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

Optimal. Leaf size=314 \[ \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 (d+e x)}-\frac {c^2 x \left (5 A c d e-3 B \left (a e^2+5 c d^2\right )\right )}{e^7}-\frac {c^2 \log (d+e x) \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8 (d+e x)^3}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{4 e^8 (d+e x)^4}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{2 e^8 (d+e x)^2}-\frac {c^3 x^2 (5 B d-A e)}{2 e^6}+\frac {B c^3 x^3}{3 e^5} \]

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Rubi [A]  time = 0.38, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \begin {gather*} \frac {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 (d+e x)}-\frac {c^2 x \left (5 A c d e-3 B \left (a e^2+5 c d^2\right )\right )}{e^7}-\frac {c^2 \log (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 {3 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 )}{2 e^8 (d+e x)^2}-\frac {\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}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{4 e^8 (d+e x)^4}-\frac {c^3 x^2 (5 B d-A e)}{2 e^6}+\frac {B c^3 x^3}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((c^2*(5*A*c*d*e - 3*B*(5*c*d^2 + a*e^2))*x)/e^7) - (c^3*(5*B*d - A*e)*x^2)/(2*e^6) + (B*c^3*x^3)/(3*e^5) + (
(B*d - A*e)*(c*d^2 + a*e^2)^3)/(4*e^8*(d + e*x)^4) - ((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) + (3*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))/(2*e^8*(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)))/(e^8*(d + e*x)) - (c^2*(3
5*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*Log[d + e*x])/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} \, dx &=\int \left (-\frac {c^2 \left (-15 B c d^2+5 A c d e-3 a B e^2\right )}{e^7}+\frac {c^3 (-5 B d+A e) x}{e^6}+\frac {B c^3 x^2}{e^5}+\frac {(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^5}+\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)^4}+\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}-\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)^2}+\frac {c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right )}{e^7 (d+e x)}\right ) \, dx\\ &=-\frac {c^2 \left (5 A c d e-3 B \left (5 c d^2+a e^2\right )\right ) x}{e^7}-\frac {c^3 (5 B d-A e) x^2}{2 e^6}+\frac {B c^3 x^3}{3 e^5}+\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{4 e^8 (d+e x)^4}-\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 )}{3 e^8 (d+e x)^3}+\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 )}{2 e^8 (d+e x)^2}+\frac {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 (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 ) \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 405, normalized size = 1.29 \begin {gather*} \frac {3 A e \left (-a^3 e^6-a^2 c e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+a c^2 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+c^3 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )-B \left (a^3 e^6 (d+4 e x)+9 a^2 c e^4 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+3 a c^2 e^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )+c^3 \left (319 d^7+856 d^6 e x+444 d^5 e^2 x^2-544 d^4 e^3 x^3-556 d^3 e^4 x^4-84 d^2 e^5 x^5+14 d e^6 x^6-4 e^7 x^7\right )\right )+12 c^2 (d+e x)^4 \log (d+e x) \left (3 A e \left (a e^2+5 c d^2\right )-5 B \left (3 a d e^2+7 c d^3\right )\right )}{12 e^8 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*A*e*(-(a^3*e^6) - a^2*c*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2) + a*c^2*d*e^2*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2
+ 48*e^3*x^3) + c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 +
2*e^6*x^6)) - B*(a^3*e^6*(d + 4*e*x) + 9*a^2*c*e^4*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + 3*a*c^2*e^2*
(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) + c^3*(319*d^7 + 856*d^6
*e*x + 444*d^5*e^2*x^2 - 544*d^4*e^3*x^3 - 556*d^3*e^4*x^4 - 84*d^2*e^5*x^5 + 14*d*e^6*x^6 - 4*e^7*x^7)) + 12*
c^2*(3*A*e*(5*c*d^2 + a*e^2) - 5*B*(7*c*d^3 + 3*a*d*e^2))*(d + e*x)^4*Log[d + e*x])/(12*e^8*(d + e*x)^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.42, size = 746, normalized size = 2.38 \begin {gather*} \frac {4 \, B c^{3} e^{7} x^{7} - 319 \, B c^{3} d^{7} + 171 \, A c^{3} d^{6} e - 231 \, B a c^{2} d^{5} e^{2} + 75 \, A a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 3 \, A a^{3} e^{7} - 2 \, {\left (7 \, B c^{3} d e^{6} - 3 \, A c^{3} e^{7}\right )} x^{6} + 12 \, {\left (7 \, B c^{3} d^{2} e^{5} - 3 \, A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} + 4 \, {\left (139 \, B c^{3} d^{3} e^{4} - 51 \, A c^{3} d^{2} e^{5} + 36 \, B a c^{2} d e^{6}\right )} x^{4} + 4 \, {\left (136 \, B c^{3} d^{4} e^{3} - 24 \, A c^{3} d^{3} e^{4} - 36 \, B a c^{2} d^{2} e^{5} + 36 \, A a c^{2} d e^{6} - 9 \, B a^{2} c e^{7}\right )} x^{3} - 6 \, {\left (74 \, B c^{3} d^{5} e^{2} - 66 \, A c^{3} d^{4} e^{3} + 126 \, B a c^{2} d^{3} e^{4} - 54 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} + 3 \, A a^{2} c e^{7}\right )} x^{2} - 4 \, {\left (214 \, B c^{3} d^{6} e - 126 \, A c^{3} d^{5} e^{2} + 186 \, B a c^{2} d^{4} e^{3} - 66 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x - 12 \, {\left (35 \, B c^{3} d^{7} - 15 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + {\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 4 \, {\left (35 \, B c^{3} d^{4} e^{3} - 15 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} - 3 \, A a c^{2} d e^{6}\right )} x^{3} + 6 \, {\left (35 \, B c^{3} d^{5} e^{2} - 15 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} - 3 \, A a c^{2} d^{2} e^{5}\right )} x^{2} + 4 \, {\left (35 \, B c^{3} d^{6} e - 15 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} - 3 \, A a c^{2} d^{3} e^{4}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\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)^5,x, algorithm="fricas")

