3.12.38 \(\int \frac {(A+B x) (a+c x^2)^2}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=206 \[ -\frac {c \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^6}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{8 e^6 (d+e x)^8}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{9 e^6 (d+e x)^9}+\frac {2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6 (d+e x)^7}+\frac {c^2 (5 B d-A e)}{5 e^6 (d+e x)^5}-\frac {B c^2}{4 e^6 (d+e x)^4} \]

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Rubi [A]  time = 0.14, antiderivative size = 206, 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 (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^6}+\frac {2 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6 (d+e x)^7}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{8 e^6 (d+e x)^8}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{9 e^6 (d+e x)^9}+\frac {c^2 (5 B d-A e)}{5 e^6 (d+e x)^5}-\frac {B c^2}{4 e^6 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^10,x]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^2)/(9*e^6*(d + e*x)^9) - ((c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(8*e
^6*(d + e*x)^8) + (2*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(7*e^6*(d + e*x)^7) - (c*(5*B*c*d^2
- 2*A*c*d*e + a*B*e^2))/(3*e^6*(d + e*x)^6) + (c^2*(5*B*d - A*e))/(5*e^6*(d + e*x)^5) - (B*c^2)/(4*e^6*(d + e*
x)^4)

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 )^2}{(d+e x)^{10}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^{10}}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^9}+\frac {2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^8}-\frac {2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 (d+e x)^7}+\frac {c^2 (-5 B d+A e)}{e^5 (d+e x)^6}+\frac {B c^2}{e^5 (d+e x)^5}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{9 e^6 (d+e x)^9}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{8 e^6 (d+e x)^8}+\frac {2 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{7 e^6 (d+e x)^7}-\frac {c \left (5 B c d^2-2 A c d e+a B e^2\right )}{3 e^6 (d+e x)^6}+\frac {c^2 (5 B d-A e)}{5 e^6 (d+e x)^5}-\frac {B c^2}{4 e^6 (d+e x)^4}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 202, normalized size = 0.98 \begin {gather*} -\frac {4 A e \left (70 a^2 e^4+5 a c e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+c^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+5 B \left (7 a^2 e^4 (d+9 e x)+2 a c e^2 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+c^2 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )}{2520 e^6 (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^10,x]

[Out]

-1/2520*(4*A*e*(70*a^2*e^4 + 5*a*c*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + c^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 +
84*d*e^3*x^3 + 126*e^4*x^4)) + 5*B*(7*a^2*e^4*(d + 9*e*x) + 2*a*c*e^2*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3
*x^3) + c^2*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)))/(e^6*(d + e*x)
^9)

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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 )^2}{(d+e x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^10,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^10, x]

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fricas [A]  time = 0.39, size = 339, normalized size = 1.65 \begin {gather*} -\frac {630 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 4 \, A c^{2} d^{4} e + 10 \, B a c d^{3} e^{2} + 20 \, A a c d^{2} e^{3} + 35 \, B a^{2} d e^{4} + 280 \, A a^{2} e^{5} + 126 \, {\left (5 \, B c^{2} d e^{4} + 4 \, A c^{2} e^{5}\right )} x^{4} + 84 \, {\left (5 \, B c^{2} d^{2} e^{3} + 4 \, A c^{2} d e^{4} + 10 \, B a c e^{5}\right )} x^{3} + 36 \, {\left (5 \, B c^{2} d^{3} e^{2} + 4 \, A c^{2} d^{2} e^{3} + 10 \, B a c d e^{4} + 20 \, A a c e^{5}\right )} x^{2} + 9 \, {\left (5 \, B c^{2} d^{4} e + 4 \, A c^{2} d^{3} e^{2} + 10 \, B a c d^{2} e^{3} + 20 \, A a c d e^{4} + 35 \, B a^{2} e^{5}\right )} x}{2520 \, {\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^10,x, algorithm="fricas")

[Out]

-1/2520*(630*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 4*A*c^2*d^4*e + 10*B*a*c*d^3*e^2 + 20*A*a*c*d^2*e^3 + 35*B*a^2*d*e^
4 + 280*A*a^2*e^5 + 126*(5*B*c^2*d*e^4 + 4*A*c^2*e^5)*x^4 + 84*(5*B*c^2*d^2*e^3 + 4*A*c^2*d*e^4 + 10*B*a*c*e^5
)*x^3 + 36*(5*B*c^2*d^3*e^2 + 4*A*c^2*d^2*e^3 + 10*B*a*c*d*e^4 + 20*A*a*c*e^5)*x^2 + 9*(5*B*c^2*d^4*e + 4*A*c^
2*d^3*e^2 + 10*B*a*c*d^2*e^3 + 20*A*a*c*d*e^4 + 35*B*a^2*e^5)*x)/(e^15*x^9 + 9*d*e^14*x^8 + 36*d^2*e^13*x^7 +
84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d^5*e^10*x^4 + 84*d^6*e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*e^
6)

