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

Optimal. Leaf size=344 \[ -\frac {2 c (d+e x)^{5/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8}+\frac {2 c^2 (d+e x)^{9/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{3 e^8}-\frac {2 c^2 (d+e x)^{7/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{7 e^8}+\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 \sqrt {d+e x}}-\frac {2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac {2 c^3 (d+e x)^{11/2} (7 B d-A e)}{11 e^8}+\frac {2 B c^3 (d+e x)^{13/2}}{13 e^8} \]

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Rubi [A]  time = 0.16, antiderivative size = 344, 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)^{5/2} \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 )}{5 e^8}+\frac {2 c^2 (d+e x)^{9/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{3 e^8}-\frac {2 c^2 (d+e x)^{7/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{7 e^8}-\frac {2 c (d+e x)^{3/2} \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 \sqrt {d+e x} \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}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 \sqrt {d+e x}}-\frac {2 c^3 (d+e x)^{11/2} (7 B d-A e)}{11 e^8}+\frac {2 B c^3 (d+e x)^{13/2}}{13 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(e^8*Sqrt[d + e*x]) + (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)
*Sqrt[d + e*x])/e^8 - (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)*(d + e*x)^(3/2))/
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)^(5/2))/(5*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)^(7/2))/(7*e^8) + (2*c^2*(7*B*c*d
^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(9/2))/(3*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(11/2))/(11*e^8) + (2*B*c^
3*(d + e*x)^(13/2))/(13*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)^{3/2}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^{3/2}}+\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 \sqrt {d+e x}}+\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 ) \sqrt {d+e x}}{e^7}-\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 ) (d+e x)^{3/2}}{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)^{5/2}}{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)^{7/2}}{e^7}+\frac {c^3 (-7 B d+A e) (d+e x)^{9/2}}{e^7}+\frac {B c^3 (d+e x)^{11/2}}{e^7}\right ) \, dx\\ &=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^3}{e^8 \sqrt {d+e x}}+\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 ) \sqrt {d+e x}}{e^8}-\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 ) (d+e x)^{3/2}}{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)^{5/2}}{5 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)^{7/2}}{7 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)^{9/2}}{3 e^8}-\frac {2 c^3 (7 B d-A e) (d+e x)^{11/2}}{11 e^8}+\frac {2 B c^3 (d+e x)^{13/2}}{13 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 373, normalized size = 1.08 \begin {gather*} \frac {2 B \left (15015 a^3 e^6 (2 d+e x)+9009 a^2 c e^4 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+715 a c^2 e^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )+35 c^3 \left (2048 d^7+1024 d^6 e x-256 d^5 e^2 x^2+128 d^4 e^3 x^3-80 d^3 e^4 x^4+56 d^2 e^5 x^5-42 d e^6 x^6+33 e^7 x^7\right )\right )-26 A e \left (1155 a^3 e^6+1155 a^2 c e^4 \left (8 d^2+4 d e x-e^2 x^2\right )+99 a c^2 e^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )+5 c^3 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )}{15015 e^8 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-26*A*e*(1155*a^3*e^6 + 1155*a^2*c*e^4*(8*d^2 + 4*d*e*x - e^2*x^2) + 99*a*c^2*e^2*(128*d^4 + 64*d^3*e*x - 16*
d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) + 5*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40
*d^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6)) + 2*B*(15015*a^3*e^6*(2*d + e*x) + 9009*a^2*c*e^4*(16*d^3 + 8*d^2*e
*x - 2*d*e^2*x^2 + e^3*x^3) + 715*a*c^2*e^2*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^
4*x^4 + 7*e^5*x^5) + 35*c^3*(2048*d^7 + 1024*d^6*e*x - 256*d^5*e^2*x^2 + 128*d^4*e^3*x^3 - 80*d^3*e^4*x^4 + 56
*d^2*e^5*x^5 - 42*d*e^6*x^6 + 33*e^7*x^7)))/(15015*e^8*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.24, size = 573, normalized size = 1.67 \begin {gather*} \frac {2 \left (-15015 a^3 A e^7+15015 a^3 B e^6 (d+e x)+15015 a^3 B d e^6-45045 a^2 A c d^2 e^5-90090 a^2 A c d e^5 (d+e x)+15015 a^2 A c e^5 (d+e x)^2+45045 a^2 B c d^3 e^4+135135 a^2 B c d^2 e^4 (d+e x)-45045 a^2 B c d e^4 (d+e x)^2+9009 a^2 B c e^4 (d+e x)^3-45045 a A c^2 d^4 e^3-180180 a A c^2 d^3 e^3 (d+e x)+90090 a A c^2 d^2 e^3 (d+e x)^2-36036 a A c^2 d e^3 (d+e x)^3+6435 a A c^2 e^3 (d+e x)^4+45045 a B c^2 d^5 e^2+225225 a B c^2 d^4 e^2 (d+e x)-150150 a B c^2 d^3 e^2 (d+e x)^2+90090 a B c^2 d^2 e^2 (d+e x)^3-32175 a B c^2 d e^2 (d+e x)^4+5005 a B c^2 e^2 (d+e x)^5-15015 A c^3 d^6 e-90090 A c^3 d^5 e (d+e x)+75075 A c^3 d^4 e (d+e x)^2-60060 A c^3 d^3 e (d+e x)^3+32175 A c^3 d^2 e (d+e x)^4-10010 A c^3 d e (d+e x)^5+1365 A c^3 e (d+e x)^6+15015 B c^3 d^7+105105 B c^3 d^6 (d+e x)-105105 B c^3 d^5 (d+e x)^2+105105 B c^3 d^4 (d+e x)^3-75075 B c^3 d^3 (d+e x)^4+35035 B c^3 d^2 (d+e x)^5-9555 B c^3 d (d+e x)^6+1155 B c^3 (d+e x)^7\right )}{15015 e^8 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(15015*B*c^3*d^7 - 15015*A*c^3*d^6*e + 45045*a*B*c^2*d^5*e^2 - 45045*a*A*c^2*d^4*e^3 + 45045*a^2*B*c*d^3*e^
4 - 45045*a^2*A*c*d^2*e^5 + 15015*a^3*B*d*e^6 - 15015*a^3*A*e^7 + 105105*B*c^3*d^6*(d + e*x) - 90090*A*c^3*d^5
*e*(d + e*x) + 225225*a*B*c^2*d^4*e^2*(d + e*x) - 180180*a*A*c^2*d^3*e^3*(d + e*x) + 135135*a^2*B*c*d^2*e^4*(d
+ e*x) - 90090*a^2*A*c*d*e^5*(d + e*x) + 15015*a^3*B*e^6*(d + e*x) - 105105*B*c^3*d^5*(d + e*x)^2 + 75075*A*c
^3*d^4*e*(d + e*x)^2 - 150150*a*B*c^2*d^3*e^2*(d + e*x)^2 + 90090*a*A*c^2*d^2*e^3*(d + e*x)^2 - 45045*a^2*B*c*
d*e^4*(d + e*x)^2 + 15015*a^2*A*c*e^5*(d + e*x)^2 + 105105*B*c^3*d^4*(d + e*x)^3 - 60060*A*c^3*d^3*e*(d + e*x)
^3 + 90090*a*B*c^2*d^2*e^2*(d + e*x)^3 - 36036*a*A*c^2*d*e^3*(d + e*x)^3 + 9009*a^2*B*c*e^4*(d + e*x)^3 - 7507
5*B*c^3*d^3*(d + e*x)^4 + 32175*A*c^3*d^2*e*(d + e*x)^4 - 32175*a*B*c^2*d*e^2*(d + e*x)^4 + 6435*a*A*c^2*e^3*(
d + e*x)^4 + 35035*B*c^3*d^2*(d + e*x)^5 - 10010*A*c^3*d*e*(d + e*x)^5 + 5005*a*B*c^2*e^2*(d + e*x)^5 - 9555*B
*c^3*d*(d + e*x)^6 + 1365*A*c^3*e*(d + e*x)^6 + 1155*B*c^3*(d + e*x)^7))/(15015*e^8*Sqrt[d + e*x])

