3.1443 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=346 \[ -\frac{2 c (d+e x)^{3/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8}+\frac{6 c^2 (d+e x)^{7/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{7 e^8}-\frac{2 c^2 (d+e x)^{5/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{5 e^8}-\frac{6 c \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 \sqrt{d+e x}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{3 e^8 (d+e x)^{3/2}}-\frac{2 c^3 (d+e x)^{9/2} (7 B d-A e)}{9 e^8}+\frac{2 B c^3 (d+e x)^{11/2}}{11 e^8} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.161879, 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} \[ -\frac{2 c (d+e x)^{3/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8}+\frac{6 c^2 (d+e x)^{7/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{7 e^8}-\frac{2 c^2 (d+e x)^{5/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{5 e^8}-\frac{6 c \sqrt{d+e x} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 \sqrt{d+e x}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{3 e^8 (d+e x)^{3/2}}-\frac{2 c^3 (d+e x)^{9/2} (7 B d-A e)}{9 e^8}+\frac{2 B c^3 (d+e x)^{11/2}}{11 e^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(3*e^8*(d + e*x)^(3/2)) - (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*
e^2))/(e^8*Sqrt[d + e*x]) - (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)*Sqrt[d + e*
x])/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)^(3/2))/
(3*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)^(5/2))/(5*e^8) + (6*c^2*(7*B
*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(7/2))/(7*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(9/2))/(9*e^8) + (2*B*
c^3*(d + e*x)^(11/2))/(11*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)^{5/2}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^{5/2}}+\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)^{3/2}}+\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 \sqrt{d+e x}}-\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 ) \sqrt{d+e x}}{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)^{3/2}}{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)^{5/2}}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^{7/2}}{e^7}+\frac{B c^3 (d+e x)^{9/2}}{e^7}\right ) \, dx\\ &=\frac{2 (B d-A e) \left (c d^2+a e^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\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 )}{e^8 \sqrt{d+e x}}-\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 ) \sqrt{d+e x}}{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)^{3/2}}{3 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)^{5/2}}{5 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)^{7/2}}{7 e^8}-\frac{2 c^3 (7 B d-A e) (d+e x)^{9/2}}{9 e^8}+\frac{2 B c^3 (d+e x)^{11/2}}{11 e^8}\\ \end{align*}

Mathematica [A]  time = 0.308825, size = 375, normalized size = 1.08 \[ \frac{22 A e \left (315 a^2 c e^4 \left (8 d^2+12 d e x+3 e^2 x^2\right )-105 a^3 e^6+63 a c^2 e^2 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )+5 c^3 \left (384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4+1536 d^5 e x+1024 d^6-12 d e^5 x^5+7 e^6 x^6\right )\right )-10 B \left (693 a^2 c e^4 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )+231 a^3 e^6 (2 d+3 e x)+99 a c^2 e^2 \left (96 d^3 e^2 x^2-16 d^2 e^3 x^3+384 d^4 e x+256 d^5+6 d e^4 x^4-3 e^5 x^5\right )+7 c^3 \left (768 d^5 e^2 x^2-128 d^4 e^3 x^3+48 d^3 e^4 x^4-24 d^2 e^5 x^5+3072 d^6 e x+2048 d^7+14 d e^6 x^6-9 e^7 x^7\right )\right )}{3465 e^8 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(22*A*e*(-105*a^3*e^6 + 315*a^2*c*e^4*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 63*a*c^2*e^2*(128*d^4 + 192*d^3*e*x + 4
8*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + 5*c^3*(1024*d^6 + 1536*d^5*e*x + 384*d^4*e^2*x^2 - 64*d^3*e^3*x^3 +
24*d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*x^6)) - 10*B*(231*a^3*e^6*(2*d + 3*e*x) + 693*a^2*c*e^4*(16*d^3 + 24*d^
2*e*x + 6*d*e^2*x^2 - e^3*x^3) + 99*a*c^2*e^2*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e
^4*x^4 - 3*e^5*x^5) + 7*c^3*(2048*d^7 + 3072*d^6*e*x + 768*d^5*e^2*x^2 - 128*d^4*e^3*x^3 + 48*d^3*e^4*x^4 - 24
*d^2*e^5*x^5 + 14*d*e^6*x^6 - 9*e^7*x^7)))/(3465*e^8*(d + e*x)^(3/2))

