3.13.71 \(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=342 \[ \frac {2 c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 \sqrt {d+e x}}+\frac {2 c^2 (d+e x)^{3/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}-\frac {2 c^2 \sqrt {d+e x} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8}-\frac {2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^{5/2}}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{7 e^8 (d+e x)^{7/2}}+\frac {2 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^{3/2}}-\frac {2 c^3 (d+e x)^{5/2} (7 B d-A e)}{5 e^8}+\frac {2 B c^3 (d+e x)^{7/2}}{7 e^8} \]

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Rubi [A]  time = 0.16, antiderivative size = 342, 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 \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.29, size = 372, normalized size = 1.09 \begin {gather*} \frac {2 A e \left (-5 a^3 e^6-a^2 c e^4 \left (8 d^2+28 d e x+35 e^2 x^2\right )+3 a c^2 e^2 \left (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\right )+c^3 \left (1024 d^6+3584 d^5 e x+4480 d^4 e^2 x^2+2240 d^3 e^3 x^3+280 d^2 e^4 x^4-28 d e^5 x^5+7 e^6 x^6\right )\right )-2 B \left (a^3 e^6 (2 d+7 e x)+3 a^2 c e^4 \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )+5 a c^2 e^2 \left (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\right )+c^3 \left (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\right )\right )}{35 e^8 (d+e x)^{7/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.30, size = 573, normalized size = 1.68 \begin {gather*} \frac {2 \left (-5 a^3 A e^7-7 a^3 B e^6 (d+e x)+5 a^3 B d e^6-15 a^2 A c d^2 e^5+42 a^2 A c d e^5 (d+e x)-35 a^2 A c e^5 (d+e x)^2+15 a^2 B c d^3 e^4-63 a^2 B c d^2 e^4 (d+e x)+105 a^2 B c d e^4 (d+e x)^2-105 a^2 B c e^4 (d+e x)^3-15 a A c^2 d^4 e^3+84 a A c^2 d^3 e^3 (d+e x)-210 a A c^2 d^2 e^3 (d+e x)^2+420 a A c^2 d e^3 (d+e x)^3+105 a A c^2 e^3 (d+e x)^4+15 a B c^2 d^5 e^2-105 a B c^2 d^4 e^2 (d+e x)+350 a B c^2 d^3 e^2 (d+e x)^2-1050 a B c^2 d^2 e^2 (d+e x)^3-525 a B c^2 d e^2 (d+e x)^4+35 a B c^2 e^2 (d+e x)^5-5 A c^3 d^6 e+42 A c^3 d^5 e (d+e x)-175 A c^3 d^4 e (d+e x)^2+700 A c^3 d^3 e (d+e x)^3+525 A c^3 d^2 e (d+e x)^4-70 A c^3 d e (d+e x)^5+7 A c^3 e (d+e x)^6+5 B c^3 d^7-49 B c^3 d^6 (d+e x)+245 B c^3 d^5 (d+e x)^2-1225 B c^3 d^4 (d+e x)^3-1225 B c^3 d^3 (d+e x)^4+245 B c^3 d^2 (d+e x)^5-49 B c^3 d (d+e x)^6+5 B c^3 (d+e x)^7\right )}{35 e^8 (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(9/2),x]

[Out]

(2*(5*B*c^3*d^7 - 5*A*c^3*d^6*e + 15*a*B*c^2*d^5*e^2 - 15*a*A*c^2*d^4*e^3 + 15*a^2*B*c*d^3*e^4 - 15*a^2*A*c*d^
2*e^5 + 5*a^3*B*d*e^6 - 5*a^3*A*e^7 - 49*B*c^3*d^6*(d + e*x) + 42*A*c^3*d^5*e*(d + e*x) - 105*a*B*c^2*d^4*e^2*
(d + e*x) + 84*a*A*c^2*d^3*e^3*(d + e*x) - 63*a^2*B*c*d^2*e^4*(d + e*x) + 42*a^2*A*c*d*e^5*(d + e*x) - 7*a^3*B
*e^6*(d + e*x) + 245*B*c^3*d^5*(d + e*x)^2 - 175*A*c^3*d^4*e*(d + e*x)^2 + 350*a*B*c^2*d^3*e^2*(d + e*x)^2 - 2
10*a*A*c^2*d^2*e^3*(d + e*x)^2 + 105*a^2*B*c*d*e^4*(d + e*x)^2 - 35*a^2*A*c*e^5*(d + e*x)^2 - 1225*B*c^3*d^4*(
d + e*x)^3 + 700*A*c^3*d^3*e*(d + e*x)^3 - 1050*a*B*c^2*d^2*e^2*(d + e*x)^3 + 420*a*A*c^2*d*e^3*(d + e*x)^3 -
105*a^2*B*c*e^4*(d + e*x)^3 - 1225*B*c^3*d^3*(d + e*x)^4 + 525*A*c^3*d^2*e*(d + e*x)^4 - 525*a*B*c^2*d*e^2*(d
+ e*x)^4 + 105*a*A*c^2*e^3*(d + e*x)^4 + 245*B*c^3*d^2*(d + e*x)^5 - 70*A*c^3*d*e*(d + e*x)^5 + 35*a*B*c^2*e^2
*(d + e*x)^5 - 49*B*c^3*d*(d + e*x)^6 + 7*A*c^3*e*(d + e*x)^6 + 5*B*c^3*(d + e*x)^7))/(35*e^8*(d + e*x)^(7/2))

