{x}^{3}+1890\,A{a}^{2}c{e}^{7}{x}^{2}+30240\,Aa{c}^{2}{d}^{2}{e}^{5}{x}^{2}+34560\,A{c}^{3}{d}^{4}{e}^{3}{x}^{2}-11340\,B{a}^{2}cd{e}^{6}{x}^{2}-60480\,Ba{c}^{2}{d}^{3}{e}^{4}{x}^{2}-53760\,B{c}^{3}{d}^{5}{e}^{2}{x}^{2}+2520\,A{a}^{2}cd{e}^{6}x+40320\,Aa{c}^{2}{d}^{3}{e}^{4}x+46080\,A{c}^{3}{d}^{5}{e}^{2}x+210\,B{a}^{3}{e}^{7}x-15120\,B{a}^{2}c{d}^{2}{e}^{5}x-80640\,Ba{c}^{2}{d}^{4}{e}^{3}x-71680\,B{c}^{3}{d}^{6}ex+126\,A{a}^{3}{e}^{7}+1008\,A{a}^{2}c{d}^{2}{e}^{5}+16128\,Aa{c}^{2}{d}^{4}{e}^{3}+18432\,A{c}^{3}{d}^{6}e+84\,B{a}^{3}d{e}^{6}-6048\,B{a}^{2}c{d}^{3}{e}^{4}-32256\,Ba{c}^{2}{d}^{5}{e}^{2}-28672\,B{c}^{3}{d}^{7}}{315\,{e}^{8}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \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)^(7/2),x)

[Out]

-2/315/(e*x+d)^(5/2)*(-35*B*c^3*e^7*x^7-45*A*c^3*e^7*x^6+70*B*c^3*d*e^6*x^6+108*A*c^3*d*e^6*x^5-189*B*a*c^2*e^
7*x^5-168*B*c^3*d^2*e^5*x^5-315*A*a*c^2*e^7*x^4-360*A*c^3*d^2*e^5*x^4+630*B*a*c^2*d*e^6*x^4+560*B*c^3*d^3*e^4*
x^4+2520*A*a*c^2*d*e^6*x^3+2880*A*c^3*d^3*e^4*x^3-945*B*a^2*c*e^7*x^3-5040*B*a*c^2*d^2*e^5*x^3-4480*B*c^3*d^4*
e^3*x^3+945*A*a^2*c*e^7*x^2+15120*A*a*c^2*d^2*e^5*x^2+17280*A*c^3*d^4*e^3*x^2-5670*B*a^2*c*d*e^6*x^2-30240*B*a
*c^2*d^3*e^4*x^2-26880*B*c^3*d^5*e^2*x^2+1260*A*a^2*c*d*e^6*x+20160*A*a*c^2*d^3*e^4*x+23040*A*c^3*d^5*e^2*x+10
5*B*a^3*e^7*x-7560*B*a^2*c*d^2*e^5*x-40320*B*a*c^2*d^4*e^3*x-35840*B*c^3*d^6*e*x+63*A*a^3*e^7+504*A*a^2*c*d^2*
e^5+8064*A*a*c^2*d^4*e^3+9216*A*c^3*d^6*e+42*B*a^3*d*e^6-3024*B*a^2*c*d^3*e^4-16128*B*a*c^2*d^5*e^2-14336*B*c^
3*d^7)/e^8

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Maxima [A]  time = 1.00836, size = 622, normalized size = 1.8 \begin{align*} \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} B c^{3} - 45 \,{\left (7 \, B c^{3} d - A c^{3} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\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{5}{2}} - 105 \,{\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{3}{2}} + 315 \,{\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 )} \sqrt{e x + d}}{e^{7}} + \frac{21 \,{\left (3 \, B c^{3} d^{7} - 3 \, A c^{3} d^{6} e + 9 \, B a c^{2} d^{5} e^{2} - 9 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} - 9 \, A a^{2} c d^{2} e^{5} + 3 \, B a^{3} d e^{6} - 3 \, A a^{3} e^{7} + 45 \,{\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 )}^{2} - 5 \,{\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 )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{7}}\right )}}{315 \, 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)^(7/2),x, algorithm="maxima")