[Out]

1/12*(4*B*c^3*e^7*x^7 - 319*B*c^3*d^7 + 171*A*c^3*d^6*e - 231*B*a*c^2*d^5*e^2 + 75*A*a*c^2*d^4*e^3 - 9*B*a^2*c
*d^3*e^4 - 3*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 3*A*a^3*e^7 - 2*(7*B*c^3*d*e^6 - 3*A*c^3*e^7)*x^6 + 12*(7*B*c^3*d
^2*e^5 - 3*A*c^3*d*e^6 + 3*B*a*c^2*e^7)*x^5 + 4*(139*B*c^3*d^3*e^4 - 51*A*c^3*d^2*e^5 + 36*B*a*c^2*d*e^6)*x^4
+ 4*(136*B*c^3*d^4*e^3 - 24*A*c^3*d^3*e^4 - 36*B*a*c^2*d^2*e^5 + 36*A*a*c^2*d*e^6 - 9*B*a^2*c*e^7)*x^3 - 6*(74
*B*c^3*d^5*e^2 - 66*A*c^3*d^4*e^3 + 126*B*a*c^2*d^3*e^4 - 54*A*a*c^2*d^2*e^5 + 9*B*a^2*c*d*e^6 + 3*A*a^2*c*e^7
)*x^2 - 4*(214*B*c^3*d^6*e - 126*A*c^3*d^5*e^2 + 186*B*a*c^2*d^4*e^3 - 66*A*a*c^2*d^3*e^4 + 9*B*a^2*c*d^2*e^5
+ 3*A*a^2*c*d*e^6 + B*a^3*e^7)*x - 12*(35*B*c^3*d^7 - 15*A*c^3*d^6*e + 15*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3
+ (35*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 - 3*A*a*c^2*e^7)*x^4 + 4*(35*B*c^3*d^4*e^3 - 15*A*c^
3*d^3*e^4 + 15*B*a*c^2*d^2*e^5 - 3*A*a*c^2*d*e^6)*x^3 + 6*(35*B*c^3*d^5*e^2 - 15*A*c^3*d^4*e^3 + 15*B*a*c^2*d^
3*e^4 - 3*A*a*c^2*d^2*e^5)*x^2 + 4*(35*B*c^3*d^6*e - 15*A*c^3*d^5*e^2 + 15*B*a*c^2*d^4*e^3 - 3*A*a*c^2*d^3*e^4
)*x)*log(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 [B]  time = 0.18, size = 647, normalized size = 2.06 \begin {gather*} \frac {1}{6} \, {\left (2 \, B c^{3} - \frac {3 \, {\left (7 \, B c^{3} d e - A c^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {18 \, {\left (7 \, B c^{3} d^{2} e^{2} - 2 \, A c^{3} d e^{3} + B a c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )} {\left (x e + d\right )}^{3} e^{\left (-8\right )} + {\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 )} e^{\left (-8\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {1}{12} \, {\left (\frac {420 \, B c^{3} d^{4} e^{36}}{x e + d} - \frac {126 \, B c^{3} d^{5} e^{36}}{{\left (x e + d\right )}^{2}} + \frac {28 \, B c^{3} d^{6} e^{36}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B c^{3} d^{7} e^{36}}{{\left (x e + d\right )}^{4}} - \frac {240 \, A c^{3} d^{3} e^{37}}{x e + d} + \frac {90 \, A c^{3} d^{4} e^{37}}{{\left (x e + d\right )}^{2}} - \frac {24 \, A c^{3} d^{5} e^{37}}{{\left (x e + d\right )}^{3}} + \frac {3 \, A c^{3} d^{6} e^{37}}{{\left (x e + d\right )}^{4}} + \frac {360 \, B a c^{2} d^{2} e^{38}}{x e + d} - \frac {180 \, B a c^{2} d^{3} e^{38}}{{\left (x e + d\right )}^{2}} + \frac {60 \, B a c^{2} d^{4} e^{38}}{{\left (x e + d\right )}^{3}} - \frac {9 \, B a c^{2} d^{5} e^{38}}{{\left (x e + d\right )}^{4}} - \frac {144 \, A a c^{2} d e^{39}}{x e + d} + \frac {108 \, A a c^{2} d^{2} e^{39}}{{\left (x e + d\right )}^{2}} - \frac {48 \, A a c^{2} d^{3} e^{39}}{{\left (x e + d\right )}^{3}} + \frac {9 \, A a c^{2} d^{4} e^{39}}{{\left (x e + d\right )}^{4}} + \frac {36 \, B a^{2} c e^{40}}{x e + d} - \frac {54 \, B a^{2} c d e^{40}}{{\left (x e + d\right )}^{2}} + \frac {36 \, B a^{2} c d^{2} e^{40}}{{\left (x e + d\right )}^{3}} - \frac {9 \, B a^{2} c d^{3} e^{40}}{{\left (x e + d\right )}^{4}} + \frac {18 \, A a^{2} c e^{41}}{{\left (x e + d\right )}^{2}} - \frac {24 \, A a^{2} c d e^{41}}{{\left (x e + d\right )}^{3}} + \frac {9 \, A a^{2} c d^{2} e^{41}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a^{3} e^{42}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a^{3} d e^{42}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a^{3} e^{43}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-44\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,x, algorithm="giac")

[Out]

