3.1445 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^{9/2}} \, dx\)
Optimal. Leaf size=342 \[ \frac{2 c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{3/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}-\frac{2 c^2 \sqrt{d+e x} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8}+\frac{2 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^{5/2}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{7 e^8 (d+e x)^{7/2}}-\frac{2 c^3 (d+e x)^{5/2} (7 B d-A e)}{5 e^8}+\frac{2 B c^3 (d+e x)^{7/2}}{7 e^8} \]
[Out]
(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(7*e^8*(d + e*x)^(7/2)) - (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*
e^2))/(5*e^8*(d + e*x)^(5/2)) + (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))/(e^8*(
d + e*x)^(3/2)) + (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)))/(e^8*Sqr
t[d + e*x]) - (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)*Sqrt[d + e*x])/e^8 + (2*c^2*(7*B*c
*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(3/2))/e^8 - (2*c^3*(7*B*d - A*e)*(d + e*x)^(5/2))/(5*e^8) + (2*B*c^3*(d
+ e*x)^(7/2))/(7*e^8)
________________________________________________________________________________________
Rubi [A] time = 0.156123, antiderivative size = 342, 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} \[ \frac{2 c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 \sqrt{d+e x}}+\frac{2 c^2 (d+e x)^{3/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}-\frac{2 c^2 \sqrt{d+e x} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8}+\frac{2 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^{3/2}}-\frac{2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^{5/2}}+\frac{2 \left (a e^2+c d^2\right )^3 (B d-A e)}{7 e^8 (d+e x)^{7/2}}-\frac{2 c^3 (d+e x)^{5/2} (7 B d-A e)}{5 e^8}+\frac{2 B c^3 (d+e x)^{7/2}}{7 e^8} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(9/2),x]
[Out]
(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(7*e^8*(d + e*x)^(7/2)) - (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*
e^2))/(5*e^8*(d + e*x)^(5/2)) + (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))/(e^8*(
d + e*x)^(3/2)) + (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)))/(e^8*Sqr
t[d + e*x]) - (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)*Sqrt[d + e*x])/e^8 + (2*c^2*(7*B*c
*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(3/2))/e^8 - (2*c^3*(7*B*d - A*e)*(d + e*x)^(5/2))/(5*e^8) + (2*B*c^3*(d
+ e*x)^(7/2))/(7*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)^{9/2}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^{9/2}}+\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)^{7/2}}+\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)^{5/2}}-\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 (d+e x)^{3/2}}+\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 )}{e^7 \sqrt{d+e x}}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) \sqrt{d+e x}}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^{3/2}}{e^7}+\frac{B c^3 (d+e x)^{5/2}}{e^7}\right ) \, dx\\ &=\frac{2 (B d-A e) \left (c d^2+a e^2\right )^3}{7 e^8 (d+e x)^{7/2}}-\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 )}{5 e^8 (d+e x)^{5/2}}+\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 )}{e^8 (d+e x)^{3/2}}+\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 )}{e^8 \sqrt{d+e x}}-\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 ) \sqrt{d+e x}}{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)^{3/2}}{e^8}-\frac{2 c^3 (7 B d-A e) (d+e x)^{5/2}}{5 e^8}+\frac{2 B c^3 (d+e x)^{7/2}}{7 e^8}\\ \end{align*}