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fricas [A]  time = 0.43, size = 463, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left (1155 \, B c^{3} e^{7} x^{7} + 71680 \, B c^{3} d^{7} - 66560 \, A c^{3} d^{6} e + 183040 \, B a c^{2} d^{5} e^{2} - 164736 \, A a c^{2} d^{4} e^{3} + 144144 \, B a^{2} c d^{3} e^{4} - 120120 \, A a^{2} c d^{2} e^{5} + 30030 \, B a^{3} d e^{6} - 15015 \, A a^{3} e^{7} - 105 \, {\left (14 \, B c^{3} d e^{6} - 13 \, A c^{3} e^{7}\right )} x^{6} + 35 \, {\left (56 \, B c^{3} d^{2} e^{5} - 52 \, A c^{3} d e^{6} + 143 \, B a c^{2} e^{7}\right )} x^{5} - 5 \, {\left (560 \, B c^{3} d^{3} e^{4} - 520 \, A c^{3} d^{2} e^{5} + 1430 \, B a c^{2} d e^{6} - 1287 \, A a c^{2} e^{7}\right )} x^{4} + {\left (4480 \, B c^{3} d^{4} e^{3} - 4160 \, A c^{3} d^{3} e^{4} + 11440 \, B a c^{2} d^{2} e^{5} - 10296 \, A a c^{2} d e^{6} + 9009 \, B a^{2} c e^{7}\right )} x^{3} - {\left (8960 \, B c^{3} d^{5} e^{2} - 8320 \, A c^{3} d^{4} e^{3} + 22880 \, B a c^{2} d^{3} e^{4} - 20592 \, A a c^{2} d^{2} e^{5} + 18018 \, B a^{2} c d e^{6} - 15015 \, A a^{2} c e^{7}\right )} x^{2} + {\left (35840 \, B c^{3} d^{6} e - 33280 \, A c^{3} d^{5} e^{2} + 91520 \, B a c^{2} d^{4} e^{3} - 82368 \, A a c^{2} d^{3} e^{4} + 72072 \, B a^{2} c d^{2} e^{5} - 60060 \, A a^{2} c d e^{6} + 15015 \, B a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{15015 \, {\left (e^{9} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/15015*(1155*B*c^3*e^7*x^7 + 71680*B*c^3*d^7 - 66560*A*c^3*d^6*e + 183040*B*a*c^2*d^5*e^2 - 164736*A*a*c^2*d^
4*e^3 + 144144*B*a^2*c*d^3*e^4 - 120120*A*a^2*c*d^2*e^5 + 30030*B*a^3*d*e^6 - 15015*A*a^3*e^7 - 105*(14*B*c^3*
d*e^6 - 13*A*c^3*e^7)*x^6 + 35*(56*B*c^3*d^2*e^5 - 52*A*c^3*d*e^6 + 143*B*a*c^2*e^7)*x^5 - 5*(560*B*c^3*d^3*e^
4 - 520*A*c^3*d^2*e^5 + 1430*B*a*c^2*d*e^6 - 1287*A*a*c^2*e^7)*x^4 + (4480*B*c^3*d^4*e^3 - 4160*A*c^3*d^3*e^4
+ 11440*B*a*c^2*d^2*e^5 - 10296*A*a*c^2*d*e^6 + 9009*B*a^2*c*e^7)*x^3 - (8960*B*c^3*d^5*e^2 - 8320*A*c^3*d^4*e
^3 + 22880*B*a*c^2*d^3*e^4 - 20592*A*a*c^2*d^2*e^5 + 18018*B*a^2*c*d*e^6 - 15015*A*a^2*c*e^7)*x^2 + (35840*B*c
^3*d^6*e - 33280*A*c^3*d^5*e^2 + 91520*B*a*c^2*d^4*e^3 - 82368*A*a*c^2*d^3*e^4 + 72072*B*a^2*c*d^2*e^5 - 60060
*A*a^2*c*d*e^6 + 15015*B*a^3*e^7)*x)*sqrt(e*x + d)/(e^9*x + d*e^8)