1/6*(2*B*c^3 - 3*(7*B*c^3*d*e - A*c^3*e^2)*e^(-1)/(x*e + d) + 18*(7*B*c^3*d^2*e^2 - 2*A*c^3*d*e^3 + B*a*c^2*e^
4)*e^(-2)/(x*e + d)^2)*(x*e + d)^3*e^(-8) + (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^(-8)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(420*B*c^3*d^4*e^36/(x*e + d) - 126*B*c^3*d^5*e^36/(x*e +
d)^2 + 28*B*c^3*d^6*e^36/(x*e + d)^3 - 3*B*c^3*d^7*e^36/(x*e + d)^4 - 240*A*c^3*d^3*e^37/(x*e + d) + 90*A*c^3*
d^4*e^37/(x*e + d)^2 - 24*A*c^3*d^5*e^37/(x*e + d)^3 + 3*A*c^3*d^6*e^37/(x*e + d)^4 + 360*B*a*c^2*d^2*e^38/(x*
e + d) - 180*B*a*c^2*d^3*e^38/(x*e + d)^2 + 60*B*a*c^2*d^4*e^38/(x*e + d)^3 - 9*B*a*c^2*d^5*e^38/(x*e + d)^4 -
144*A*a*c^2*d*e^39/(x*e + d) + 108*A*a*c^2*d^2*e^39/(x*e + d)^2 - 48*A*a*c^2*d^3*e^39/(x*e + d)^3 + 9*A*a*c^2
*d^4*e^39/(x*e + d)^4 + 36*B*a^2*c*e^40/(x*e + d) - 54*B*a^2*c*d*e^40/(x*e + d)^2 + 36*B*a^2*c*d^2*e^40/(x*e +
d)^3 - 9*B*a^2*c*d^3*e^40/(x*e + d)^4 + 18*A*a^2*c*e^41/(x*e + d)^2 - 24*A*a^2*c*d*e^41/(x*e + d)^3 + 9*A*a^2
*c*d^2*e^41/(x*e + d)^4 + 4*B*a^3*e^42/(x*e + d)^3 - 3*B*a^3*d*e^42/(x*e + d)^4 + 3*A*a^3*e^43/(x*e + d)^4)*e^
(-44)

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maple [B]  time = 0.07, size = 632, normalized size = 2.01 \begin {gather*} -\frac {A \,a^{3}}{4 \left (e x +d \right )^{4} e}-\frac {3 A \,a^{2} c \,d^{2}}{4 \left (e x +d \right )^{4} e^{3}}-\frac {3 A a \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}-\frac {A \,c^{3} d^{6}}{4 \left (e x +d \right )^{4} e^{7}}+\frac {B \,a^{3} d}{4 \left (e x +d \right )^{4} e^{2}}+\frac {3 B \,a^{2} c \,d^{3}}{4 \left (e x +d \right )^{4} e^{4}}+\frac {3 B a \,c^{2} d^{5}}{4 \left (e x +d \right )^{4} e^{6}}+\frac {B \,c^{3} d^{7}}{4 \left (e x +d \right )^{4} e^{8}}+\frac {2 A \,a^{2} c d}{\left (e x +d \right )^{3} e^{3}}+\frac {4 A a \,c^{2} d^{3}}{\left (e x +d \right )^{3} e^{5}}+\frac {2 A \,c^{3} d^{5}}{\left (e x +d \right )^{3} e^{7}}-\frac {B \,a^{3}}{3 \left (e x +d \right )^{3} e^{2}}-\frac {3 B \,a^{2} c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}-\frac {5 B a \,c^{2} d^{4}}{\left (e x +d \right )^{3} e^{6}}-\frac {7 B \,c^{3} d^{6}}{3 \left (e x +d \right )^{3} e^{8}}+\frac {B \,c^{3} x^{3}}{3 e^{5}}-\frac {3 A \,a^{2} c}{2 \left (e x +d \right )^{2} e^{3}}-\frac {9 A a \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{5}}-\frac {15 A \,c^{3} d^{4}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {A \,c^{3} x^{2}}{2 e^{5}}+\frac {9 B \,a^{2} c d}{2 \left (e x +d \right )^{2} e^{4}}+\frac {15 B a \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{6}}+\frac {21 B \,c^{3} d^{5}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {5 B \,c^{3} d \,x^{2}}{2 e^{6}}+\frac {12 A a \,c^{2} d}{\left (e x +d \right ) e^{5}}+\frac {3 A a \,c^{2} \ln \left (e x +d \right )}{e^{5}}+\frac {20 A \,c^{3} d^{3}}{\left (e x +d \right ) e^{7}}+\frac {15 A \,c^{3} d^{2} \ln \left (e x +d \right )}{e^{7}}-\frac {5 A \,c^{3} d x}{e^{6}}-\frac {3 B \,a^{2} c}{\left (e x +d \right ) e^{4}}-\frac {30 B a \,c^{2} d^{2}}{\left (e x +d \right ) e^{6}}-\frac {15 B a \,c^{2} d \ln \left (e x +d \right )}{e^{6}}+\frac {3 B a \,c^{2} x}{e^{5}}-\frac {35 B \,c^{3} d^{4}}{\left (e x +d \right ) e^{8}}-\frac {35 B \,c^{3} d^{3} \ln \left (e x +d \right )}{e^{8}}+\frac {15 B \,c^{3} d^{2} x}{e^{7}} \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,x)