Mathematica [A] time = 0.30636, size = 372, normalized size = 1.09 \[ \frac{2 A e \left (-a^2 c e^4 \left (8 d^2+28 d e x+35 e^2 x^2\right )-5 a^3 e^6+3 a c^2 e^2 \left (560 d^2 e^2 x^2+448 d^3 e x+128 d^4+280 d e^3 x^3+35 e^4 x^4\right )+c^3 \left (4480 d^4 e^2 x^2+2240 d^3 e^3 x^3+280 d^2 e^4 x^4+3584 d^5 e x+1024 d^6-28 d e^5 x^5+7 e^6 x^6\right )\right )-2 B \left (3 a^2 c e^4 \left (56 d^2 e x+16 d^3+70 d e^2 x^2+35 e^3 x^3\right )+a^3 e^6 (2 d+7 e x)+5 a c^2 e^2 \left (1120 d^3 e^2 x^2+560 d^2 e^3 x^3+896 d^4 e x+256 d^5+70 d e^4 x^4-7 e^5 x^5\right )+c^3 \left (8960 d^5 e^2 x^2+4480 d^4 e^3 x^3+560 d^3 e^4 x^4-56 d^2 e^5 x^5+7168 d^6 e x+2048 d^7+14 d e^6 x^6-5 e^7 x^7\right )\right )}{35 e^8 (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(9/2),x]
[Out]
(2*A*e*(-5*a^3*e^6 - a^2*c*e^4*(8*d^2 + 28*d*e*x + 35*e^2*x^2) + 3*a*c^2*e^2*(128*d^4 + 448*d^3*e*x + 560*d^2*
e^2*x^2 + 280*d*e^3*x^3 + 35*e^4*x^4) + c^3*(1024*d^6 + 3584*d^5*e*x + 4480*d^4*e^2*x^2 + 2240*d^3*e^3*x^3 + 2
80*d^2*e^4*x^4 - 28*d*e^5*x^5 + 7*e^6*x^6)) - 2*B*(a^3*e^6*(2*d + 7*e*x) + 3*a^2*c*e^4*(16*d^3 + 56*d^2*e*x +
70*d*e^2*x^2 + 35*e^3*x^3) + 5*a*c^2*e^2*(256*d^5 + 896*d^4*e*x + 1120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 70*d*e^
4*x^4 - 7*e^5*x^5) + c^3*(2048*d^7 + 7168*d^6*e*x + 8960*d^5*e^2*x^2 + 4480*d^4*e^3*x^3 + 560*d^3*e^4*x^4 - 56
*d^2*e^5*x^5 + 14*d*e^6*x^6 - 5*e^7*x^7)))/(35*e^8*(d + e*x)^(7/2))
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Maple [A] time = 0.009, size = 489, normalized size = 1.4 \begin{align*} -{\frac{-10\,B{c}^{3}{x}^{7}{e}^{7}-14\,A{c}^{3}{e}^{7}{x}^{6}+28\,B{c}^{3}d{e}^{6}{x}^{6}+56\,A{c}^{3}d{e}^{6}{x}^{5}-70\,Ba{c}^{2}{e}^{7}{x}^{5}-112\,B{c}^{3}{d}^{2}{e}^{5}{x}^{5}-210\,Aa{c}^{2}{e}^{7}{x}^{4}-560\,A{c}^{3}{d}^{2}{e}^{5}{x}^{4}+700\,Ba{c}^{2}d{e}^{6}{x}^{4}+1120\,B{c}^{3}{d}^{3}{e}^{4}{x}^{4}-1680\,Aa{c}^{2}d{e}^{6}{x}^{3}-4480\,A{c}^{3}{d}^{3}{e}^{4}{x}^{3}+210\,B{a}^{2}c{e}^{7}{x}^{3}+5600\,Ba{c}^{2}{d}^{2}{e}^{5}{x}^{3}+8960\,B{c}^{3}{d}^{4}{e}^{3}{x}^{3}+70\,A{a}^{2}c{e}^{7}{x}^{2}-3360\,Aa{c}^{2}{d}^{2}{e}^{5}{x}^{2}-8960\,A{c}^{3}{d}^{4}{e}^{3}{x}^{2}+420\,B{a}^{2}cd{e}^{6}{x}^{2}+11200\,Ba{c}^{2}{d}^{3}{e}^{4}{x}^{2}+17920\,B{c}^{3}{d}^{5}{e}^{2}{x}^{2}+56\,A{a}^{2}cd{e}^{6}x-2688\,Aa{c}^{2}{d}^{3}{e}^{4}x-7168\,A{c}^{3}{d}^{5}{e}^{2}x+14\,B{a}^{3}{e}^{7}x+336\,B{a}^{2}c{d}^{2}{e}^{5}x+8960\,Ba{c}^{2}{d}^{4}{e}^{3}x+14336\,B{c}^{3}{d}^{6}ex+10\,A{a}^{3}{e}^{7}+16\,A{a}^{2}c{d}^{2}{e}^{5}-768\,Aa{c}^{2}{d}^{4}{e}^{3}-2048\,A{c}^{3}{d}^{6}e+4\,B{a}^{3}d{e}^{6}+96\,B{a}^{2}c{d}^{3}{e}^{4}+2560\,Ba{c}^{2}{d}^{5}{e}^{2}+4096\,B{c}^{3}{d}^{7}}{35\,{e}^{8}} \left ( ex+d \right ) ^{-{\frac{7}{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)^(9/2),x)