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giac [A]  time = 0.29, size = 615, normalized size = 1.79 \begin {gather*} \frac {2}{15015} \, {\left (1155 \, {\left (x e + d\right )}^{\frac {13}{2}} B c^{3} e^{96} - 9555 \, {\left (x e + d\right )}^{\frac {11}{2}} B c^{3} d e^{96} + 35035 \, {\left (x e + d\right )}^{\frac {9}{2}} B c^{3} d^{2} e^{96} - 75075 \, {\left (x e + d\right )}^{\frac {7}{2}} B c^{3} d^{3} e^{96} + 105105 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{3} d^{4} e^{96} - 105105 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{3} d^{5} e^{96} + 105105 \, \sqrt {x e + d} B c^{3} d^{6} e^{96} + 1365 \, {\left (x e + d\right )}^{\frac {11}{2}} A c^{3} e^{97} - 10010 \, {\left (x e + d\right )}^{\frac {9}{2}} A c^{3} d e^{97} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} A c^{3} d^{2} e^{97} - 60060 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{3} d^{3} e^{97} + 75075 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{3} d^{4} e^{97} - 90090 \, \sqrt {x e + d} A c^{3} d^{5} e^{97} + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} B a c^{2} e^{98} - 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} B a c^{2} d e^{98} + 90090 \, {\left (x e + d\right )}^{\frac {5}{2}} B a c^{2} d^{2} e^{98} - 150150 \, {\left (x e + d\right )}^{\frac {3}{2}} B a c^{2} d^{3} e^{98} + 225225 \, \sqrt {x e + d} B a c^{2} d^{4} e^{98} + 6435 \, {\left (x e + d\right )}^{\frac {7}{2}} A a c^{2} e^{99} - 36036 \, {\left (x e + d\right )}^{\frac {5}{2}} A a c^{2} d e^{99} + 90090 \, {\left (x e + d\right )}^{\frac {3}{2}} A a c^{2} d^{2} e^{99} - 180180 \, \sqrt {x e + d} A a c^{2} d^{3} e^{99} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} c e^{100} - 45045 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} c d e^{100} + 135135 \, \sqrt {x e + d} B a^{2} c d^{2} e^{100} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} c e^{101} - 90090 \, \sqrt {x e + d} A a^{2} c d e^{101} + 15015 \, \sqrt {x e + d} B a^{3} e^{102}\right )} e^{\left (-104\right )} + \frac {2 \, {\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}\right )} e^{\left (-8\right )}}{\sqrt {x e + d}} \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)^(3/2),x, algorithm="giac")

