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

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

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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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.28, size = 375, normalized size = 1.08 \begin {gather*} \frac {22 A e \left (-105 a^3 e^6+315 a^2 c e^4 \left (8 d^2+12 d e x+3 e^2 x^2\right )+63 a c^2 e^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+5 c^3 \left (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\right )\right )-10 B \left (231 a^3 e^6 (2 d+3 e x)+693 a^2 c e^4 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )+99 a c^2 e^2 \left (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\right )+7 c^3 \left (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\right )\right )}{3465 e^8 (d+e x)^{3/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.23, size = 573, normalized size = 1.66 \begin {gather*} \frac {2 \left (-1155 a^3 A e^7-3465 a^3 B e^6 (d+e x)+1155 a^3 B d e^6-3465 a^2 A c d^2 e^5+20790 a^2 A c d e^5 (d+e x)+10395 a^2 A c e^5 (d+e x)^2+3465 a^2 B c d^3 e^4-31185 a^2 B c d^2 e^4 (d+e x)-31185 a^2 B c d e^4 (d+e x)^2+3465 a^2 B c e^4 (d+e x)^3-3465 a A c^2 d^4 e^3+41580 a A c^2 d^3 e^3 (d+e x)+62370 a A c^2 d^2 e^3 (d+e x)^2-13860 a A c^2 d e^3 (d+e x)^3+2079 a A c^2 e^3 (d+e x)^4+3465 a B c^2 d^5 e^2-51975 a B c^2 d^4 e^2 (d+e x)-103950 a B c^2 d^3 e^2 (d+e x)^2+34650 a B c^2 d^2 e^2 (d+e x)^3-10395 a B c^2 d e^2 (d+e x)^4+1485 a B c^2 e^2 (d+e x)^5-1155 A c^3 d^6 e+20790 A c^3 d^5 e (d+e x)+51975 A c^3 d^4 e (d+e x)^2-23100 A c^3 d^3 e (d+e x)^3+10395 A c^3 d^2 e (d+e x)^4-2970 A c^3 d e (d+e x)^5+385 A c^3 e (d+e x)^6+1155 B c^3 d^7-24255 B c^3 d^6 (d+e x)-72765 B c^3 d^5 (d+e x)^2+40425 B c^3 d^4 (d+e x)^3-24255 B c^3 d^3 (d+e x)^4+10395 B c^3 d^2 (d+e x)^5-2695 B c^3 d (d+e x)^6+315 B c^3 (d+e x)^7\right )}{3465 e^8 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1155*B*c^3*d^7 - 1155*A*c^3*d^6*e + 3465*a*B*c^2*d^5*e^2 - 3465*a*A*c^2*d^4*e^3 + 3465*a^2*B*c*d^3*e^4 - 3
465*a^2*A*c*d^2*e^5 + 1155*a^3*B*d*e^6 - 1155*a^3*A*e^7 - 24255*B*c^3*d^6*(d + e*x) + 20790*A*c^3*d^5*e*(d + e
*x) - 51975*a*B*c^2*d^4*e^2*(d + e*x) + 41580*a*A*c^2*d^3*e^3*(d + e*x) - 31185*a^2*B*c*d^2*e^4*(d + e*x) + 20
790*a^2*A*c*d*e^5*(d + e*x) - 3465*a^3*B*e^6*(d + e*x) - 72765*B*c^3*d^5*(d + e*x)^2 + 51975*A*c^3*d^4*e*(d +
e*x)^2 - 103950*a*B*c^2*d^3*e^2*(d + e*x)^2 + 62370*a*A*c^2*d^2*e^3*(d + e*x)^2 - 31185*a^2*B*c*d*e^4*(d + e*x
)^2 + 10395*a^2*A*c*e^5*(d + e*x)^2 + 40425*B*c^3*d^4*(d + e*x)^3 - 23100*A*c^3*d^3*e*(d + e*x)^3 + 34650*a*B*
c^2*d^2*e^2*(d + e*x)^3 - 13860*a*A*c^2*d*e^3*(d + e*x)^3 + 3465*a^2*B*c*e^4*(d + e*x)^3 - 24255*B*c^3*d^3*(d
+ e*x)^4 + 10395*A*c^3*d^2*e*(d + e*x)^4 - 10395*a*B*c^2*d*e^2*(d + e*x)^4 + 2079*a*A*c^2*e^3*(d + e*x)^4 + 10
395*B*c^3*d^2*(d + e*x)^5 - 2970*A*c^3*d*e*(d + e*x)^5 + 1485*a*B*c^2*e^2*(d + e*x)^5 - 2695*B*c^3*d*(d + e*x)
^6 + 385*A*c^3*e*(d + e*x)^6 + 315*B*c^3*(d + e*x)^7))/(3465*e^8*(d + e*x)^(3/2))

