3.1446 \(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^{11/2}} \, dx\)

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

[Out]

2/9*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^(9/2)-2/7*(a*e^2+c*d^2)^2*(-6*A*c*d*e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d
)^(7/2)+6/5*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/(e*x+d)^(5/2)+2/3*c*(4*A*c*d*e*(3
*a*e^2+5*c*d^2)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))/e^8/(e*x+d)^(3/2)-2/3*c^3*(-A*e+7*B*d)*(e*x+d)^(3/2)/
e^8+2/5*B*c^3*(e*x+d)^(5/2)/e^8+2*c^2*(-3*A*a*e^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3)/e^8/(e*x+d)^(1/2)+6*c^
2*(-2*A*c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(1/2)/e^8

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Rubi [A]  time = 0.16, antiderivative size = 346, normalized size of antiderivative = 1.00, 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} \[ \frac {2 c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8 (d+e x)^{3/2}}+\frac {6 c^2 \sqrt {d+e x} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac {2 c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 \sqrt {d+e x}}+\frac {6 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8 (d+e x)^{5/2}}-\frac {2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8 (d+e x)^{7/2}}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{9 e^8 (d+e x)^{9/2}}-\frac {2 c^3 (d+e x)^{3/2} (7 B d-A e)}{3 e^8}+\frac {2 B c^3 (d+e x)^{5/2}}{5 e^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(11/2),x]

[Out]

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

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Mathematica [A]  time = 0.30, size = 375, normalized size = 1.08 \[ \frac {2 B \left (-5 a^3 e^6 (2 d+9 e x)-3 a^2 c e^4 \left (16 d^3+72 d^2 e x+126 d e^2 x^2+105 e^3 x^3\right )+15 a c^2 e^2 \left (256 d^5+1152 d^4 e x+2016 d^3 e^2 x^2+1680 d^2 e^3 x^3+630 d e^4 x^4+63 e^5 x^5\right )+7 c^3 \left (2048 d^7+9216 d^6 e x+16128 d^5 e^2 x^2+13440 d^4 e^3 x^3+5040 d^3 e^4 x^4+504 d^2 e^5 x^5-42 d e^6 x^6+9 e^7 x^7\right )\right )-2 A e \left (35 a^3 e^6+3 a^2 c e^4 \left (8 d^2+36 d e x+63 e^2 x^2\right )+3 a c^2 e^2 \left (128 d^4+576 d^3 e x+1008 d^2 e^2 x^2+840 d e^3 x^3+315 e^4 x^4\right )+5 c^3 \left (1024 d^6+4608 d^5 e x+8064 d^4 e^2 x^2+6720 d^3 e^3 x^3+2520 d^2 e^4 x^4+252 d e^5 x^5-21 e^6 x^6\right )\right )}{315 e^8 (d+e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(11/2),x]

[Out]

(-2*A*e*(35*a^3*e^6 + 3*a^2*c*e^4*(8*d^2 + 36*d*e*x + 63*e^2*x^2) + 3*a*c^2*e^2*(128*d^4 + 576*d^3*e*x + 1008*
d^2*e^2*x^2 + 840*d*e^3*x^3 + 315*e^4*x^4) + 5*c^3*(1024*d^6 + 4608*d^5*e*x + 8064*d^4*e^2*x^2 + 6720*d^3*e^3*
x^3 + 2520*d^2*e^4*x^4 + 252*d*e^5*x^5 - 21*e^6*x^6)) + 2*B*(-5*a^3*e^6*(2*d + 9*e*x) - 3*a^2*c*e^4*(16*d^3 +
72*d^2*e*x + 126*d*e^2*x^2 + 105*e^3*x^3) + 15*a*c^2*e^2*(256*d^5 + 1152*d^4*e*x + 2016*d^3*e^2*x^2 + 1680*d^2
*e^3*x^3 + 630*d*e^4*x^4 + 63*e^5*x^5) + 7*c^3*(2048*d^7 + 9216*d^6*e*x + 16128*d^5*e^2*x^2 + 13440*d^4*e^3*x^
3 + 5040*d^3*e^4*x^4 + 504*d^2*e^5*x^5 - 42*d*e^6*x^6 + 9*e^7*x^7)))/(315*e^8*(d + e*x)^(9/2))

