3.15.44 \(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^{7/2}} \, dx\) [1444]

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

[Out]

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

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Rubi [A]
time = 0.10, 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 = {786} \begin {gather*} -\frac {2 c \sqrt {d+e x} \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}+\frac {6 c^2 (d+e x)^{5/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8}-\frac {2 c^2 (d+e x)^{3/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{3 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 )}{3 e^8 (d+e x)^{3/2}}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^{5/2}}+\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 )}{e^8 \sqrt {d+e x}}-\frac {2 c^3 (d+e x)^{7/2} (7 B d-A e)}{7 e^8}+\frac {2 B c^3 (d+e x)^{9/2}}{9 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(7/2),x]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(5*e^8*(d + e*x)^(5/2)) - (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*
e^2))/(3*e^8*(d + e*x)^(3/2)) + (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))/(e^8*S
qrt[d + e*x]) - (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))*Sqrt[d + e*
x])/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)^(3/2))/(3*e^8) + (6*c^2*(7*B
*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(5/2))/(5*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(7/2))/(7*e^8) + (2*B*
c^3*(d + e*x)^(9/2))/(9*e^8)

Rule 786

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

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Mathematica [A]
time = 0.28, size = 373, normalized size = 1.08 \begin {gather*} \frac {2 \left (-9 A e \left (7 a^3 e^6+7 a^2 c e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+7 a c^2 e^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )+7 B \left (-3 a^3 e^6 (2 d+5 e x)+27 a^2 c e^4 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+9 a c^2 e^2 \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )+c^3 \left (2048 d^7+5120 d^6 e x+3840 d^5 e^2 x^2+640 d^4 e^3 x^3-80 d^3 e^4 x^4+24 d^2 e^5 x^5-10 d e^6 x^6+5 e^7 x^7\right )\right )\right )}{315 e^8 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(7/2),x]

[Out]

(2*(-9*A*e*(7*a^3*e^6 + 7*a^2*c*e^4*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 7*a*c^2*e^2*(128*d^4 + 320*d^3*e*x + 240
*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) + c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 -
40*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6)) + 7*B*(-3*a^3*e^6*(2*d + 5*e*x) + 27*a^2*c*e^4*(16*d^3 + 40*d^2*e
*x + 30*d*e^2*x^2 + 5*e^3*x^3) + 9*a*c^2*e^2*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*
e^4*x^4 + 3*e^5*x^5) + c^3*(2048*d^7 + 5120*d^6*e*x + 3840*d^5*e^2*x^2 + 640*d^4*e^3*x^3 - 80*d^3*e^4*x^4 + 24
*d^2*e^5*x^5 - 10*d*e^6*x^6 + 5*e^7*x^7))))/(315*e^8*(d + e*x)^(5/2))

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Maple [A]
time = 0.71, size = 506, normalized size = 1.46 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/e^8*(1/9*B*c^3*(e*x+d)^(9/2)+1/7*A*c^3*e*(e*x+d)^(7/2)-B*c^3*d*(e*x+d)^(7/2)-6/5*A*c^3*d*e*(e*x+d)^(5/2)+3/5
*B*a*c^2*e^2*(e*x+d)^(5/2)+21/5*B*c^3*d^2*(e*x+d)^(5/2)+A*a*c^2*e^3*(e*x+d)^(3/2)+5*A*c^3*d^2*e*(e*x+d)^(3/2)-
5*B*a*c^2*d*e^2*(e*x+d)^(3/2)-35/3*B*c^3*d^3*(e*x+d)^(3/2)-12*A*a*c^2*d*e^3*(e*x+d)^(1/2)-20*A*c^3*d^3*e*(e*x+
d)^(1/2)+3*B*a^2*c*e^4*(e*x+d)^(1/2)+30*B*a*c^2*d^2*e^2*(e*x+d)^(1/2)+35*B*c^3*d^4*(e*x+d)^(1/2)-3*c*(A*a^2*e^
5+6*A*a*c*d^2*e^3+5*A*c^2*d^4*e-3*B*a^2*d*e^4-10*B*a*c*d^3*e^2-7*B*c^2*d^5)/(e*x+d)^(1/2)-1/5*(A*a^3*e^7+3*A*a
^2*c*d^2*e^5+3*A*a*c^2*d^4*e^3+A*c^3*d^6*e-B*a^3*d*e^6-3*B*a^2*c*d^3*e^4-3*B*a*c^2*d^5*e^2-B*c^3*d^7)/(e*x+d)^
(5/2)-1/3*(-6*A*a^2*c*d*e^5-12*A*a*c^2*d^3*e^3-6*A*c^3*d^5*e+B*a^3*e^6+9*B*a^2*c*d^2*e^4+15*B*a*c^2*d^4*e^2+7*
B*c^3*d^6)/(e*x+d)^(3/2))