[Out]

2/15015*(1155*(x*e + d)^(13/2)*B*c^3*e^96 - 9555*(x*e + d)^(11/2)*B*c^3*d*e^96 + 35035*(x*e + d)^(9/2)*B*c^3*d
^2*e^96 - 75075*(x*e + d)^(7/2)*B*c^3*d^3*e^96 + 105105*(x*e + d)^(5/2)*B*c^3*d^4*e^96 - 105105*(x*e + d)^(3/2
)*B*c^3*d^5*e^96 + 105105*sqrt(x*e + d)*B*c^3*d^6*e^96 + 1365*(x*e + d)^(11/2)*A*c^3*e^97 - 10010*(x*e + d)^(9
/2)*A*c^3*d*e^97 + 32175*(x*e + d)^(7/2)*A*c^3*d^2*e^97 - 60060*(x*e + d)^(5/2)*A*c^3*d^3*e^97 + 75075*(x*e +
d)^(3/2)*A*c^3*d^4*e^97 - 90090*sqrt(x*e + d)*A*c^3*d^5*e^97 + 5005*(x*e + d)^(9/2)*B*a*c^2*e^98 - 32175*(x*e
+ d)^(7/2)*B*a*c^2*d*e^98 + 90090*(x*e + d)^(5/2)*B*a*c^2*d^2*e^98 - 150150*(x*e + d)^(3/2)*B*a*c^2*d^3*e^98 +
225225*sqrt(x*e + d)*B*a*c^2*d^4*e^98 + 6435*(x*e + d)^(7/2)*A*a*c^2*e^99 - 36036*(x*e + d)^(5/2)*A*a*c^2*d*e
^99 + 90090*(x*e + d)^(3/2)*A*a*c^2*d^2*e^99 - 180180*sqrt(x*e + d)*A*a*c^2*d^3*e^99 + 9009*(x*e + d)^(5/2)*B*
a^2*c*e^100 - 45045*(x*e + d)^(3/2)*B*a^2*c*d*e^100 + 135135*sqrt(x*e + d)*B*a^2*c*d^2*e^100 + 15015*(x*e + d)
^(3/2)*A*a^2*c*e^101 - 90090*sqrt(x*e + d)*A*a^2*c*d*e^101 + 15015*sqrt(x*e + d)*B*a^3*e^102)*e^(-104) + 2*(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)*e^(-8)/sqrt(x*e + d)