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fricas [A]  time = 0.69, size = 509, normalized size = 1.47 \[ \frac {2 \, {\left (63 \, B c^{3} e^{7} x^{7} + 14336 \, B c^{3} d^{7} - 5120 \, A c^{3} d^{6} e + 3840 \, B a c^{2} d^{5} e^{2} - 384 \, A a c^{2} d^{4} e^{3} - 48 \, B a^{2} c d^{3} e^{4} - 24 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 35 \, A a^{3} e^{7} - 21 \, {\left (14 \, B c^{3} d e^{6} - 5 \, A c^{3} e^{7}\right )} x^{6} + 63 \, {\left (56 \, B c^{3} d^{2} e^{5} - 20 \, A c^{3} d e^{6} + 15 \, B a c^{2} e^{7}\right )} x^{5} + 315 \, {\left (112 \, B c^{3} d^{3} e^{4} - 40 \, A c^{3} d^{2} e^{5} + 30 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 105 \, {\left (896 \, B c^{3} d^{4} e^{3} - 320 \, A c^{3} d^{3} e^{4} + 240 \, B a c^{2} d^{2} e^{5} - 24 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 63 \, {\left (1792 \, B c^{3} d^{5} e^{2} - 640 \, A c^{3} d^{4} e^{3} + 480 \, B a c^{2} d^{3} e^{4} - 48 \, A a c^{2} d^{2} e^{5} - 6 \, B a^{2} c d e^{6} - 3 \, A a^{2} c e^{7}\right )} x^{2} + 9 \, {\left (7168 \, B c^{3} d^{6} e - 2560 \, A c^{3} d^{5} e^{2} + 1920 \, B a c^{2} d^{4} e^{3} - 192 \, A a c^{2} d^{3} e^{4} - 24 \, B a^{2} c d^{2} e^{5} - 12 \, A a^{2} c d e^{6} - 5 \, B a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

2/315*(63*B*c^3*e^7*x^7 + 14336*B*c^3*d^7 - 5120*A*c^3*d^6*e + 3840*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 - 24*A*a^2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 35*A*a^3*e^7 - 21*(14*B*c^3*d*e^6 - 5*A*c^3*e^7)*x^6
+ 63*(56*B*c^3*d^2*e^5 - 20*A*c^3*d*e^6 + 15*B*a*c^2*e^7)*x^5 + 315*(112*B*c^3*d^3*e^4 - 40*A*c^3*d^2*e^5 + 30
*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 + 105*(896*B*c^3*d^4*e^3 - 320*A*c^3*d^3*e^4 + 240*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 + 63*(1792*B*c^3*d^5*e^2 - 640*A*c^3*d^4*e^3 + 480*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 - 3*A*a^2*c*e^7)*x^2 + 9*(7168*B*c^3*d^6*e - 2560*A*c^3*d^5*e^2 + 1920*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 - 12*A*a^2*c*d*e^6 - 5*B*a^3*e^7)*x)*sqrt(e*x + d)/(e^13*x^5
+ 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*e^8)