[Out]

2/315*((35*(e*x + d)^(9/2)*B*c^3 - 45*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(7/2) + 189*(7*B*c^3*d^2 - 2*A*c^3*d*e +
B*a*c^2*e^2)*(e*x + d)^(5/2) - 105*(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)^(3/2) + 315*(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)*sqrt(e*
x + d))/e^7 + 21*(3*B*c^3*d^7 - 3*A*c^3*d^6*e + 9*B*a*c^2*d^5*e^2 - 9*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e^4 - 9*
A*a^2*c*d^2*e^5 + 3*B*a^3*d*e^6 - 3*A*a^3*e^7 + 45*(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)^2 - 5*(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)*(e*x + d))/((e*x + d)^(5/2)*e^7))/e

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Fricas [A]  time = 1.62192, size = 1118, normalized size = 3.23 \begin{align*} \frac{2 \,{\left (35 \, B c^{3} e^{7} x^{7} + 14336 \, B c^{3} d^{7} - 9216 \, A c^{3} d^{6} e + 16128 \, B a c^{2} d^{5} e^{2} - 8064 \, A a c^{2} d^{4} e^{3} + 3024 \, B a^{2} c d^{3} e^{4} - 504 \, A a^{2} c d^{2} e^{5} - 42 \, B a^{3} d e^{6} - 63 \, A a^{3} e^{7} - 5 \,{\left (14 \, B c^{3} d e^{6} - 9 \, A c^{3} e^{7}\right )} x^{6} + 3 \,{\left (56 \, B c^{3} d^{2} e^{5} - 36 \, A c^{3} d e^{6} + 63 \, B a c^{2} e^{7}\right )} x^{5} - 5 \,{\left (112 \, B c^{3} d^{3} e^{4} - 72 \, A c^{3} d^{2} e^{5} + 126 \, B a c^{2} d e^{6} - 63 \, A a c^{2} e^{7}\right )} x^{4} + 5 \,{\left (896 \, B c^{3} d^{4} e^{3} - 576 \, A c^{3} d^{3} e^{4} + 1008 \, B a c^{2} d^{2} e^{5} - 504 \, A a c^{2} d e^{6} + 189 \, B a^{2} c e^{7}\right )} x^{3} + 15 \,{\left (1792 \, B c^{3} d^{5} e^{2} - 1152 \, A c^{3} d^{4} e^{3} + 2016 \, B a c^{2} d^{3} e^{4} - 1008 \, A a c^{2} d^{2} e^{5} + 378 \, B a^{2} c d e^{6} - 63 \, A a^{2} c e^{7}\right )} x^{2} + 5 \,{\left (7168 \, B c^{3} d^{6} e - 4608 \, A c^{3} d^{5} e^{2} + 8064 \, B a c^{2} d^{4} e^{3} - 4032 \, A a c^{2} d^{3} e^{4} + 1512 \, B a^{2} c d^{2} e^{5} - 252 \, A a^{2} c d e^{6} - 21 \, B a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/315*(35*B*c^3*e^7*x^7 + 14336*B*c^3*d^7 - 9216*A*c^3*d^6*e + 16128*B*a*c^2*d^5*e^2 - 8064*A*a*c^2*d^4*e^3 +
3024*B*a^2*c*d^3*e^4 - 504*A*a^2*c*d^2*e^5 - 42*B*a^3*d*e^6 - 63*A*a^3*e^7 - 5*(14*B*c^3*d*e^6 - 9*A*c^3*e^7)*
x^6 + 3*(56*B*c^3*d^2*e^5 - 36*A*c^3*d*e^6 + 63*B*a*c^2*e^7)*x^5 - 5*(112*B*c^3*d^3*e^4 - 72*A*c^3*d^2*e^5 + 1
26*B*a*c^2*d*e^6 - 63*A*a*c^2*e^7)*x^4 + 5*(896*B*c^3*d^4*e^3 - 576*A*c^3*d^3*e^4 + 1008*B*a*c^2*d^2*e^5 - 504
*A*a*c^2*d*e^6 + 189*B*a^2*c*e^7)*x^3 + 15*(1792*B*c^3*d^5*e^2 - 1152*A*c^3*d^4*e^3 + 2016*B*a*c^2*d^3*e^4 - 1
008*A*a*c^2*d^2*e^5 + 378*B*a^2*c*d*e^6 - 63*A*a^2*c*e^7)*x^2 + 5*(7168*B*c^3*d^6*e - 4608*A*c^3*d^5*e^2 + 806
4*B*a*c^2*d^4*e^3 - 4032*A*a*c^2*d^3*e^4 + 1512*B*a^2*c*d^2*e^5 - 252*A*a^2*c*d*e^6 - 21*B*a^3*e^7)*x)*sqrt(e*
x + d)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8)