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Maple [A]  time = 0.009, size = 489, normalized size = 1.4 \begin{align*} -{\frac{-630\,B{c}^{3}{x}^{7}{e}^{7}-770\,A{c}^{3}{e}^{7}{x}^{6}+980\,B{c}^{3}d{e}^{6}{x}^{6}+1320\,A{c}^{3}d{e}^{6}{x}^{5}-2970\,Ba{c}^{2}{e}^{7}{x}^{5}-1680\,B{c}^{3}{d}^{2}{e}^{5}{x}^{5}-4158\,Aa{c}^{2}{e}^{7}{x}^{4}-2640\,A{c}^{3}{d}^{2}{e}^{5}{x}^{4}+5940\,Ba{c}^{2}d{e}^{6}{x}^{4}+3360\,B{c}^{3}{d}^{3}{e}^{4}{x}^{4}+11088\,Aa{c}^{2}d{e}^{6}{x}^{3}+7040\,A{c}^{3}{d}^{3}{e}^{4}{x}^{3}-6930\,B{a}^{2}c{e}^{7}{x}^{3}-15840\,Ba{c}^{2}{d}^{2}{e}^{5}{x}^{3}-8960\,B{c}^{3}{d}^{4}{e}^{3}{x}^{3}-20790\,A{a}^{2}c{e}^{7}{x}^{2}-66528\,Aa{c}^{2}{d}^{2}{e}^{5}{x}^{2}-42240\,A{c}^{3}{d}^{4}{e}^{3}{x}^{2}+41580\,B{a}^{2}cd{e}^{6}{x}^{2}+95040\,Ba{c}^{2}{d}^{3}{e}^{4}{x}^{2}+53760\,B{c}^{3}{d}^{5}{e}^{2}{x}^{2}-83160\,A{a}^{2}cd{e}^{6}x-266112\,Aa{c}^{2}{d}^{3}{e}^{4}x-168960\,A{c}^{3}{d}^{5}{e}^{2}x+6930\,B{a}^{3}{e}^{7}x+166320\,B{a}^{2}c{d}^{2}{e}^{5}x+380160\,Ba{c}^{2}{d}^{4}{e}^{3}x+215040\,B{c}^{3}{d}^{6}ex+2310\,A{a}^{3}{e}^{7}-55440\,A{a}^{2}c{d}^{2}{e}^{5}-177408\,Aa{c}^{2}{d}^{4}{e}^{3}-112640\,A{c}^{3}{d}^{6}e+4620\,B{a}^{3}d{e}^{6}+110880\,B{a}^{2}c{d}^{3}{e}^{4}+253440\,Ba{c}^{2}{d}^{5}{e}^{2}+143360\,B{c}^{3}{d}^{7}}{3465\,{e}^{8}} \left ( ex+d \right ) ^{-{\frac{3}{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)^(5/2),x)

[Out]

-2/3465/(e*x+d)^(3/2)*(-315*B*c^3*e^7*x^7-385*A*c^3*e^7*x^6+490*B*c^3*d*e^6*x^6+660*A*c^3*d*e^6*x^5-1485*B*a*c
^2*e^7*x^5-840*B*c^3*d^2*e^5*x^5-2079*A*a*c^2*e^7*x^4-1320*A*c^3*d^2*e^5*x^4+2970*B*a*c^2*d*e^6*x^4+1680*B*c^3
*d^3*e^4*x^4+5544*A*a*c^2*d*e^6*x^3+3520*A*c^3*d^3*e^4*x^3-3465*B*a^2*c*e^7*x^3-7920*B*a*c^2*d^2*e^5*x^3-4480*
B*c^3*d^4*e^3*x^3-10395*A*a^2*c*e^7*x^2-33264*A*a*c^2*d^2*e^5*x^2-21120*A*c^3*d^4*e^3*x^2+20790*B*a^2*c*d*e^6*
x^2+47520*B*a*c^2*d^3*e^4*x^2+26880*B*c^3*d^5*e^2*x^2-41580*A*a^2*c*d*e^6*x-133056*A*a*c^2*d^3*e^4*x-84480*A*c
^3*d^5*e^2*x+3465*B*a^3*e^7*x+83160*B*a^2*c*d^2*e^5*x+190080*B*a*c^2*d^4*e^3*x+107520*B*c^3*d^6*e*x+1155*A*a^3
*e^7-27720*A*a^2*c*d^2*e^5-88704*A*a*c^2*d^4*e^3-56320*A*c^3*d^6*e+2310*B*a^3*d*e^6+55440*B*a^2*c*d^3*e^4+1267
20*B*a*c^2*d^5*e^2+71680*B*c^3*d^7)/e^8