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giac [A]  time = 0.24, size = 595, normalized size = 1.72 \[ \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{3} e^{32} - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{3} d e^{32} + 315 \, \sqrt {x e + d} B c^{3} d^{2} e^{32} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{3} e^{33} - 90 \, \sqrt {x e + d} A c^{3} d e^{33} + 45 \, \sqrt {x e + d} B a c^{2} e^{34}\right )} e^{\left (-40\right )} + \frac {2 \, {\left (11025 \, {\left (x e + d\right )}^{4} B c^{3} d^{3} - 3675 \, {\left (x e + d\right )}^{3} B c^{3} d^{4} + 1323 \, {\left (x e + d\right )}^{2} B c^{3} d^{5} - 315 \, {\left (x e + d\right )} B c^{3} d^{6} + 35 \, B c^{3} d^{7} - 4725 \, {\left (x e + d\right )}^{4} A c^{3} d^{2} e + 2100 \, {\left (x e + d\right )}^{3} A c^{3} d^{3} e - 945 \, {\left (x e + d\right )}^{2} A c^{3} d^{4} e + 270 \, {\left (x e + d\right )} A c^{3} d^{5} e - 35 \, A c^{3} d^{6} e + 4725 \, {\left (x e + d\right )}^{4} B a c^{2} d e^{2} - 3150 \, {\left (x e + d\right )}^{3} B a c^{2} d^{2} e^{2} + 1890 \, {\left (x e + d\right )}^{2} B a c^{2} d^{3} e^{2} - 675 \, {\left (x e + d\right )} B a c^{2} d^{4} e^{2} + 105 \, B a c^{2} d^{5} e^{2} - 945 \, {\left (x e + d\right )}^{4} A a c^{2} e^{3} + 1260 \, {\left (x e + d\right )}^{3} A a c^{2} d e^{3} - 1134 \, {\left (x e + d\right )}^{2} A a c^{2} d^{2} e^{3} + 540 \, {\left (x e + d\right )} A a c^{2} d^{3} e^{3} - 105 \, A a c^{2} d^{4} e^{3} - 315 \, {\left (x e + d\right )}^{3} B a^{2} c e^{4} + 567 \, {\left (x e + d\right )}^{2} B a^{2} c d e^{4} - 405 \, {\left (x e + d\right )} B a^{2} c d^{2} e^{4} + 105 \, B a^{2} c d^{3} e^{4} - 189 \, {\left (x e + d\right )}^{2} A a^{2} c e^{5} + 270 \, {\left (x e + d\right )} A a^{2} c d e^{5} - 105 \, A a^{2} c d^{2} e^{5} - 45 \, {\left (x e + d\right )} B a^{3} e^{6} + 35 \, B a^{3} d e^{6} - 35 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{315 \, {\left (x e + d\right )}^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*B*c^3*e^32 - 35*(x*e + d)^(3/2)*B*c^3*d*e^32 + 315*sqrt(x*e + d)*B*c^3*d^2*e^32 + 5*(x
*e + d)^(3/2)*A*c^3*e^33 - 90*sqrt(x*e + d)*A*c^3*d*e^33 + 45*sqrt(x*e + d)*B*a*c^2*e^34)*e^(-40) + 2/315*(110
25*(x*e + d)^4*B*c^3*d^3 - 3675*(x*e + d)^3*B*c^3*d^4 + 1323*(x*e + d)^2*B*c^3*d^5 - 315*(x*e + d)*B*c^3*d^6 +
35*B*c^3*d^7 - 4725*(x*e + d)^4*A*c^3*d^2*e + 2100*(x*e + d)^3*A*c^3*d^3*e - 945*(x*e + d)^2*A*c^3*d^4*e + 27
0*(x*e + d)*A*c^3*d^5*e - 35*A*c^3*d^6*e + 4725*(x*e + d)^4*B*a*c^2*d*e^2 - 3150*(x*e + d)^3*B*a*c^2*d^2*e^2 +
1890*(x*e + d)^2*B*a*c^2*d^3*e^2 - 675*(x*e + d)*B*a*c^2*d^4*e^2 + 105*B*a*c^2*d^5*e^2 - 945*(x*e + d)^4*A*a*
c^2*e^3 + 1260*(x*e + d)^3*A*a*c^2*d*e^3 - 1134*(x*e + d)^2*A*a*c^2*d^2*e^3 + 540*(x*e + d)*A*a*c^2*d^3*e^3 -
105*A*a*c^2*d^4*e^3 - 315*(x*e + d)^3*B*a^2*c*e^4 + 567*(x*e + d)^2*B*a^2*c*d*e^4 - 405*(x*e + d)*B*a^2*c*d^2*
e^4 + 105*B*a^2*c*d^3*e^4 - 189*(x*e + d)^2*A*a^2*c*e^5 + 270*(x*e + d)*A*a^2*c*d*e^5 - 105*A*a^2*c*d^2*e^5 -
45*(x*e + d)*B*a^3*e^6 + 35*B*a^3*d*e^6 - 35*A*a^3*e^7)*e^(-8)/(x*e + d)^(9/2)