[Out]
-2/35/(e*x+d)^(7/2)*(-5*B*c^3*e^7*x^7-7*A*c^3*e^7*x^6+14*B*c^3*d*e^6*x^6+28*A*c^3*d*e^6*x^5-35*B*a*c^2*e^7*x^5
-56*B*c^3*d^2*e^5*x^5-105*A*a*c^2*e^7*x^4-280*A*c^3*d^2*e^5*x^4+350*B*a*c^2*d*e^6*x^4+560*B*c^3*d^3*e^4*x^4-84
0*A*a*c^2*d*e^6*x^3-2240*A*c^3*d^3*e^4*x^3+105*B*a^2*c*e^7*x^3+2800*B*a*c^2*d^2*e^5*x^3+4480*B*c^3*d^4*e^3*x^3
+35*A*a^2*c*e^7*x^2-1680*A*a*c^2*d^2*e^5*x^2-4480*A*c^3*d^4*e^3*x^2+210*B*a^2*c*d*e^6*x^2+5600*B*a*c^2*d^3*e^4
*x^2+8960*B*c^3*d^5*e^2*x^2+28*A*a^2*c*d*e^6*x-1344*A*a*c^2*d^3*e^4*x-3584*A*c^3*d^5*e^2*x+7*B*a^3*e^7*x+168*B
*a^2*c*d^2*e^5*x+4480*B*a*c^2*d^4*e^3*x+7168*B*c^3*d^6*e*x+5*A*a^3*e^7+8*A*a^2*c*d^2*e^5-384*A*a*c^2*d^4*e^3-1
024*A*c^3*d^6*e+2*B*a^3*d*e^6+48*B*a^2*c*d^3*e^4+1280*B*a*c^2*d^5*e^2+2048*B*c^3*d^7)/e^8
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Maxima [A] time = 1.00994, size = 621, normalized size = 1.82 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} B c^{3} - 7 \,{\left (7 \, B c^{3} d - A c^{3} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\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{3}{2}} - 35 \,{\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 )} \sqrt{e x + d}}{e^{7}} + \frac{5 \, B c^{3} d^{7} - 5 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} - 15 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} - 15 \, A a^{2} c d^{2} e^{5} + 5 \, B a^{3} d e^{6} - 5 \, A a^{3} e^{7} - 35 \,{\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 )}^{3} + 35 \,{\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} - 7 \,{\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 )}}{{\left (e x + d\right )}^{\frac{7}{2}} e^{7}}\right )}}{35 \, 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)^(9/2),x, algorithm="maxima")
[Out]
2/35*((5*(e*x + d)^(7/2)*B*c^3 - 7*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(5/2) + 35*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a
*c^2*e^2)*(e*x + d)^(3/2) - 35*(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)*sqrt(e*x + d
))/e^7 + (5*B*c^3*d^7 - 5*A*c^3*d^6*e + 15*B*a*c^2*d^5*e^2 - 15*A*a*c^2*d^4*e^3 + 15*B*a^2*c*d^3*e^4 - 15*A*a^
2*c*d^2*e^5 + 5*B*a^3*d*e^6 - 5*A*a^3*e^7 - 35*(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 + 35*(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 - 7*(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)^(7/2)*e^7))/e
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Fricas [A] time = 1.53064, size = 1080, normalized size = 3.16 \begin{align*} \frac{2 \,{\left (5 \, B c^{3} e^{7} x^{7} - 2048 \, B c^{3} d^{7} + 1024 \, A c^{3} d^{6} e - 1280 \, B a c^{2} d^{5} e^{2} + 384 \, A a c^{2} d^{4} e^{3} - 48 \, B a^{2} c d^{3} e^{4} - 8 \, A a^{2} c d^{2} e^{5} - 2 \, B a^{3} d e^{6} - 5 \, A a^{3} e^{7} - 7 \,{\left (2 \, B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 7 \,{\left (8 \, B c^{3} d^{2} e^{5} - 4 \, A c^{3} d e^{6} + 5 \, B a c^{2} e^{7}\right )} x^{5} - 35 \,{\left (16 \, B c^{3} d^{3} e^{4} - 8 \, A c^{3} d^{2} e^{5} + 10 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} - 35 \,{\left (128 \, B c^{3} d^{4} e^{3} - 64 \, A c^{3} d^{3} e^{4} + 80 \, B a c^{2} d^{2} e^{5} - 24 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} - 35 \,{\left (256 \, B c^{3} d^{5} e^{2} - 128 \, A c^{3} d^{4} e^{3} + 160 \, B a c^{2} d^{3} e^{4} - 48 \, A a c^{2} d^{2} e^{5} + 6 \, B a^{2} c d e^{6} + A a^{2} c e^{7}\right )} x^{2} - 7 \,{\left (1024 \, B c^{3} d^{6} e - 512 \, A c^{3} d^{5} e^{2} + 640 \, B a c^{2} d^{4} e^{3} - 192 \, A a c^{2} d^{3} e^{4} + 24 \, B a^{2} c d^{2} e^{5} + 4 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} 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)^(9/2),x, algorithm="fricas")
[Out]
2/35*(5*B*c^3*e^7*x^7 - 2048*B*c^3*d^7 + 1024*A*c^3*d^6*e - 1280*B*a*c^2*d^5*e^2 + 384*A*a*c^2*d^4*e^3 - 48*B*
a^2*c*d^3*e^4 - 8*A*a^2*c*d^2*e^5 - 2*B*a^3*d*e^6 - 5*A*a^3*e^7 - 7*(2*B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 7*(8*B*c
^3*d^2*e^5 - 4*A*c^3*d*e^6 + 5*B*a*c^2*e^7)*x^5 - 35*(16*B*c^3*d^3*e^4 - 8*A*c^3*d^2*e^5 + 10*B*a*c^2*d*e^6 -
3*A*a*c^2*e^7)*x^4 - 35*(128*B*c^3*d^4*e^3 - 64*A*c^3*d^3*e^4 + 80*B*a*c^2*d^2*e^5 - 24*A*a*c^2*d*e^6 + 3*B*a^
2*c*e^7)*x^3 - 35*(256*B*c^3*d^5*e^2 - 128*A*c^3*d^4*e^3 + 160*B*a*c^2*d^3*e^4 - 48*A*a*c^2*d^2*e^5 + 6*B*a^2*
c*d*e^6 + A*a^2*c*e^7)*x^2 - 7*(1024*B*c^3*d^6*e - 512*A*c^3*d^5*e^2 + 640*B*a*c^2*d^4*e^3 - 192*A*a*c^2*d^3*e
^4 + 24*B*a^2*c*d^2*e^5 + 4*A*a^2*c*d*e^6 + B*a^3*e^7)*x)*sqrt(e*x + d)/(e^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*
x^2 + 4*d^3*e^9*x + d^4*e^8)
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Sympy [A] time = 11.784, size = 3218, normalized size = 9.41 \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)**(9/2),x)
[Out]
Piecewise((-10*A*a**3*e**7/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt
(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 16*A*a**2*c*d**2*e**5/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x
*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 56*A*a**2*c*d*e**6*x/(35*d**3
*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d +
e*x)) - 70*A*a**2*c*e**7*x**2/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*s
qrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 768*A*a*c**2*d**4*e**3/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e*
*9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 2688*A*a*c**2*d**3*e**4*x
/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*
sqrt(d + e*x)) + 3360*A*a*c**2*d**2*e**5*x**2/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 10
5*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 1680*A*a*c**2*d*e**6*x**3/(35*d**3*e**8*sqrt(d +
e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 210*A*
a*c**2*e**7*x**4/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x)
+ 35*e**11*x**3*sqrt(d + e*x)) + 2048*A*c**3*d**6*e/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x
) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 7168*A*c**3*d**5*e**2*x/(35*d**3*e**8*sqrt
(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 89
60*A*c**3*d**4*e**3*x**2/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d