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Maxima [A]  time = 1.02233, size = 620, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (\frac{315 \,{\left (e x + d\right )}^{\frac{11}{2}} B c^{3} - 385 \,{\left (7 \, B c^{3} d - A c^{3} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 1485 \,{\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{7}{2}} - 693 \,{\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{5}{2}} + 1155 \,{\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{3}{2}} - 10395 \,{\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 )} \sqrt{e x + d}}{e^{7}} + \frac{1155 \,{\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} - 3 \,{\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{3}{2}} e^{7}}\right )}}{3465 \, 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)^(5/2),x, algorithm="maxima")

[Out]

2/3465*((315*(e*x + d)^(11/2)*B*c^3 - 385*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(9/2) + 1485*(7*B*c^3*d^2 - 2*A*c^3*
d*e + B*a*c^2*e^2)*(e*x + d)^(7/2) - 693*(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)^(5/2) + 1155*(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)^(3/2) - 10395*(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)*sqrt(e*x + d))/e^7 + 1155*(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 - 3*(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)^(3/2)*e^7
))/e

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Fricas [A]  time = 1.56369, size = 1138, normalized size = 3.29 \begin{align*} \frac{2 \,{\left (315 \, B c^{3} e^{7} x^{7} - 71680 \, B c^{3} d^{7} + 56320 \, A c^{3} d^{6} e - 126720 \, B a c^{2} d^{5} e^{2} + 88704 \, A a c^{2} d^{4} e^{3} - 55440 \, B a^{2} c d^{3} e^{4} + 27720 \, A a^{2} c d^{2} e^{5} - 2310 \, B a^{3} d e^{6} - 1155 \, A a^{3} e^{7} - 35 \,{\left (14 \, B c^{3} d e^{6} - 11 \, A c^{3} e^{7}\right )} x^{6} + 15 \,{\left (56 \, B c^{3} d^{2} e^{5} - 44 \, A c^{3} d e^{6} + 99 \, B a c^{2} e^{7}\right )} x^{5} - 3 \,{\left (560 \, B c^{3} d^{3} e^{4} - 440 \, A c^{3} d^{2} e^{5} + 990 \, B a c^{2} d e^{6} - 693 \, A a c^{2} e^{7}\right )} x^{4} +{\left (4480 \, B c^{3} d^{4} e^{3} - 3520 \, A c^{3} d^{3} e^{4} + 7920 \, B a c^{2} d^{2} e^{5} - 5544 \, A a c^{2} d e^{6} + 3465 \, B a^{2} c e^{7}\right )} x^{3} - 3 \,{\left (8960 \, B c^{3} d^{5} e^{2} - 7040 \, A c^{3} d^{4} e^{3} + 15840 \, B a c^{2} d^{3} e^{4} - 11088 \, A a c^{2} d^{2} e^{5} + 6930 \, B a^{2} c d e^{6} - 3465 \, A a^{2} c e^{7}\right )} x^{2} - 3 \,{\left (35840 \, B c^{3} d^{6} e - 28160 \, A c^{3} d^{5} e^{2} + 63360 \, B a c^{2} d^{4} e^{3} - 44352 \, A a c^{2} d^{3} e^{4} + 27720 \, B a^{2} c d^{2} e^{5} - 13860 \, A a^{2} c d e^{6} + 1155 \, B a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{3465 \,{\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*c^3*e^7*x^7 - 71680*B*c^3*d^7 + 56320*A*c^3*d^6*e - 126720*B*a*c^2*d^5*e^2 + 88704*A*a*c^2*d^4*e
^3 - 55440*B*a^2*c*d^3*e^4 + 27720*A*a^2*c*d^2*e^5 - 2310*B*a^3*d*e^6 - 1155*A*a^3*e^7 - 35*(14*B*c^3*d*e^6 -
11*A*c^3*e^7)*x^6 + 15*(56*B*c^3*d^2*e^5 - 44*A*c^3*d*e^6 + 99*B*a*c^2*e^7)*x^5 - 3*(560*B*c^3*d^3*e^4 - 440*A
*c^3*d^2*e^5 + 990*B*a*c^2*d*e^6 - 693*A*a*c^2*e^7)*x^4 + (4480*B*c^3*d^4*e^3 - 3520*A*c^3*d^3*e^4 + 7920*B*a*
c^2*d^2*e^5 - 5544*A*a*c^2*d*e^6 + 3465*B*a^2*c*e^7)*x^3 - 3*(8960*B*c^3*d^5*e^2 - 7040*A*c^3*d^4*e^3 + 15840*
B*a*c^2*d^3*e^4 - 11088*A*a*c^2*d^2*e^5 + 6930*B*a^2*c*d*e^6 - 3465*A*a^2*c*e^7)*x^2 - 3*(35840*B*c^3*d^6*e -
28160*A*c^3*d^5*e^2 + 63360*B*a*c^2*d^4*e^3 - 44352*A*a*c^2*d^3*e^4 + 27720*B*a^2*c*d^2*e^5 - 13860*A*a^2*c*d*
e^6 + 1155*B*a^3*e^7)*x)*sqrt(e*x + d)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)