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maple [A]  time = 0.05, size = 489, normalized size = 1.41 \[ -\frac {2 \left (-63 B \,c^{3} x^{7} e^{7}-105 A \,c^{3} e^{7} x^{6}+294 B \,c^{3} d \,e^{6} x^{6}+1260 A \,c^{3} d \,e^{6} x^{5}-945 B a \,c^{2} e^{7} x^{5}-3528 B \,c^{3} d^{2} e^{5} x^{5}+945 A a \,c^{2} e^{7} x^{4}+12600 A \,c^{3} d^{2} e^{5} x^{4}-9450 B a \,c^{2} d \,e^{6} x^{4}-35280 B \,c^{3} d^{3} e^{4} x^{4}+2520 A a \,c^{2} d \,e^{6} x^{3}+33600 A \,c^{3} d^{3} e^{4} x^{3}+315 B \,a^{2} c \,e^{7} x^{3}-25200 B a \,c^{2} d^{2} e^{5} x^{3}-94080 B \,c^{3} d^{4} e^{3} x^{3}+189 A \,a^{2} c \,e^{7} x^{2}+3024 A a \,c^{2} d^{2} e^{5} x^{2}+40320 A \,c^{3} d^{4} e^{3} x^{2}+378 B \,a^{2} c d \,e^{6} x^{2}-30240 B a \,c^{2} d^{3} e^{4} x^{2}-112896 B \,c^{3} d^{5} e^{2} x^{2}+108 A \,a^{2} c d \,e^{6} x +1728 A a \,c^{2} d^{3} e^{4} x +23040 A \,c^{3} d^{5} e^{2} x +45 B \,a^{3} e^{7} x +216 B \,a^{2} c \,d^{2} e^{5} x -17280 B a \,c^{2} d^{4} e^{3} x -64512 B \,c^{3} d^{6} e x +35 A \,a^{3} e^{7}+24 A \,d^{2} a^{2} c \,e^{5}+384 A a \,c^{2} d^{4} e^{3}+5120 A \,c^{3} d^{6} e +10 B \,a^{3} d \,e^{6}+48 B \,d^{3} a^{2} c \,e^{4}-3840 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{315 \left (e x +d \right )^{\frac {9}{2}} e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x)

[Out]

-2/315/(e*x+d)^(9/2)*(-63*B*c^3*e^7*x^7-105*A*c^3*e^7*x^6+294*B*c^3*d*e^6*x^6+1260*A*c^3*d*e^6*x^5-945*B*a*c^2
*e^7*x^5-3528*B*c^3*d^2*e^5*x^5+945*A*a*c^2*e^7*x^4+12600*A*c^3*d^2*e^5*x^4-9450*B*a*c^2*d*e^6*x^4-35280*B*c^3
*d^3*e^4*x^4+2520*A*a*c^2*d*e^6*x^3+33600*A*c^3*d^3*e^4*x^3+315*B*a^2*c*e^7*x^3-25200*B*a*c^2*d^2*e^5*x^3-9408
0*B*c^3*d^4*e^3*x^3+189*A*a^2*c*e^7*x^2+3024*A*a*c^2*d^2*e^5*x^2+40320*A*c^3*d^4*e^3*x^2+378*B*a^2*c*d*e^6*x^2
-30240*B*a*c^2*d^3*e^4*x^2-112896*B*c^3*d^5*e^2*x^2+108*A*a^2*c*d*e^6*x+1728*A*a*c^2*d^3*e^4*x+23040*A*c^3*d^5
*e^2*x+45*B*a^3*e^7*x+216*B*a^2*c*d^2*e^5*x-17280*B*a*c^2*d^4*e^3*x-64512*B*c^3*d^6*e*x+35*A*a^3*e^7+24*A*a^2*
c*d^2*e^5+384*A*a*c^2*d^4*e^3+5120*A*c^3*d^6*e+10*B*a^3*d*e^6+48*B*a^2*c*d^3*e^4-3840*B*a*c^2*d^5*e^2-14336*B*
c^3*d^7)/e^8