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Maxima [A]
time = 0.30, size = 452, normalized size = 1.31 \begin {gather*} \frac {2}{315} \, {\left ({\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B c^{3} - 45 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 189 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 105 \, {\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 (x e + d\right )}^{\frac {3}{2}} + 315 \, {\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 )} \sqrt {x e + d}\right )} e^{\left (-7\right )} + \frac {21 \, {\left (3 \, B c^{3} d^{7} - 3 \, A c^{3} d^{6} e + 9 \, B a c^{2} d^{5} e^{2} - 9 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} - 9 \, A a^{2} c d^{2} e^{5} + 3 \, B a^{3} d e^{6} - 3 \, A a^{3} e^{7} + 45 \, {\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 (x e + d\right )}^{2} - 5 \, {\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 (x e + d\right )}\right )} e^{\left (-7\right )}}{{\left (x e + d\right )}^{\frac {5}{2}}}\right )} e^{\left (-1\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)^(7/2),x, algorithm="maxima")

[Out]

2/315*((35*(x*e + d)^(9/2)*B*c^3 - 45*(7*B*c^3*d - A*c^3*e)*(x*e + d)^(7/2) + 189*(7*B*c^3*d^2 - 2*A*c^3*d*e +
B*a*c^2*e^2)*(x*e + d)^(5/2) - 105*(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)*(x*e +
d)^(3/2) + 315*(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)*sqrt(x*
e + d))*e^(-7) + 21*(3*B*c^3*d^7 - 3*A*c^3*d^6*e + 9*B*a*c^2*d^5*e^2 - 9*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e^4 -
9*A*a^2*c*d^2*e^5 + 3*B*a^3*d*e^6 - 3*A*a^3*e^7 + 45*(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)*(x*e + d)^2 - 5*(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)*(x*e + d))*e^(-7)/(x*e + d)^(5/2))*
e^(-1)

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Fricas [A]
time = 2.89, size = 450, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (14336 \, B c^{3} d^{7} + {\left (35 \, B c^{3} x^{7} + 45 \, A c^{3} x^{6} + 189 \, B a c^{2} x^{5} + 315 \, A a c^{2} x^{4} + 945 \, B a^{2} c x^{3} - 945 \, A a^{2} c x^{2} - 105 \, B a^{3} x - 63 \, A a^{3}\right )} e^{7} - 2 \, {\left (35 \, B c^{3} d x^{6} + 54 \, A c^{3} d x^{5} + 315 \, B a c^{2} d x^{4} + 1260 \, A a c^{2} d x^{3} - 2835 \, B a^{2} c d x^{2} + 630 \, A a^{2} c d x + 21 \, B a^{3} d\right )} e^{6} + 24 \, {\left (7 \, B c^{3} d^{2} x^{5} + 15 \, A c^{3} d^{2} x^{4} + 210 \, B a c^{2} d^{2} x^{3} - 630 \, A a c^{2} d^{2} x^{2} + 315 \, B a^{2} c d^{2} x - 21 \, A a^{2} c d^{2}\right )} e^{5} - 16 \, {\left (35 \, B c^{3} d^{3} x^{4} + 180 \, A c^{3} d^{3} x^{3} - 1890 \, B a c^{2} d^{3} x^{2} + 1260 \, A a c^{2} d^{3} x - 189 \, B a^{2} c d^{3}\right )} e^{4} + 128 \, {\left (35 \, B c^{3} d^{4} x^{3} - 135 \, A c^{3} d^{4} x^{2} + 315 \, B a c^{2} d^{4} x - 63 \, A a c^{2} d^{4}\right )} e^{3} + 768 \, {\left (35 \, B c^{3} d^{5} x^{2} - 30 \, A c^{3} d^{5} x + 21 \, B a c^{2} d^{5}\right )} e^{2} + 1024 \, {\left (35 \, B c^{3} d^{6} x - 9 \, A c^{3} d^{6}\right )} e\right )} \sqrt {x e + d}}{315 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} 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)^(7/2),x, algorithm="fricas")