[Out]

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

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maxima [A]  time = 0.61, size = 487, normalized size = 1.55 \begin {gather*} -\frac {319 \, B c^{3} d^{7} - 171 \, A c^{3} d^{6} e + 231 \, B a c^{2} d^{5} e^{2} - 75 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} + 3 \, A a^{3} e^{7} + 12 \, {\left (35 \, B c^{3} d^{4} e^{3} - 20 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 18 \, {\left (63 \, B c^{3} d^{5} e^{2} - 35 \, A c^{3} d^{4} e^{3} + 50 \, B a c^{2} d^{3} e^{4} - 18 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + A a^{2} c e^{7}\right )} x^{2} + 4 \, {\left (259 \, B c^{3} d^{6} e - 141 \, A c^{3} d^{5} e^{2} + 195 \, B a c^{2} d^{4} e^{3} - 66 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + B a^{3} e^{7}\right )} x}{12 \, {\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 )}} + \frac {2 \, B c^{3} e^{2} x^{3} - 3 \, {\left (5 \, B c^{3} d e - A c^{3} e^{2}\right )} x^{2} + 6 \, {\left (15 \, B c^{3} d^{2} - 5 \, A c^{3} d e + 3 \, B a c^{2} e^{2}\right )} x}{6 \, e^{7}} - \frac {{\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 )} \log \left (e x + d\right )}{e^{8}} \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,x, algorithm="maxima")

[Out]

-1/12*(319*B*c^3*d^7 - 171*A*c^3*d^6*e + 231*B*a*c^2*d^5*e^2 - 75*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e^4 + 3*A*a^
2*c*d^2*e^5 + B*a^3*d*e^6 + 3*A*a^3*e^7 + 12*(35*B*c^3*d^4*e^3 - 20*A*c^3*d^3*e^4 + 30*B*a*c^2*d^2*e^5 - 12*A*
a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 + 18*(63*B*c^3*d^5*e^2 - 35*A*c^3*d^4*e^3 + 50*B*a*c^2*d^3*e^4 - 18*A*a*c^2*d
^2*e^5 + 3*B*a^2*c*d*e^6 + A*a^2*c*e^7)*x^2 + 4*(259*B*c^3*d^6*e - 141*A*c^3*d^5*e^2 + 195*B*a*c^2*d^4*e^3 - 6
6*A*a*c^2*d^3*e^4 + 9*B*a^2*c*d^2*e^5 + 3*A*a^2*c*d*e^6 + B*a^3*e^7)*x)/(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) + 1/6*(2*B*c^3*e^2*x^3 - 3*(5*B*c^3*d*e - A*c^3*e^2)*x^2 + 6*(15*B*c^3*d^2 - 5*A*
c^3*d*e + 3*B*a*c^2*e^2)*x)/e^7 - (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)*log(e*x +
d)/e^8