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maxima [A]  time = 0.58, size = 461, normalized size = 1.33 \[ \frac {2 \, {\left (\frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{3} - 5 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 45 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} \sqrt {e x + d}\right )}}{e^{7}} + \frac {35 \, B c^{3} d^{7} - 35 \, A c^{3} d^{6} e + 105 \, B a c^{2} d^{5} e^{2} - 105 \, A a c^{2} d^{4} e^{3} + 105 \, B a^{2} c d^{3} e^{4} - 105 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} - 35 \, A a^{3} e^{7} + 315 \, {\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} {\left (e x + d\right )}^{4} - 105 \, {\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} {\left (e x + d\right )}^{3} + 189 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} {\left (e x + d\right )}^{2} - 45 \, {\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {9}{2}} e^{7}}\right )}}{315 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.89, size = 454, normalized size = 1.31 \[ \frac {{\left (d+e\,x\right )}^4\,\left (70\,B\,c^3\,d^3-30\,A\,c^3\,d^2\,e+30\,B\,a\,c^2\,d\,e^2-6\,A\,a\,c^2\,e^3\right )-\left (d+e\,x\right )\,\left (\frac {2\,B\,a^3\,e^6}{7}+\frac {18\,B\,a^2\,c\,d^2\,e^4}{7}-\frac {12\,A\,a^2\,c\,d\,e^5}{7}+\frac {30\,B\,a\,c^2\,d^4\,e^2}{7}-\frac {24\,A\,a\,c^2\,d^3\,e^3}{7}+2\,B\,c^3\,d^6-\frac {12\,A\,c^3\,d^5\,e}{7}\right )-{\left (d+e\,x\right )}^3\,\left (2\,B\,a^2\,c\,e^4+20\,B\,a\,c^2\,d^2\,e^2-8\,A\,a\,c^2\,d\,e^3+\frac {70\,B\,c^3\,d^4}{3}-\frac {40\,A\,c^3\,d^3\,e}{3}\right )+{\left (d+e\,x\right )}^2\,\left (\frac {18\,B\,a^2\,c\,d\,e^4}{5}-\frac {6\,A\,a^2\,c\,e^5}{5}+12\,B\,a\,c^2\,d^3\,e^2-\frac {36\,A\,a\,c^2\,d^2\,e^3}{5}+\frac {42\,B\,c^3\,d^5}{5}-6\,A\,c^3\,d^4\,e\right )-\frac {2\,A\,a^3\,e^7}{9}+\frac {2\,B\,c^3\,d^7}{9}+\frac {2\,B\,a^3\,d\,e^6}{9}-\frac {2\,A\,c^3\,d^6\,e}{9}-\frac {2\,A\,a\,c^2\,d^4\,e^3}{3}-\frac {2\,A\,a^2\,c\,d^2\,e^5}{3}+\frac {2\,B\,a\,c^2\,d^5\,e^2}{3}+\frac {2\,B\,a^2\,c\,d^3\,e^4}{3}}{e^8\,{\left (d+e\,x\right )}^{9/2}}+\frac {\sqrt {d+e\,x}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{e^8}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^(11/2),x)

[Out]

((d + e*x)^4*(70*B*c^3*d^3 - 6*A*a*c^2*e^3 - 30*A*c^3*d^2*e + 30*B*a*c^2*d*e^2) - (d + e*x)*((2*B*a^3*e^6)/7 +
2*B*c^3*d^6 - (12*A*c^3*d^5*e)/7 - (24*A*a*c^2*d^3*e^3)/7 + (30*B*a*c^2*d^4*e^2)/7 + (18*B*a^2*c*d^2*e^4)/7 -
(12*A*a^2*c*d*e^5)/7) - (d + e*x)^3*((70*B*c^3*d^4)/3 + 2*B*a^2*c*e^4 - (40*A*c^3*d^3*e)/3 + 20*B*a*c^2*d^2*e
^2 - 8*A*a*c^2*d*e^3) + (d + e*x)^2*((42*B*c^3*d^5)/5 - (6*A*a^2*c*e^5)/5 - 6*A*c^3*d^4*e - (36*A*a*c^2*d^2*e^
3)/5 + 12*B*a*c^2*d^3*e^2 + (18*B*a^2*c*d*e^4)/5) - (2*A*a^3*e^7)/9 + (2*B*c^3*d^7)/9 + (2*B*a^3*d*e^6)/9 - (2
*A*c^3*d^6*e)/9 - (2*A*a*c^2*d^4*e^3)/3 - (2*A*a^2*c*d^2*e^5)/3 + (2*B*a*c^2*d^5*e^2)/3 + (2*B*a^2*c*d^3*e^4)/
3)/(e^8*(d + e*x)^(9/2)) + ((d + e*x)^(1/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/e^8 + (2*B*c^3*(d +
e*x)^(5/2))/(5*e^8) + (2*c^3*(A*e - 7*B*d)*(d + e*x)^(3/2))/(3*e^8)