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giac [A]  time = 0.19, size = 242, normalized size = 1.17 \begin {gather*} -\frac {{\left (630 \, B c^{2} x^{5} e^{5} + 630 \, B c^{2} d x^{4} e^{4} + 420 \, B c^{2} d^{2} x^{3} e^{3} + 180 \, B c^{2} d^{3} x^{2} e^{2} + 45 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 504 \, A c^{2} x^{4} e^{5} + 336 \, A c^{2} d x^{3} e^{4} + 144 \, A c^{2} d^{2} x^{2} e^{3} + 36 \, A c^{2} d^{3} x e^{2} + 4 \, A c^{2} d^{4} e + 840 \, B a c x^{3} e^{5} + 360 \, B a c d x^{2} e^{4} + 90 \, B a c d^{2} x e^{3} + 10 \, B a c d^{3} e^{2} + 720 \, A a c x^{2} e^{5} + 180 \, A a c d x e^{4} + 20 \, A a c d^{2} e^{3} + 315 \, B a^{2} x e^{5} + 35 \, B a^{2} d e^{4} + 280 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{2520 \, {\left (x e + d\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/2520*(630*B*c^2*x^5*e^5 + 630*B*c^2*d*x^4*e^4 + 420*B*c^2*d^2*x^3*e^3 + 180*B*c^2*d^3*x^2*e^2 + 45*B*c^2*d^
4*x*e + 5*B*c^2*d^5 + 504*A*c^2*x^4*e^5 + 336*A*c^2*d*x^3*e^4 + 144*A*c^2*d^2*x^2*e^3 + 36*A*c^2*d^3*x*e^2 + 4
*A*c^2*d^4*e + 840*B*a*c*x^3*e^5 + 360*B*a*c*d*x^2*e^4 + 90*B*a*c*d^2*x*e^3 + 10*B*a*c*d^3*e^2 + 720*A*a*c*x^2
*e^5 + 180*A*a*c*d*x*e^4 + 20*A*a*c*d^2*e^3 + 315*B*a^2*x*e^5 + 35*B*a^2*d*e^4 + 280*A*a^2*e^5)*e^(-6)/(x*e +
d)^9

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maple [A]  time = 0.05, size = 249, normalized size = 1.21 \begin {gather*} -\frac {B \,c^{2}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {\left (A e -5 B d \right ) c^{2}}{5 \left (e x +d \right )^{5} e^{6}}+\frac {\left (2 A c d e -B a \,e^{2}-5 B c \,d^{2}\right ) c}{3 \left (e x +d \right )^{6} e^{6}}-\frac {2 \left (a A \,e^{3}+3 A c \,d^{2} e -3 a B d \,e^{2}-5 B c \,d^{3}\right ) c}{7 \left (e x +d \right )^{7} e^{6}}-\frac {A \,a^{2} e^{5}+2 A \,d^{2} a c \,e^{3}+A \,c^{2} d^{4} e -B d \,a^{2} e^{4}-2 B \,d^{3} a c \,e^{2}-B \,d^{5} c^{2}}{9 \left (e x +d \right )^{9} e^{6}}-\frac {-4 A d a c \,e^{3}-4 A \,c^{2} d^{3} e +B \,a^{2} e^{4}+6 B \,d^{2} a c \,e^{2}+5 B \,d^{4} c^{2}}{8 \left (e x +d \right )^{8} e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^10,x)

[Out]

-1/9*(A*a^2*e^5+2*A*a*c*d^2*e^3+A*c^2*d^4*e-B*a^2*d*e^4-2*B*a*c*d^3*e^2-B*c^2*d^5)/e^6/(e*x+d)^9-1/4*B*c^2/e^6
/(e*x+d)^4-1/8*(-4*A*a*c*d*e^3-4*A*c^2*d^3*e+B*a^2*e^4+6*B*a*c*d^2*e^2+5*B*c^2*d^4)/e^6/(e*x+d)^8+1/3*c*(2*A*c
*d*e-B*a*e^2-5*B*c*d^2)/e^6/(e*x+d)^6-2/7*c*(A*a*e^3+3*A*c*d^2*e-3*B*a*d*e^2-5*B*c*d^3)/e^6/(e*x+d)^7-1/5*c^2*
(A*e-5*B*d)/e^6/(e*x+d)^5