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fricas [A]  time = 0.43, size = 496, normalized size = 1.45 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.25, size = 597, normalized size = 1.75 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 489, normalized size = 1.43 \begin {gather*} -\frac {2 \left (-5 B \,c^{3} x^{7} e^{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}-840 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 \,d^{2} a^{2} c \,e^{5}-384 A a \,c^{2} d^{4} e^{3}-1024 A \,c^{3} d^{6} e +2 B \,a^{3} d \,e^{6}+48 B \,d^{3} a^{2} c \,e^{4}+1280 B a \,c^{2} d^{5} e^{2}+2048 B \,c^{3} d^{7}\right )}{35 \left (e x +d \right )^{\frac {7}{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)^(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 = 0.63, size = 460, normalized size = 1.35 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.85, size = 452, normalized size = 1.32 \begin {gather*} \frac {{\left (d+e\,x\right )}^{3/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{3\,e^8}-\frac {\left (d+e\,x\right )\,\left (\frac {2\,B\,a^3\,e^6}{5}+\frac {18\,B\,a^2\,c\,d^2\,e^4}{5}-\frac {12\,A\,a^2\,c\,d\,e^5}{5}+6\,B\,a\,c^2\,d^4\,e^2-\frac {24\,A\,a\,c^2\,d^3\,e^3}{5}+\frac {14\,B\,c^3\,d^6}{5}-\frac {12\,A\,c^3\,d^5\,e}{5}\right )+{\left (d+e\,x\right )}^3\,\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 )-{\left (d+e\,x\right )}^2\,\left (6\,B\,a^2\,c\,d\,e^4-2\,A\,a^2\,c\,e^5+20\,B\,a\,c^2\,d^3\,e^2-12\,A\,a\,c^2\,d^2\,e^3+14\,B\,c^3\,d^5-10\,A\,c^3\,d^4\,e\right )+\frac {2\,A\,a^3\,e^7}{7}-\frac {2\,B\,c^3\,d^7}{7}-\frac {2\,B\,a^3\,d\,e^6}{7}+\frac {2\,A\,c^3\,d^6\,e}{7}+\frac {6\,A\,a\,c^2\,d^4\,e^3}{7}+\frac {6\,A\,a^2\,c\,d^2\,e^5}{7}-\frac {6\,B\,a\,c^2\,d^5\,e^2}{7}-\frac {6\,B\,a^2\,c\,d^3\,e^4}{7}}{e^8\,{\left (d+e\,x\right )}^{7/2}}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8}+\frac {2\,c^2\,\sqrt {d+e\,x}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,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)^(9/2),x)

[Out]

((d + e*x)^(3/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/(3*e^8) - ((d + e*x)*((2*B*a^3*e^6)/5 + (14*B*
c^3*d^6)/5 - (12*A*c^3*d^5*e)/5 - (24*A*a*c^2*d^3*e^3)/5 + 6*B*a*c^2*d^4*e^2 + (18*B*a^2*c*d^2*e^4)/5 - (12*A*
a^2*c*d*e^5)/5) + (d + e*x)^3*(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) - (d + e*x)^2*(14*B*c^3*d^5 - 2*A*a^2*c*e^5 - 10*A*c^3*d^4*e - 12*A*a*c^2*d^2*e^3 + 20*B*a*c^2*d^3*e^2
+ 6*B*a^2*c*d*e^4) + (2*A*a^3*e^7)/7 - (2*B*c^3*d^7)/7 - (2*B*a^3*d*e^6)/7 + (2*A*c^3*d^6*e)/7 + (6*A*a*c^2*d
^4*e^3)/7 + (6*A*a^2*c*d^2*e^5)/7 - (6*B*a*c^2*d^5*e^2)/7 - (6*B*a^2*c*d^3*e^4)/7)/(e^8*(d + e*x)^(7/2)) + (2*
B*c^3*(d + e*x)^(7/2))/(7*e^8) + (2*c^2*(d + e*x)^(1/2)*(3*A*a*e^3 - 35*B*c*d^3 - 15*B*a*d*e^2 + 15*A*c*d^2*e)
)/e^8 + (2*c^3*(A*e - 7*B*d)*(d + e*x)^(5/2))/(5*e^8)

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sympy [A]  time = 9.86, size = 3218, normalized size = 9.41

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)**(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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