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Sympy [A]  time = 87.7128, size = 406, normalized size = 1.17 \begin{align*} \frac{2 B c^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{8}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (2 A c^{3} e - 14 B c^{3} d\right )}{9 e^{8}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (- 12 A c^{3} d e + 6 B a c^{2} e^{2} + 42 B c^{3} d^{2}\right )}{7 e^{8}} + \frac{\left (d + e x\right )^{\frac{5}{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 )}{5 e^{8}} + \frac{\left (d + e x\right )^{\frac{3}{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 )}{3 e^{8}} + \frac{\sqrt{d + e x} \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 )}{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 )}{e^{8} \sqrt{d + e x}} + \frac{2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{3 e^{8} \left (d + e x\right )^{\frac{3}{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)**(5/2),x)

[Out]

2*B*c**3*(d + e*x)**(11/2)/(11*e**8) + (d + e*x)**(9/2)*(2*A*c**3*e - 14*B*c**3*d)/(9*e**8) + (d + e*x)**(7/2)
*(-12*A*c**3*d*e + 6*B*a*c**2*e**2 + 42*B*c**3*d**2)/(7*e**8) + (d + e*x)**(5/2)*(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)/(5*e**8) + (d + e*x)**(3/2)*(-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)/(3*e**8) + sqrt(d + e*x)*(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)/e**8 - 2
*(a*e**2 + c*d**2)**2*(-6*A*c*d*e + B*a*e**2 + 7*B*c*d**2)/(e**8*sqrt(d + e*x)) + 2*(-A*e + B*d)*(a*e**2 + c*d
**2)**3/(3*e**8*(d + e*x)**(3/2))