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mupad [B]  time = 1.80, size = 501, normalized size = 1.60 \begin {gather*} x^2\,\left (\frac {A\,c^3}{2\,e^5}-\frac {5\,B\,c^3\,d}{2\,e^6}\right )-x\,\left (\frac {5\,d\,\left (\frac {A\,c^3}{e^5}-\frac {5\,B\,c^3\,d}{e^6}\right )}{e}-\frac {3\,B\,a\,c^2}{e^5}+\frac {10\,B\,c^3\,d^2}{e^7}\right )-\frac {\frac {B\,a^3\,d\,e^6+3\,A\,a^3\,e^7+9\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5+231\,B\,a\,c^2\,d^5\,e^2-75\,A\,a\,c^2\,d^4\,e^3+319\,B\,c^3\,d^7-171\,A\,c^3\,d^6\,e}{12\,e}+x^2\,\left (\frac {9\,B\,a^2\,c\,d\,e^5}{2}+\frac {3\,A\,a^2\,c\,e^6}{2}+75\,B\,a\,c^2\,d^3\,e^3-27\,A\,a\,c^2\,d^2\,e^4+\frac {189\,B\,c^3\,d^5\,e}{2}-\frac {105\,A\,c^3\,d^4\,e^2}{2}\right )+x^3\,\left (3\,B\,a^2\,c\,e^6+30\,B\,a\,c^2\,d^2\,e^4-12\,A\,a\,c^2\,d\,e^5+35\,B\,c^3\,d^4\,e^2-20\,A\,c^3\,d^3\,e^3\right )+x\,\left (\frac {B\,a^3\,e^6}{3}+3\,B\,a^2\,c\,d^2\,e^4+A\,a^2\,c\,d\,e^5+65\,B\,a\,c^2\,d^4\,e^2-22\,A\,a\,c^2\,d^3\,e^3+\frac {259\,B\,c^3\,d^6}{3}-47\,A\,c^3\,d^5\,e\right )}{d^4\,e^7+4\,d^3\,e^8\,x+6\,d^2\,e^9\,x^2+4\,d\,e^{10}\,x^3+e^{11}\,x^4}-\frac {\ln \left (d+e\,x\right )\,\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 )}{e^8}+\frac {B\,c^3\,x^3}{3\,e^5} \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,x)

[Out]

x^2*((A*c^3)/(2*e^5) - (5*B*c^3*d)/(2*e^6)) - x*((5*d*((A*c^3)/e^5 - (5*B*c^3*d)/e^6))/e - (3*B*a*c^2)/e^5 + (
10*B*c^3*d^2)/e^7) - ((3*A*a^3*e^7 + 319*B*c^3*d^7 + B*a^3*d*e^6 - 171*A*c^3*d^6*e - 75*A*a*c^2*d^4*e^3 + 3*A*
a^2*c*d^2*e^5 + 231*B*a*c^2*d^5*e^2 + 9*B*a^2*c*d^3*e^4)/(12*e) + x^2*((3*A*a^2*c*e^6)/2 + (189*B*c^3*d^5*e)/2
- (105*A*c^3*d^4*e^2)/2 - 27*A*a*c^2*d^2*e^4 + 75*B*a*c^2*d^3*e^3 + (9*B*a^2*c*d*e^5)/2) + x^3*(3*B*a^2*c*e^6
- 20*A*c^3*d^3*e^3 + 35*B*c^3*d^4*e^2 + 30*B*a*c^2*d^2*e^4 - 12*A*a*c^2*d*e^5) + x*((B*a^3*e^6)/3 + (259*B*c^
3*d^6)/3 - 47*A*c^3*d^5*e - 22*A*a*c^2*d^3*e^3 + 65*B*a*c^2*d^4*e^2 + 3*B*a^2*c*d^2*e^4 + A*a^2*c*d*e^5))/(d^4
*e^7 + e^11*x^4 + 4*d^3*e^8*x + 4*d*e^10*x^3 + 6*d^2*e^9*x^2) - (log(d + e*x)*(35*B*c^3*d^3 - 3*A*a*c^2*e^3 -
15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2))/e^8 + (B*c^3*x^3)/(3*e^5)