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maple [A]  time = 0.05, size = 489, normalized size = 1.42 \begin {gather*} -\frac {2 \left (-1155 B \,c^{3} x^{7} e^{7}-1365 A \,c^{3} e^{7} x^{6}+1470 B \,c^{3} d \,e^{6} x^{6}+1820 A \,c^{3} d \,e^{6} x^{5}-5005 B a \,c^{2} e^{7} x^{5}-1960 B \,c^{3} d^{2} e^{5} x^{5}-6435 A a \,c^{2} e^{7} x^{4}-2600 A \,c^{3} d^{2} e^{5} x^{4}+7150 B a \,c^{2} d \,e^{6} x^{4}+2800 B \,c^{3} d^{3} e^{4} x^{4}+10296 A a \,c^{2} d \,e^{6} x^{3}+4160 A \,c^{3} d^{3} e^{4} x^{3}-9009 B \,a^{2} c \,e^{7} x^{3}-11440 B a \,c^{2} d^{2} e^{5} x^{3}-4480 B \,c^{3} d^{4} e^{3} x^{3}-15015 A \,a^{2} c \,e^{7} x^{2}-20592 A a \,c^{2} d^{2} e^{5} x^{2}-8320 A \,c^{3} d^{4} e^{3} x^{2}+18018 B \,a^{2} c d \,e^{6} x^{2}+22880 B a \,c^{2} d^{3} e^{4} x^{2}+8960 B \,c^{3} d^{5} e^{2} x^{2}+60060 A \,a^{2} c d \,e^{6} x +82368 A a \,c^{2} d^{3} e^{4} x +33280 A \,c^{3} d^{5} e^{2} x -15015 B \,a^{3} e^{7} x -72072 B \,a^{2} c \,d^{2} e^{5} x -91520 B a \,c^{2} d^{4} e^{3} x -35840 B \,c^{3} d^{6} e x +15015 A \,a^{3} e^{7}+120120 A \,d^{2} a^{2} c \,e^{5}+164736 A a \,c^{2} d^{4} e^{3}+66560 A \,c^{3} d^{6} e -30030 B \,a^{3} d \,e^{6}-144144 B \,d^{3} a^{2} c \,e^{4}-183040 B a \,c^{2} d^{5} e^{2}-71680 B \,c^{3} d^{7}\right )}{15015 \sqrt {e x +d}\, 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)^(3/2),x)

[Out]

-2/15015/(e*x+d)^(1/2)*(-1155*B*c^3*e^7*x^7-1365*A*c^3*e^7*x^6+1470*B*c^3*d*e^6*x^6+1820*A*c^3*d*e^6*x^5-5005*
B*a*c^2*e^7*x^5-1960*B*c^3*d^2*e^5*x^5-6435*A*a*c^2*e^7*x^4-2600*A*c^3*d^2*e^5*x^4+7150*B*a*c^2*d*e^6*x^4+2800
*B*c^3*d^3*e^4*x^4+10296*A*a*c^2*d*e^6*x^3+4160*A*c^3*d^3*e^4*x^3-9009*B*a^2*c*e^7*x^3-11440*B*a*c^2*d^2*e^5*x
^3-4480*B*c^3*d^4*e^3*x^3-15015*A*a^2*c*e^7*x^2-20592*A*a*c^2*d^2*e^5*x^2-8320*A*c^3*d^4*e^3*x^2+18018*B*a^2*c
*d*e^6*x^2+22880*B*a*c^2*d^3*e^4*x^2+8960*B*c^3*d^5*e^2*x^2+60060*A*a^2*c*d*e^6*x+82368*A*a*c^2*d^3*e^4*x+3328
0*A*c^3*d^5*e^2*x-15015*B*a^3*e^7*x-72072*B*a^2*c*d^2*e^5*x-91520*B*a*c^2*d^4*e^3*x-35840*B*c^3*d^6*e*x+15015*
A*a^3*e^7+120120*A*a^2*c*d^2*e^5+164736*A*a*c^2*d^4*e^3+66560*A*c^3*d^6*e-30030*B*a^3*d*e^6-144144*B*a^2*c*d^3
*e^4-183040*B*a*c^2*d^5*e^2-71680*B*c^3*d^7)/e^8

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maxima [A]  time = 0.55, size = 461, normalized size = 1.34 \begin {gather*} \frac {2 \, {\left (\frac {1155 \, {\left (e x + d\right )}^{\frac {13}{2}} B c^{3} - 1365 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\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 {9}{2}} - 2145 \, {\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 {7}{2}} + 3003 \, {\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 {5}{2}} - 15015 \, {\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 )}^{\frac {3}{2}} + 15015 \, {\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 )} \sqrt {e x + d}}{e^{7}} + \frac {15015 \, {\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}\right )}}{\sqrt {e x + d} e^{7}}\right )}}{15015 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/15015*((1155*(e*x + d)^(13/2)*B*c^3 - 1365*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(11/2) + 5005*(7*B*c^3*d^2 - 2*A*
c^3*d*e + B*a*c^2*e^2)*(e*x + d)^(9/2) - 2145*(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)^(7/2) + 3003*(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)^(5/2) - 15015*(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)^(3/2) + 15015*(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)*sqrt(e*x + d))/e^7 + 15015*(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)/(sqr
t(e*x + d)*e^7))/e