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Giac [A]  time = 1.17076, size = 809, normalized size = 2.34 \begin{align*} \frac{2}{3465} \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} B c^{3} e^{80} - 2695 \,{\left (x e + d\right )}^{\frac{9}{2}} B c^{3} d e^{80} + 10395 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{3} d^{2} e^{80} - 24255 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{3} d^{3} e^{80} + 40425 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{3} d^{4} e^{80} - 72765 \, \sqrt{x e + d} B c^{3} d^{5} e^{80} + 385 \,{\left (x e + d\right )}^{\frac{9}{2}} A c^{3} e^{81} - 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} A c^{3} d e^{81} + 10395 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{3} d^{2} e^{81} - 23100 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{3} d^{3} e^{81} + 51975 \, \sqrt{x e + d} A c^{3} d^{4} e^{81} + 1485 \,{\left (x e + d\right )}^{\frac{7}{2}} B a c^{2} e^{82} - 10395 \,{\left (x e + d\right )}^{\frac{5}{2}} B a c^{2} d e^{82} + 34650 \,{\left (x e + d\right )}^{\frac{3}{2}} B a c^{2} d^{2} e^{82} - 103950 \, \sqrt{x e + d} B a c^{2} d^{3} e^{82} + 2079 \,{\left (x e + d\right )}^{\frac{5}{2}} A a c^{2} e^{83} - 13860 \,{\left (x e + d\right )}^{\frac{3}{2}} A a c^{2} d e^{83} + 62370 \, \sqrt{x e + d} A a c^{2} d^{2} e^{83} + 3465 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} c e^{84} - 31185 \, \sqrt{x e + d} B a^{2} c d e^{84} + 10395 \, \sqrt{x e + d} A a^{2} c e^{85}\right )} e^{\left (-88\right )} - \frac{2 \,{\left (21 \,{\left (x e + d\right )} B c^{3} d^{6} - B c^{3} d^{7} - 18 \,{\left (x e + d\right )} A c^{3} d^{5} e + A c^{3} d^{6} e + 45 \,{\left (x e + d\right )} B a c^{2} d^{4} e^{2} - 3 \, B a c^{2} d^{5} e^{2} - 36 \,{\left (x e + d\right )} A a c^{2} d^{3} e^{3} + 3 \, A a c^{2} d^{4} e^{3} + 27 \,{\left (x e + d\right )} B a^{2} c d^{2} e^{4} - 3 \, B a^{2} c d^{3} e^{4} - 18 \,{\left (x e + d\right )} A a^{2} c d e^{5} + 3 \, A a^{2} c d^{2} e^{5} + 3 \,{\left (x e + d\right )} B a^{3} e^{6} - B a^{3} d e^{6} + A a^{3} e^{7}\right )} e^{\left (-8\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

2/3465*(315*(x*e + d)^(11/2)*B*c^3*e^80 - 2695*(x*e + d)^(9/2)*B*c^3*d*e^80 + 10395*(x*e + d)^(7/2)*B*c^3*d^2*
e^80 - 24255*(x*e + d)^(5/2)*B*c^3*d^3*e^80 + 40425*(x*e + d)^(3/2)*B*c^3*d^4*e^80 - 72765*sqrt(x*e + d)*B*c^3
*d^5*e^80 + 385*(x*e + d)^(9/2)*A*c^3*e^81 - 2970*(x*e + d)^(7/2)*A*c^3*d*e^81 + 10395*(x*e + d)^(5/2)*A*c^3*d
^2*e^81 - 23100*(x*e + d)^(3/2)*A*c^3*d^3*e^81 + 51975*sqrt(x*e + d)*A*c^3*d^4*e^81 + 1485*(x*e + d)^(7/2)*B*a
*c^2*e^82 - 10395*(x*e + d)^(5/2)*B*a*c^2*d*e^82 + 34650*(x*e + d)^(3/2)*B*a*c^2*d^2*e^82 - 103950*sqrt(x*e +
d)*B*a*c^2*d^3*e^82 + 2079*(x*e + d)^(5/2)*A*a*c^2*e^83 - 13860*(x*e + d)^(3/2)*A*a*c^2*d*e^83 + 62370*sqrt(x*
e + d)*A*a*c^2*d^2*e^83 + 3465*(x*e + d)^(3/2)*B*a^2*c*e^84 - 31185*sqrt(x*e + d)*B*a^2*c*d*e^84 + 10395*sqrt(
x*e + d)*A*a^2*c*e^85)*e^(-88) - 2/3*(21*(x*e + d)*B*c^3*d^6 - B*c^3*d^7 - 18*(x*e + d)*A*c^3*d^5*e + A*c^3*d^
6*e + 45*(x*e + d)*B*a*c^2*d^4*e^2 - 3*B*a*c^2*d^5*e^2 - 36*(x*e + d)*A*a*c^2*d^3*e^3 + 3*A*a*c^2*d^4*e^3 + 27
*(x*e + d)*B*a^2*c*d^2*e^4 - 3*B*a^2*c*d^3*e^4 - 18*(x*e + d)*A*a^2*c*d*e^5 + 3*A*a^2*c*d^2*e^5 + 3*(x*e + d)*
B*a^3*e^6 - B*a^3*d*e^6 + A*a^3*e^7)*e^(-8)/(x*e + d)^(3/2)