[Out]

2/315*(14336*B*c^3*d^7 + (35*B*c^3*x^7 + 45*A*c^3*x^6 + 189*B*a*c^2*x^5 + 315*A*a*c^2*x^4 + 945*B*a^2*c*x^3 -
945*A*a^2*c*x^2 - 105*B*a^3*x - 63*A*a^3)*e^7 - 2*(35*B*c^3*d*x^6 + 54*A*c^3*d*x^5 + 315*B*a*c^2*d*x^4 + 1260*
A*a*c^2*d*x^3 - 2835*B*a^2*c*d*x^2 + 630*A*a^2*c*d*x + 21*B*a^3*d)*e^6 + 24*(7*B*c^3*d^2*x^5 + 15*A*c^3*d^2*x^
4 + 210*B*a*c^2*d^2*x^3 - 630*A*a*c^2*d^2*x^2 + 315*B*a^2*c*d^2*x - 21*A*a^2*c*d^2)*e^5 - 16*(35*B*c^3*d^3*x^4
+ 180*A*c^3*d^3*x^3 - 1890*B*a*c^2*d^3*x^2 + 1260*A*a*c^2*d^3*x - 189*B*a^2*c*d^3)*e^4 + 128*(35*B*c^3*d^4*x^
3 - 135*A*c^3*d^4*x^2 + 315*B*a*c^2*d^4*x - 63*A*a*c^2*d^4)*e^3 + 768*(35*B*c^3*d^5*x^2 - 30*A*c^3*d^5*x + 21*
B*a*c^2*d^5)*e^2 + 1024*(35*B*c^3*d^6*x - 9*A*c^3*d^6)*e)*sqrt(x*e + d)/(x^3*e^11 + 3*d*x^2*e^10 + 3*d^2*x*e^9
+ d^3*e^8)

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Sympy [A]
time = 68.87, size = 377, normalized size = 1.09 \begin {gather*} \frac {2 B c^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{8}} + \frac {6 c \left (a e^{2} + c d^{2}\right ) \left (- A a e^{3} - 5 A c d^{2} e + 3 B a d e^{2} + 7 B c d^{3}\right )}{e^{8} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 A c^{3} e - 14 B c^{3} d\right )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 12 A c^{3} d e + 6 B a c^{2} e^{2} + 42 B c^{3} d^{2}\right )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \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 )}{3 e^{8}} + \frac {\sqrt {d + e x} \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 )}{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 )}{3 e^{8} \left (d + e x\right )^{\frac {3}{2}}} + \frac {2 \left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{5 e^{8} \left (d + e x\right )^{\frac {5}{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)**(7/2),x)

[Out]

2*B*c**3*(d + e*x)**(9/2)/(9*e**8) + 6*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*sqrt(d + e*x)) + (d + e*x)**(7/2)*(2*A*c**3*e - 14*B*c**3*d)/(7*e**8) + (d + e*x)**(5/2)*(-12*A*c**3
*d*e + 6*B*a*c**2*e**2 + 42*B*c**3*d**2)/(5*e**8) + (d + e*x)**(3/2)*(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)/(3*e**8) + sqrt(d + e*x)*(-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)/e**8 - 2*(a*e**2 + c*d**2)**2*(-6*A*c*d*e + B*a*e**2 + 7*B*c*
d**2)/(3*e**8*(d + e*x)**(3/2)) + 2*(-A*e + B*d)*(a*e**2 + c*d**2)**3/(5*e**8*(d + e*x)**(5/2))