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Sympy [A]  time = 124.059, size = 377, normalized size = 1.09 \begin{align*} \frac{2 B c^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{8}} + \frac{6 c \left (a e^{2} + c d^{2}\right ) \left (- A a e^{3} - 5 A c d^{2} e + 3 B a d e^{2} + 7 B c d^{3}\right )}{e^{8} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 A c^{3} e - 14 B c^{3} d\right )}{7 e^{8}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 12 A c^{3} d e + 6 B a c^{2} e^{2} + 42 B c^{3} d^{2}\right )}{5 e^{8}} + \frac{\left (d + e x\right )^{\frac{3}{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 )}{3 e^{8}} + \frac{\sqrt{d + e x} \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 )}{e^{8}} - \frac{2 \left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{3 e^{8} \left (d + e x\right )^{\frac{3}{2}}} + \frac{2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{5 e^{8} \left (d + e x\right )^{\frac{5}{2}}} \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)**(7/2),x)

[Out]

2*B*c**3*(d + e*x)**(9/2)/(9*e**8) + 6*c*(a*e**2 + c*d**2)*(-A*a*e**3 - 5*A*c*d**2*e + 3*B*a*d*e**2 + 7*B*c*d*
*3)/(e**8*sqrt(d + e*x)) + (d + e*x)**(7/2)*(2*A*c**3*e - 14*B*c**3*d)/(7*e**8) + (d + e*x)**(5/2)*(-12*A*c**3
*d*e + 6*B*a*c**2*e**2 + 42*B*c**3*d**2)/(5*e**8) + (d + e*x)**(3/2)*(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)/(3*e**8) + sqrt(d + e*x)*(-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)/e**8 - 2*(a*e**2 + c*d**2)**2*(-6*A*c*d*e + B*a*e**2 + 7*B*c*
d**2)/(3*e**8*(d + e*x)**(3/2)) + 2*(-A*e + B*d)*(a*e**2 + c*d**2)**3/(5*e**8*(d + e*x)**(5/2))