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sympy [A]  time = 63.11, size = 537, normalized size = 1.71 \begin {gather*} \frac {B c^{3} x^{3}}{3 e^{5}} - \frac {c^{2} \left (- 3 A a e^{3} - 15 A c d^{2} e + 15 B a d e^{2} + 35 B c d^{3}\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{2} \left (\frac {A c^{3}}{2 e^{5}} - \frac {5 B c^{3} d}{2 e^{6}}\right ) + x \left (- \frac {5 A c^{3} d}{e^{6}} + \frac {3 B a c^{2}}{e^{5}} + \frac {15 B c^{3} d^{2}}{e^{7}}\right ) + \frac {- 3 A a^{3} e^{7} - 3 A a^{2} c d^{2} e^{5} + 75 A a c^{2} d^{4} e^{3} + 171 A c^{3} d^{6} e - B a^{3} d e^{6} - 9 B a^{2} c d^{3} e^{4} - 231 B a c^{2} d^{5} e^{2} - 319 B c^{3} d^{7} + x^{3} \left (144 A a c^{2} d e^{6} + 240 A c^{3} d^{3} e^{4} - 36 B a^{2} c e^{7} - 360 B a c^{2} d^{2} e^{5} - 420 B c^{3} d^{4} e^{3}\right ) + x^{2} \left (- 18 A a^{2} c e^{7} + 324 A a c^{2} d^{2} e^{5} + 630 A c^{3} d^{4} e^{3} - 54 B a^{2} c d e^{6} - 900 B a c^{2} d^{3} e^{4} - 1134 B c^{3} d^{5} e^{2}\right ) + x \left (- 12 A a^{2} c d e^{6} + 264 A a c^{2} d^{3} e^{4} + 564 A c^{3} d^{5} e^{2} - 4 B a^{3} e^{7} - 36 B a^{2} c d^{2} e^{5} - 780 B a c^{2} d^{4} e^{3} - 1036 B c^{3} d^{6} e\right )}{12 d^{4} e^{8} + 48 d^{3} e^{9} x + 72 d^{2} e^{10} x^{2} + 48 d e^{11} x^{3} + 12 e^{12} x^{4}} \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,x)

[Out]

B*c**3*x**3/(3*e**5) - c**2*(-3*A*a*e**3 - 15*A*c*d**2*e + 15*B*a*d*e**2 + 35*B*c*d**3)*log(d + e*x)/e**8 + x*
*2*(A*c**3/(2*e**5) - 5*B*c**3*d/(2*e**6)) + x*(-5*A*c**3*d/e**6 + 3*B*a*c**2/e**5 + 15*B*c**3*d**2/e**7) + (-
3*A*a**3*e**7 - 3*A*a**2*c*d**2*e**5 + 75*A*a*c**2*d**4*e**3 + 171*A*c**3*d**6*e - B*a**3*d*e**6 - 9*B*a**2*c*
d**3*e**4 - 231*B*a*c**2*d**5*e**2 - 319*B*c**3*d**7 + x**3*(144*A*a*c**2*d*e**6 + 240*A*c**3*d**3*e**4 - 36*B
*a**2*c*e**7 - 360*B*a*c**2*d**2*e**5 - 420*B*c**3*d**4*e**3) + x**2*(-18*A*a**2*c*e**7 + 324*A*a*c**2*d**2*e*
*5 + 630*A*c**3*d**4*e**3 - 54*B*a**2*c*d*e**6 - 900*B*a*c**2*d**3*e**4 - 1134*B*c**3*d**5*e**2) + x*(-12*A*a*
*2*c*d*e**6 + 264*A*a*c**2*d**3*e**4 + 564*A*c**3*d**5*e**2 - 4*B*a**3*e**7 - 36*B*a**2*c*d**2*e**5 - 780*B*a*
c**2*d**4*e**3 - 1036*B*c**3*d**6*e))/(12*d**4*e**8 + 48*d**3*e**9*x + 72*d**2*e**10*x**2 + 48*d*e**11*x**3 +
12*e**12*x**4)

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