+ e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 4480*A*c**3*d**3*e**4*x**3/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**
9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 560*A*c**3*d**2*e**5*x**4/
(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*s
qrt(d + e*x)) - 56*A*c**3*d*e**6*x**5/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**1
0*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 14*A*c**3*e**7*x**6/(35*d**3*e**8*sqrt(d + e*x) + 105*d*
*2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 4*B*a**3*d*e**6/(35*
d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(
d + e*x)) - 14*B*a**3*e**7*x/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sq
rt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 96*B*a**2*c*d**3*e**4/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9
*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 336*B*a**2*c*d**2*e**5*x/(3
5*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqr
t(d + e*x)) - 420*B*a**2*c*d*e**6*x**2/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**
10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 210*B*a**2*c*e**7*x**3/(35*d**3*e**8*sqrt(d + e*x) + 10
5*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 2560*B*a*c**2*d*
*5*e**2/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**1
1*x**3*sqrt(d + e*x)) - 8960*B*a*c**2*d**4*e**3*x/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x)
+ 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 11200*B*a*c**2*d**3*e**4*x**2/(35*d**3*e**8*
sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x))
- 5600*B*a*c**2*d**2*e**5*x**3/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*
sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 700*B*a*c**2*d*e**6*x**4/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2
*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 70*B*a*c**2*e**7*x**5/
(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*s
qrt(d + e*x)) - 4096*B*c**3*d**7/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**
2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 14336*B*c**3*d**6*e*x/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*
e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 17920*B*c**3*d**5*e**2*
x**2/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x
**3*sqrt(d + e*x)) - 8960*B*c**3*d**4*e**3*x**3/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) +
105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 1120*B*c**3*d**3*e**4*x**4/(35*d**3*e**8*sqrt(
d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 112
*B*c**3*d**2*e**5*x**5/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d +
e*x) + 35*e**11*x**3*sqrt(d + e*x)) - 28*B*c**3*d*e**6*x**6/(35*d**3*e**8*sqrt(d + e*x) + 105*d**2*e**9*x*sqr
t(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x)) + 10*B*c**3*e**7*x**7/(35*d**3*e**8
*sqrt(d + e*x) + 105*d**2*e**9*x*sqrt(d + e*x) + 105*d*e**10*x**2*sqrt(d + e*x) + 35*e**11*x**3*sqrt(d + e*x))
, Ne(e, 0)), ((A*a**3*x + A*a**2*c*x**3 + 3*A*a*c**2*x**5/5 + A*c**3*x**7/7 + B*a**3*x**2/2 + 3*B*a**2*c*x**4/
4 + B*a*c**2*x**6/2 + B*c**3*x**8/8)/d**(9/2), True))