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Giac [A]  time = 1.21446, size = 809, normalized size = 2.34 \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} B c^{3} e^{64} - 315 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{3} d e^{64} + 1323 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{3} d^{2} e^{64} - 3675 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{3} d^{3} e^{64} + 11025 \, \sqrt{x e + d} B c^{3} d^{4} e^{64} + 45 \,{\left (x e + d\right )}^{\frac{7}{2}} A c^{3} e^{65} - 378 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{3} d e^{65} + 1575 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{3} d^{2} e^{65} - 6300 \, \sqrt{x e + d} A c^{3} d^{3} e^{65} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} B a c^{2} e^{66} - 1575 \,{\left (x e + d\right )}^{\frac{3}{2}} B a c^{2} d e^{66} + 9450 \, \sqrt{x e + d} B a c^{2} d^{2} e^{66} + 315 \,{\left (x e + d\right )}^{\frac{3}{2}} A a c^{2} e^{67} - 3780 \, \sqrt{x e + d} A a c^{2} d e^{67} + 945 \, \sqrt{x e + d} B a^{2} c e^{68}\right )} e^{\left (-72\right )} + \frac{2 \,{\left (315 \,{\left (x e + d\right )}^{2} B c^{3} d^{5} - 35 \,{\left (x e + d\right )} B c^{3} d^{6} + 3 \, B c^{3} d^{7} - 225 \,{\left (x e + d\right )}^{2} A c^{3} d^{4} e + 30 \,{\left (x e + d\right )} A c^{3} d^{5} e - 3 \, A c^{3} d^{6} e + 450 \,{\left (x e + d\right )}^{2} B a c^{2} d^{3} e^{2} - 75 \,{\left (x e + d\right )} B a c^{2} d^{4} e^{2} + 9 \, B a c^{2} d^{5} e^{2} - 270 \,{\left (x e + d\right )}^{2} A a c^{2} d^{2} e^{3} + 60 \,{\left (x e + d\right )} A a c^{2} d^{3} e^{3} - 9 \, A a c^{2} d^{4} e^{3} + 135 \,{\left (x e + d\right )}^{2} B a^{2} c d e^{4} - 45 \,{\left (x e + d\right )} B a^{2} c d^{2} e^{4} + 9 \, B a^{2} c d^{3} e^{4} - 45 \,{\left (x e + d\right )}^{2} A a^{2} c e^{5} + 30 \,{\left (x e + d\right )} A a^{2} c d e^{5} - 9 \, A a^{2} c d^{2} e^{5} - 5 \,{\left (x e + d\right )} B a^{3} e^{6} + 3 \, B a^{3} d e^{6} - 3 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \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)^(7/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*B*c^3*e^64 - 315*(x*e + d)^(7/2)*B*c^3*d*e^64 + 1323*(x*e + d)^(5/2)*B*c^3*d^2*e^64
- 3675*(x*e + d)^(3/2)*B*c^3*d^3*e^64 + 11025*sqrt(x*e + d)*B*c^3*d^4*e^64 + 45*(x*e + d)^(7/2)*A*c^3*e^65 - 3
78*(x*e + d)^(5/2)*A*c^3*d*e^65 + 1575*(x*e + d)^(3/2)*A*c^3*d^2*e^65 - 6300*sqrt(x*e + d)*A*c^3*d^3*e^65 + 18
9*(x*e + d)^(5/2)*B*a*c^2*e^66 - 1575*(x*e + d)^(3/2)*B*a*c^2*d*e^66 + 9450*sqrt(x*e + d)*B*a*c^2*d^2*e^66 + 3
15*(x*e + d)^(3/2)*A*a*c^2*e^67 - 3780*sqrt(x*e + d)*A*a*c^2*d*e^67 + 945*sqrt(x*e + d)*B*a^2*c*e^68)*e^(-72)
+ 2/15*(315*(x*e + d)^2*B*c^3*d^5 - 35*(x*e + d)*B*c^3*d^6 + 3*B*c^3*d^7 - 225*(x*e + d)^2*A*c^3*d^4*e + 30*(x
*e + d)*A*c^3*d^5*e - 3*A*c^3*d^6*e + 450*(x*e + d)^2*B*a*c^2*d^3*e^2 - 75*(x*e + d)*B*a*c^2*d^4*e^2 + 9*B*a*c
^2*d^5*e^2 - 270*(x*e + d)^2*A*a*c^2*d^2*e^3 + 60*(x*e + d)*A*a*c^2*d^3*e^3 - 9*A*a*c^2*d^4*e^3 + 135*(x*e + d
)^2*B*a^2*c*d*e^4 - 45*(x*e + d)*B*a^2*c*d^2*e^4 + 9*B*a^2*c*d^3*e^4 - 45*(x*e + d)^2*A*a^2*c*e^5 + 30*(x*e +
d)*A*a^2*c*d*e^5 - 9*A*a^2*c*d^2*e^5 - 5*(x*e + d)*B*a^3*e^6 + 3*B*a^3*d*e^6 - 3*A*a^3*e^7)*e^(-8)/(x*e + d)^(
5/2)