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mupad [B]  time = 1.84, size = 394, normalized size = 1.15 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\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 )}{5\,e^8}-\frac {-2\,B\,a^3\,d\,e^6+2\,A\,a^3\,e^7-6\,B\,a^2\,c\,d^3\,e^4+6\,A\,a^2\,c\,d^2\,e^5-6\,B\,a\,c^2\,d^5\,e^2+6\,A\,a\,c^2\,d^4\,e^3-2\,B\,c^3\,d^7+2\,A\,c^3\,d^6\,e}{e^8\,\sqrt {d+e\,x}}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{9\,e^8}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,\sqrt {d+e\,x}\,\left (7\,B\,c\,d^2-6\,A\,c\,d\,e+B\,a\,e^2\right )}{e^8}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,c^2\,{\left (d+e\,x\right )}^{7/2}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{7\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {2\,c\,\left (c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}\,\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)^(3/2),x)

[Out]

((d + e*x)^(5/2)*(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))/(5*e
^8) - (2*A*a^3*e^7 - 2*B*c^3*d^7 - 2*B*a^3*d*e^6 + 2*A*c^3*d^6*e + 6*A*a*c^2*d^4*e^3 + 6*A*a^2*c*d^2*e^5 - 6*B
*a*c^2*d^5*e^2 - 6*B*a^2*c*d^3*e^4)/(e^8*(d + e*x)^(1/2)) + ((d + e*x)^(9/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*
B*a*c^2*e^2))/(9*e^8) + (2*(a*e^2 + c*d^2)^2*(d + e*x)^(1/2)*(B*a*e^2 + 7*B*c*d^2 - 6*A*c*d*e))/e^8 + (2*B*c^3
*(d + e*x)^(13/2))/(13*e^8) + (2*c^2*(d + e*x)^(7/2)*(3*A*a*e^3 - 35*B*c*d^3 - 15*B*a*d*e^2 + 15*A*c*d^2*e))/(
7*e^8) + (2*c^3*(A*e - 7*B*d)*(d + e*x)^(11/2))/(11*e^8) + (2*c*(a*e^2 + c*d^2)*(d + e*x)^(3/2)*(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 = 101.30, size = 461, normalized size = 1.34 \begin {gather*} \frac {2 B c^{3} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{8}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (2 A c^{3} e - 14 B c^{3} d\right )}{11 e^{8}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 12 A c^{3} d e + 6 B a c^{2} e^{2} + 42 B c^{3} d^{2}\right )}{9 e^{8}} + \frac {\left (d + e x\right )^{\frac {7}{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 )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{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 )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \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 )}{3 e^{8}} + \frac {\sqrt {d + e x} \left (- 12 A a^{2} c d e^{5} - 24 A a c^{2} d^{3} e^{3} - 12 A c^{3} d^{5} e + 2 B a^{3} e^{6} + 18 B a^{2} c d^{2} e^{4} + 30 B a c^{2} d^{4} e^{2} + 14 B c^{3} d^{6}\right )}{e^{8}} + \frac {2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{e^{8} \sqrt {d + e x}} \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)**(3/2),x)

[Out]

2*B*c**3*(d + e*x)**(13/2)/(13*e**8) + (d + e*x)**(11/2)*(2*A*c**3*e - 14*B*c**3*d)/(11*e**8) + (d + e*x)**(9/
2)*(-12*A*c**3*d*e + 6*B*a*c**2*e**2 + 42*B*c**3*d**2)/(9*e**8) + (d + e*x)**(7/2)*(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)/(7*e**8) + (d + e*x)**(5/2)*(-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)/(5*e**8) + (d + e*x)**(3/2)*(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)/(3*
e**8) + sqrt(d + e*x)*(-12*A*a**2*c*d*e**5 - 24*A*a*c**2*d**3*e**3 - 12*A*c**3*d**5*e + 2*B*a**3*e**6 + 18*B*a
**2*c*d**2*e**4 + 30*B*a*c**2*d**4*e**2 + 14*B*c**3*d**6)/e**8 + 2*(-A*e + B*d)*(a*e**2 + c*d**2)**3/(e**8*sqr
t(d + e*x))

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