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Giac [A]
time = 2.74, size = 599, normalized size = 1.73 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B c^{3} e^{64} - 315 \, {\left (x e + d\right )}^{\frac {7}{2}} B c^{3} d e^{64} + 1323 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{3} d^{2} e^{64} - 3675 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{3} d^{3} e^{64} + 11025 \, \sqrt {x e + d} B c^{3} d^{4} e^{64} + 45 \, {\left (x e + d\right )}^{\frac {7}{2}} A c^{3} e^{65} - 378 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{3} d e^{65} + 1575 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{3} d^{2} e^{65} - 6300 \, \sqrt {x e + d} A c^{3} d^{3} e^{65} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} B a c^{2} e^{66} - 1575 \, {\left (x e + d\right )}^{\frac {3}{2}} B a c^{2} d e^{66} + 9450 \, \sqrt {x e + d} B a c^{2} d^{2} e^{66} + 315 \, {\left (x e + d\right )}^{\frac {3}{2}} A a c^{2} e^{67} - 3780 \, \sqrt {x e + d} A a c^{2} d e^{67} + 945 \, \sqrt {x e + d} B a^{2} c e^{68}\right )} e^{\left (-72\right )} + \frac {2 \, {\left (315 \, {\left (x e + d\right )}^{2} B c^{3} d^{5} - 35 \, {\left (x e + d\right )} B c^{3} d^{6} + 3 \, B c^{3} d^{7} - 225 \, {\left (x e + d\right )}^{2} A c^{3} d^{4} e + 30 \, {\left (x e + d\right )} A c^{3} d^{5} e - 3 \, A c^{3} d^{6} e + 450 \, {\left (x e + d\right )}^{2} B a c^{2} d^{3} e^{2} - 75 \, {\left (x e + d\right )} B a c^{2} d^{4} e^{2} + 9 \, B a c^{2} d^{5} e^{2} - 270 \, {\left (x e + d\right )}^{2} A a c^{2} d^{2} e^{3} + 60 \, {\left (x e + d\right )} A a c^{2} d^{3} e^{3} - 9 \, A a c^{2} d^{4} e^{3} + 135 \, {\left (x e + d\right )}^{2} B a^{2} c d e^{4} - 45 \, {\left (x e + d\right )} B a^{2} c d^{2} e^{4} + 9 \, B a^{2} c d^{3} e^{4} - 45 \, {\left (x e + d\right )}^{2} A a^{2} c e^{5} + 30 \, {\left (x e + d\right )} A a^{2} c d e^{5} - 9 \, A a^{2} c d^{2} e^{5} - 5 \, {\left (x e + d\right )} B a^{3} e^{6} + 3 \, B a^{3} d e^{6} - 3 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{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)^(7/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*B*c^3*e^64 - 315*(x*e + d)^(7/2)*B*c^3*d*e^64 + 1323*(x*e + d)^(5/2)*B*c^3*d^2*e^64
- 3675*(x*e + d)^(3/2)*B*c^3*d^3*e^64 + 11025*sqrt(x*e + d)*B*c^3*d^4*e^64 + 45*(x*e + d)^(7/2)*A*c^3*e^65 - 3
78*(x*e + d)^(5/2)*A*c^3*d*e^65 + 1575*(x*e + d)^(3/2)*A*c^3*d^2*e^65 - 6300*sqrt(x*e + d)*A*c^3*d^3*e^65 + 18
9*(x*e + d)^(5/2)*B*a*c^2*e^66 - 1575*(x*e + d)^(3/2)*B*a*c^2*d*e^66 + 9450*sqrt(x*e + d)*B*a*c^2*d^2*e^66 + 3
15*(x*e + d)^(3/2)*A*a*c^2*e^67 - 3780*sqrt(x*e + d)*A*a*c^2*d*e^67 + 945*sqrt(x*e + d)*B*a^2*c*e^68)*e^(-72)
+ 2/15*(315*(x*e + d)^2*B*c^3*d^5 - 35*(x*e + d)*B*c^3*d^6 + 3*B*c^3*d^7 - 225*(x*e + d)^2*A*c^3*d^4*e + 30*(x
*e + d)*A*c^3*d^5*e - 3*A*c^3*d^6*e + 450*(x*e + d)^2*B*a*c^2*d^3*e^2 - 75*(x*e + d)*B*a*c^2*d^4*e^2 + 9*B*a*c
^2*d^5*e^2 - 270*(x*e + d)^2*A*a*c^2*d^2*e^3 + 60*(x*e + d)*A*a*c^2*d^3*e^3 - 9*A*a*c^2*d^4*e^3 + 135*(x*e + d
)^2*B*a^2*c*d*e^4 - 45*(x*e + d)*B*a^2*c*d^2*e^4 + 9*B*a^2*c*d^3*e^4 - 45*(x*e + d)^2*A*a^2*c*e^5 + 30*(x*e +
d)*A*a^2*c*d*e^5 - 9*A*a^2*c*d^2*e^5 - 5*(x*e + d)*B*a^3*e^6 + 3*B*a^3*d*e^6 - 3*A*a^3*e^7)*e^(-8)/(x*e + d)^(
5/2)