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sympy [A]  time = 16.44, size = 3952, normalized size = 11.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(11/2),x)

[Out]

Piecewise((-70*A*a**3*e**7/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**
2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 48*A*a**2*c*d**2*e**5/(315
*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*
x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 216*A*a**2*c*d*e**6*x/(315*d**4*e**8*sqrt(d + e*x) + 1260
*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*
x**4*sqrt(d + e*x)) - 378*A*a**2*c*e**7*x**2/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1
890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 768*A*a*
c**2*d**4*e**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e
*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 3456*A*a*c**2*d**3*e**4*x/(315*d**4*e*
*8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqr
t(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 6048*A*a*c**2*d**2*e**5*x**2/(315*d**4*e**8*sqrt(d + e*x) + 1260*
d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x
**4*sqrt(d + e*x)) - 5040*A*a*c**2*d*e**6*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) +
1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 1890*A
*a*c**2*e**7*x**4/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d
+ e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 10240*A*c**3*d**6*e/(315*d**4*e**8*
sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d
+ e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 46080*A*c**3*d**5*e**2*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e*
*9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqr
t(d + e*x)) - 80640*A*c**3*d**4*e**3*x**2/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890
*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 67200*A*c**
3*d**3*e**4*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d +
e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 25200*A*c**3*d**2*e**5*x**4/(315*d**
4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3
*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 2520*A*c**3*d*e**6*x**5/(315*d**4*e**8*sqrt(d + e*x) + 1260*d
**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x*
*4*sqrt(d + e*x)) + 210*A*c**3*e**7*x**6/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*
d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 20*B*a**3*d*
e**6/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260
*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 90*B*a**3*e**7*x/(315*d**4*e**8*sqrt(d + e*x) +
1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e*
*12*x**4*sqrt(d + e*x)) - 96*B*a**2*c*d**3*e**4/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x)
+ 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 432*B
*a**2*c*d**2*e**5*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(
d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 756*B*a**2*c*d*e**6*x**2/(315*d**
4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3
*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) - 630*B*a**2*c*e**7*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d*
*3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**
4*sqrt(d + e*x)) + 7680*B*a*c**2*d**5*e**2/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 189
0*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 34560*B*a*
c**2*d**4*e**3*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d +
e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 60480*B*a*c**2*d**3*e**4*x**2/(315*d
**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x*
*3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 50400*B*a*c**2*d**2*e**5*x**3/(315*d**4*e**8*sqrt(d + e*x)
+ 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*
e**12*x**4*sqrt(d + e*x)) + 18900*B*a*c**2*d*e**6*x**4/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d
+ e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x))
+ 1890*B*a*c**2*e**7*x**5/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2
*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 28672*B*c**3*d**7/(315*d**4
*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*
sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 129024*B*c**3*d**6*e*x/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**
3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4
*sqrt(d + e*x)) + 225792*B*c**3*d**5*e**2*x**2/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) +
1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 188160
*B*c**3*d**4*e**3*x**3/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sq
rt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 70560*B*c**3*d**3*e**4*x**4/(3
15*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**1
1*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 7056*B*c**3*d**2*e**5*x**5/(315*d**4*e**8*sqrt(d + e*x)
+ 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315
*e**12*x**4*sqrt(d + e*x)) - 588*B*c**3*d*e**6*x**6/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e
*x) + 1890*d**2*e**10*x**2*sqrt(d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*sqrt(d + e*x)) + 1
26*B*c**3*e**7*x**7/(315*d**4*e**8*sqrt(d + e*x) + 1260*d**3*e**9*x*sqrt(d + e*x) + 1890*d**2*e**10*x**2*sqrt(
d + e*x) + 1260*d*e**11*x**3*sqrt(d + e*x) + 315*e**12*x**4*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**(11/2), True))

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