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Giac [A] time = 1.22736, size = 806, normalized size = 2.36 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{3} e^{48} - 49 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{3} d e^{48} + 245 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{3} d^{2} e^{48} - 1225 \, \sqrt{x e + d} B c^{3} d^{3} e^{48} + 7 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{3} e^{49} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{3} d e^{49} + 525 \, \sqrt{x e + d} A c^{3} d^{2} e^{49} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B a c^{2} e^{50} - 525 \, \sqrt{x e + d} B a c^{2} d e^{50} + 105 \, \sqrt{x e + d} A a c^{2} e^{51}\right )} e^{\left (-56\right )} - \frac{2 \,{\left (1225 \,{\left (x e + d\right )}^{3} B c^{3} d^{4} - 245 \,{\left (x e + d\right )}^{2} B c^{3} d^{5} + 49 \,{\left (x e + d\right )} B c^{3} d^{6} - 5 \, B c^{3} d^{7} - 700 \,{\left (x e + d\right )}^{3} A c^{3} d^{3} e + 175 \,{\left (x e + d\right )}^{2} A c^{3} d^{4} e - 42 \,{\left (x e + d\right )} A c^{3} d^{5} e + 5 \, A c^{3} d^{6} e + 1050 \,{\left (x e + d\right )}^{3} B a c^{2} d^{2} e^{2} - 350 \,{\left (x e + d\right )}^{2} B a c^{2} d^{3} e^{2} + 105 \,{\left (x e + d\right )} B a c^{2} d^{4} e^{2} - 15 \, B a c^{2} d^{5} e^{2} - 420 \,{\left (x e + d\right )}^{3} A a c^{2} d e^{3} + 210 \,{\left (x e + d\right )}^{2} A a c^{2} d^{2} e^{3} - 84 \,{\left (x e + d\right )} A a c^{2} d^{3} e^{3} + 15 \, A a c^{2} d^{4} e^{3} + 105 \,{\left (x e + d\right )}^{3} B a^{2} c e^{4} - 105 \,{\left (x e + d\right )}^{2} B a^{2} c d e^{4} + 63 \,{\left (x e + d\right )} B a^{2} c d^{2} e^{4} - 15 \, B a^{2} c d^{3} e^{4} + 35 \,{\left (x e + d\right )}^{2} A a^{2} c e^{5} - 42 \,{\left (x e + d\right )} A a^{2} c d e^{5} + 15 \, A a^{2} c d^{2} e^{5} + 7 \,{\left (x e + d\right )} B a^{3} e^{6} - 5 \, B a^{3} d e^{6} + 5 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{35 \,{\left (x e + d\right )}^{\frac{7}{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)^(9/2),x, algorithm="giac")
[Out]
2/35*(5*(x*e + d)^(7/2)*B*c^3*e^48 - 49*(x*e + d)^(5/2)*B*c^3*d*e^48 + 245*(x*e + d)^(3/2)*B*c^3*d^2*e^48 - 12
25*sqrt(x*e + d)*B*c^3*d^3*e^48 + 7*(x*e + d)^(5/2)*A*c^3*e^49 - 70*(x*e + d)^(3/2)*A*c^3*d*e^49 + 525*sqrt(x*
e + d)*A*c^3*d^2*e^49 + 35*(x*e + d)^(3/2)*B*a*c^2*e^50 - 525*sqrt(x*e + d)*B*a*c^2*d*e^50 + 105*sqrt(x*e + d)
*A*a*c^2*e^51)*e^(-56) - 2/35*(1225*(x*e + d)^3*B*c^3*d^4 - 245*(x*e + d)^2*B*c^3*d^5 + 49*(x*e + d)*B*c^3*d^6
- 5*B*c^3*d^7 - 700*(x*e + d)^3*A*c^3*d^3*e + 175*(x*e + d)^2*A*c^3*d^4*e - 42*(x*e + d)*A*c^3*d^5*e + 5*A*c^
3*d^6*e + 1050*(x*e + d)^3*B*a*c^2*d^2*e^2 - 350*(x*e + d)^2*B*a*c^2*d^3*e^2 + 105*(x*e + d)*B*a*c^2*d^4*e^2 -
15*B*a*c^2*d^5*e^2 - 420*(x*e + d)^3*A*a*c^2*d*e^3 + 210*(x*e + d)^2*A*a*c^2*d^2*e^3 - 84*(x*e + d)*A*a*c^2*d
^3*e^3 + 15*A*a*c^2*d^4*e^3 + 105*(x*e + d)^3*B*a^2*c*e^4 - 105*(x*e + d)^2*B*a^2*c*d*e^4 + 63*(x*e + d)*B*a^2
*c*d^2*e^4 - 15*B*a^2*c*d^3*e^4 + 35*(x*e + d)^2*A*a^2*c*e^5 - 42*(x*e + d)*A*a^2*c*d*e^5 + 15*A*a^2*c*d^2*e^5
+ 7*(x*e + d)*B*a^3*e^6 - 5*B*a^3*d*e^6 + 5*A*a^3*e^7)*e^(-8)/(x*e + d)^(7/2)