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maxima [A]  time = 0.78, size = 339, normalized size = 1.65 \begin {gather*} -\frac {630 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 4 \, A c^{2} d^{4} e + 10 \, B a c d^{3} e^{2} + 20 \, A a c d^{2} e^{3} + 35 \, B a^{2} d e^{4} + 280 \, A a^{2} e^{5} + 126 \, {\left (5 \, B c^{2} d e^{4} + 4 \, A c^{2} e^{5}\right )} x^{4} + 84 \, {\left (5 \, B c^{2} d^{2} e^{3} + 4 \, A c^{2} d e^{4} + 10 \, B a c e^{5}\right )} x^{3} + 36 \, {\left (5 \, B c^{2} d^{3} e^{2} + 4 \, A c^{2} d^{2} e^{3} + 10 \, B a c d e^{4} + 20 \, A a c e^{5}\right )} x^{2} + 9 \, {\left (5 \, B c^{2} d^{4} e + 4 \, A c^{2} d^{3} e^{2} + 10 \, B a c d^{2} e^{3} + 20 \, A a c d e^{4} + 35 \, B a^{2} e^{5}\right )} x}{2520 \, {\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^10,x, algorithm="maxima")

[Out]

-1/2520*(630*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 4*A*c^2*d^4*e + 10*B*a*c*d^3*e^2 + 20*A*a*c*d^2*e^3 + 35*B*a^2*d*e^
4 + 280*A*a^2*e^5 + 126*(5*B*c^2*d*e^4 + 4*A*c^2*e^5)*x^4 + 84*(5*B*c^2*d^2*e^3 + 4*A*c^2*d*e^4 + 10*B*a*c*e^5
)*x^3 + 36*(5*B*c^2*d^3*e^2 + 4*A*c^2*d^2*e^3 + 10*B*a*c*d*e^4 + 20*A*a*c*e^5)*x^2 + 9*(5*B*c^2*d^4*e + 4*A*c^
2*d^3*e^2 + 10*B*a*c*d^2*e^3 + 20*A*a*c*d*e^4 + 35*B*a^2*e^5)*x)/(e^15*x^9 + 9*d*e^14*x^8 + 36*d^2*e^13*x^7 +
84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d^5*e^10*x^4 + 84*d^6*e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*e^
6)

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mupad [B]  time = 1.76, size = 321, normalized size = 1.56 \begin {gather*} -\frac {\frac {35\,B\,a^2\,d\,e^4+280\,A\,a^2\,e^5+10\,B\,a\,c\,d^3\,e^2+20\,A\,a\,c\,d^2\,e^3+5\,B\,c^2\,d^5+4\,A\,c^2\,d^4\,e}{2520\,e^6}+\frac {x\,\left (35\,B\,a^2\,e^4+10\,B\,a\,c\,d^2\,e^2+20\,A\,a\,c\,d\,e^3+5\,B\,c^2\,d^4+4\,A\,c^2\,d^3\,e\right )}{280\,e^5}+\frac {c\,x^3\,\left (5\,B\,c\,d^2+4\,A\,c\,d\,e+10\,B\,a\,e^2\right )}{30\,e^3}+\frac {c^2\,x^4\,\left (4\,A\,e+5\,B\,d\right )}{20\,e^2}+\frac {c\,x^2\,\left (5\,B\,c\,d^3+4\,A\,c\,d^2\,e+10\,B\,a\,d\,e^2+20\,A\,a\,e^3\right )}{70\,e^4}+\frac {B\,c^2\,x^5}{4\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^2*(A + B*x))/(d + e*x)^10,x)

[Out]

-((280*A*a^2*e^5 + 5*B*c^2*d^5 + 35*B*a^2*d*e^4 + 4*A*c^2*d^4*e + 20*A*a*c*d^2*e^3 + 10*B*a*c*d^3*e^2)/(2520*e
^6) + (x*(35*B*a^2*e^4 + 5*B*c^2*d^4 + 4*A*c^2*d^3*e + 20*A*a*c*d*e^3 + 10*B*a*c*d^2*e^2))/(280*e^5) + (c*x^3*
(10*B*a*e^2 + 5*B*c*d^2 + 4*A*c*d*e))/(30*e^3) + (c^2*x^4*(4*A*e + 5*B*d))/(20*e^2) + (c*x^2*(20*A*a*e^3 + 5*B
*c*d^3 + 10*B*a*d*e^2 + 4*A*c*d^2*e))/(70*e^4) + (B*c^2*x^5)/(4*e))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*
x^2 + 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**10,x)

[Out]

Timed out

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