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fricas [A]  time = 0.42, size = 475, normalized size = 1.37 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.23, size = 599, normalized size = 1.73 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.06, size = 489, normalized size = 1.41 \begin {gather*} -\frac {2 \left (-315 B \,c^{3} x^{7} e^{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 \,d^{2} a^{2} c \,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 \,d^{3} a^{2} c \,e^{4}+126720 B a \,c^{2} d^{5} e^{2}+71680 B \,c^{3} d^{7}\right )}{3465 \left (e x +d \right )^{\frac {3}{2}} e^{8}} \end {gather*}

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 = 0.65, size = 459, normalized size = 1.33 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.83, size = 434, normalized size = 1.25 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (6\,B\,a^2\,c\,e^4+60\,B\,a\,c^2\,d^2\,e^2-24\,A\,a\,c^2\,d\,e^3+70\,B\,c^3\,d^4-40\,A\,c^3\,d^3\,e\right )}{3\,e^8}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{7\,e^8}-\frac {\left (d+e\,x\right )\,\left (2\,B\,a^3\,e^6+18\,B\,a^2\,c\,d^2\,e^4-12\,A\,a^2\,c\,d\,e^5+30\,B\,a\,c^2\,d^4\,e^2-24\,A\,a\,c^2\,d^3\,e^3+14\,B\,c^3\,d^6-12\,A\,c^3\,d^5\,e\right )+\frac {2\,A\,a^3\,e^7}{3}-\frac {2\,B\,c^3\,d^7}{3}-\frac {2\,B\,a^3\,d\,e^6}{3}+\frac {2\,A\,c^3\,d^6\,e}{3}+2\,A\,a\,c^2\,d^4\,e^3+2\,A\,a^2\,c\,d^2\,e^5-2\,B\,a\,c^2\,d^5\,e^2-2\,B\,a^2\,c\,d^3\,e^4}{e^8\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {2\,c^2\,{\left (d+e\,x\right )}^{5/2}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{5\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {6\,c\,\left (c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}\,\left (-7\,B\,c\,d^3+5\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(3/2)*(70*B*c^3*d^4 + 6*B*a^2*c*e^4 - 40*A*c^3*d^3*e + 60*B*a*c^2*d^2*e^2 - 24*A*a*c^2*d*e^3))/(3*e
^8) + ((d + e*x)^(7/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/(7*e^8) - ((d + e*x)*(2*B*a^3*e^6 + 14*B
*c^3*d^6 - 12*A*c^3*d^5*e - 24*A*a*c^2*d^3*e^3 + 30*B*a*c^2*d^4*e^2 + 18*B*a^2*c*d^2*e^4 - 12*A*a^2*c*d*e^5) +
(2*A*a^3*e^7)/3 - (2*B*c^3*d^7)/3 - (2*B*a^3*d*e^6)/3 + (2*A*c^3*d^6*e)/3 + 2*A*a*c^2*d^4*e^3 + 2*A*a^2*c*d^2
*e^5 - 2*B*a*c^2*d^5*e^2 - 2*B*a^2*c*d^3*e^4)/(e^8*(d + e*x)^(3/2)) + (2*B*c^3*(d + e*x)^(11/2))/(11*e^8) + (2
*c^2*(d + e*x)^(5/2)*(3*A*a*e^3 - 35*B*c*d^3 - 15*B*a*d*e^2 + 15*A*c*d^2*e))/(5*e^8) + (2*c^3*(A*e - 7*B*d)*(d
+ e*x)^(9/2))/(9*e^8) + (6*c*(a*e^2 + c*d^2)*(d + e*x)^(1/2)*(A*a*e^3 - 7*B*c*d^3 - 3*B*a*d*e^2 + 5*A*c*d^2*e
))/e^8

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sympy [A]  time = 113.12, size = 406, normalized size = 1.17 \begin {gather*} \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 {gather*}

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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