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Mupad [B]
time = 1.84, size = 455, normalized size = 1.32 \begin {gather*} \frac {\sqrt {d+e\,x}\,\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 )}{e^8}-\frac {\left (d+e\,x\right )\,\left (\frac {2\,B\,a^3\,e^6}{3}+6\,B\,a^2\,c\,d^2\,e^4-4\,A\,a^2\,c\,d\,e^5+10\,B\,a\,c^2\,d^4\,e^2-8\,A\,a\,c^2\,d^3\,e^3+\frac {14\,B\,c^3\,d^6}{3}-4\,A\,c^3\,d^5\,e\right )-{\left (d+e\,x\right )}^2\,\left (18\,B\,a^2\,c\,d\,e^4-6\,A\,a^2\,c\,e^5+60\,B\,a\,c^2\,d^3\,e^2-36\,A\,a\,c^2\,d^2\,e^3+42\,B\,c^3\,d^5-30\,A\,c^3\,d^4\,e\right )+\frac {2\,A\,a^3\,e^7}{5}-\frac {2\,B\,c^3\,d^7}{5}-\frac {2\,B\,a^3\,d\,e^6}{5}+\frac {2\,A\,c^3\,d^6\,e}{5}+\frac {6\,A\,a\,c^2\,d^4\,e^3}{5}+\frac {6\,A\,a^2\,c\,d^2\,e^5}{5}-\frac {6\,B\,a\,c^2\,d^5\,e^2}{5}-\frac {6\,B\,a^2\,c\,d^3\,e^4}{5}}{e^8\,{\left (d+e\,x\right )}^{5/2}}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{5\,e^8}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {2\,c^2\,{\left (d+e\,x\right )}^{3/2}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{3\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,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)^(7/2),x)

[Out]

((d + e*x)^(1/2)*(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))/e^8
- ((d + e*x)*((2*B*a^3*e^6)/3 + (14*B*c^3*d^6)/3 - 4*A*c^3*d^5*e - 8*A*a*c^2*d^3*e^3 + 10*B*a*c^2*d^4*e^2 + 6*
B*a^2*c*d^2*e^4 - 4*A*a^2*c*d*e^5) - (d + e*x)^2*(42*B*c^3*d^5 - 6*A*a^2*c*e^5 - 30*A*c^3*d^4*e - 36*A*a*c^2*d
^2*e^3 + 60*B*a*c^2*d^3*e^2 + 18*B*a^2*c*d*e^4) + (2*A*a^3*e^7)/5 - (2*B*c^3*d^7)/5 - (2*B*a^3*d*e^6)/5 + (2*A
*c^3*d^6*e)/5 + (6*A*a*c^2*d^4*e^3)/5 + (6*A*a^2*c*d^2*e^5)/5 - (6*B*a*c^2*d^5*e^2)/5 - (6*B*a^2*c*d^3*e^4)/5)
/(e^8*(d + e*x)^(5/2)) + ((d + e*x)^(5/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/(5*e^8) + (2*B*c^3*(d
+ e*x)^(9/2))/(9*e^8) + (2*c^2*(d + e*x)^(3/2)*(3*A*a*e^3 - 35*B*c*d^3 - 15*B*a*d*e^2 + 15*A*c*d^2*e))/(3*e^8
) + (2*c^3*(A*e - 7*B*d)*(d + e*x)^(7/2))/(7*e^8)

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