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

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

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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 {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^5}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\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 )}{7 e^6 (d+e x)^7}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{8 e^6 (d+e x)^8}+\frac {c^2 (5 B d-A e)}{4 e^6 (d+e x)^4}-\frac {B c^2}{3 e^6 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.09, size = 202, normalized size = 0.98 \begin {gather*} -\frac {A e \left (105 a^2 e^4+10 a c e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+3 c^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+B \left (15 a^2 e^4 (d+8 e x)+6 a c e^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+5 c^2 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{840 e^6 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/840*(A*e*(105*a^2*e^4 + 10*a*c*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*c^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 +
56*d*e^3*x^3 + 70*e^4*x^4)) + B*(15*a^2*e^4*(d + 8*e*x) + 6*a*c*e^2*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*
x^3) + 5*c^2*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)))/(e^6*(d + e*x)^
8)

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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)^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 328, normalized size = 1.59 \begin {gather*} -\frac {280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 3 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} + 10 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} + 105 \, A a^{2} e^{5} + 70 \, {\left (5 \, B c^{2} d e^{4} + 3 \, A c^{2} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B c^{2} d^{2} e^{3} + 3 \, A c^{2} d e^{4} + 6 \, B a c e^{5}\right )} x^{3} + 28 \, {\left (5 \, B c^{2} d^{3} e^{2} + 3 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} + 10 \, A a c e^{5}\right )} x^{2} + 8 \, {\left (5 \, B c^{2} d^{4} e + 3 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 10 \, A a c d e^{4} + 15 \, B a^{2} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} 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)^9,x, algorithm="fricas")

[Out]

-1/840*(280*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 3*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 + 10*A*a*c*d^2*e^3 + 15*B*a^2*d*e^4
+ 105*A*a^2*e^5 + 70*(5*B*c^2*d*e^4 + 3*A*c^2*e^5)*x^4 + 56*(5*B*c^2*d^2*e^3 + 3*A*c^2*d*e^4 + 6*B*a*c*e^5)*x^
3 + 28*(5*B*c^2*d^3*e^2 + 3*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 + 10*A*a*c*e^5)*x^2 + 8*(5*B*c^2*d^4*e + 3*A*c^2*d^3
*e^2 + 6*B*a*c*d^2*e^3 + 10*A*a*c*d*e^4 + 15*B*a^2*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7 + 28*d^2*e^12*x^6 + 56*d^3
*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)

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giac [A]  time = 0.19, size = 242, normalized size = 1.17 \begin {gather*} -\frac {{\left (280 \, B c^{2} x^{5} e^{5} + 350 \, B c^{2} d x^{4} e^{4} + 280 \, B c^{2} d^{2} x^{3} e^{3} + 140 \, B c^{2} d^{3} x^{2} e^{2} + 40 \, B c^{2} d^{4} x e + 5 \, B c^{2} d^{5} + 210 \, A c^{2} x^{4} e^{5} + 168 \, A c^{2} d x^{3} e^{4} + 84 \, A c^{2} d^{2} x^{2} e^{3} + 24 \, A c^{2} d^{3} x e^{2} + 3 \, A c^{2} d^{4} e + 336 \, B a c x^{3} e^{5} + 168 \, B a c d x^{2} e^{4} + 48 \, B a c d^{2} x e^{3} + 6 \, B a c d^{3} e^{2} + 280 \, A a c x^{2} e^{5} + 80 \, A a c d x e^{4} + 10 \, A a c d^{2} e^{3} + 120 \, B a^{2} x e^{5} + 15 \, B a^{2} d e^{4} + 105 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{840 \, {\left (x e + d\right )}^{8}} \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)^9,x, algorithm="giac")

[Out]

-1/840*(280*B*c^2*x^5*e^5 + 350*B*c^2*d*x^4*e^4 + 280*B*c^2*d^2*x^3*e^3 + 140*B*c^2*d^3*x^2*e^2 + 40*B*c^2*d^4
*x*e + 5*B*c^2*d^5 + 210*A*c^2*x^4*e^5 + 168*A*c^2*d*x^3*e^4 + 84*A*c^2*d^2*x^2*e^3 + 24*A*c^2*d^3*x*e^2 + 3*A
*c^2*d^4*e + 336*B*a*c*x^3*e^5 + 168*B*a*c*d*x^2*e^4 + 48*B*a*c*d^2*x*e^3 + 6*B*a*c*d^3*e^2 + 280*A*a*c*x^2*e^
5 + 80*A*a*c*d*x*e^4 + 10*A*a*c*d^2*e^3 + 120*B*a^2*x*e^5 + 15*B*a^2*d*e^4 + 105*A*a^2*e^5)*e^(-6)/(x*e + d)^8

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maple [A]  time = 0.06, size = 249, normalized size = 1.21 \begin {gather*} -\frac {B \,c^{2}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {\left (A e -5 B d \right ) c^{2}}{4 \left (e x +d \right )^{4} e^{6}}+\frac {2 \left (2 A c d e -B a \,e^{2}-5 B c \,d^{2}\right ) c}{5 \left (e x +d \right )^{5} e^{6}}-\frac {\left (a A \,e^{3}+3 A c \,d^{2} e -3 a B d \,e^{2}-5 B c \,d^{3}\right ) c}{3 \left (e x +d \right )^{6} 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}}{8 \left (e x +d \right )^{8} 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}}{7 \left (e x +d \right )^{7} 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)^9,x)

[Out]

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

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maxima [A]  time = 0.55, size = 328, normalized size = 1.59 \begin {gather*} -\frac {280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 3 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} + 10 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} + 105 \, A a^{2} e^{5} + 70 \, {\left (5 \, B c^{2} d e^{4} + 3 \, A c^{2} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B c^{2} d^{2} e^{3} + 3 \, A c^{2} d e^{4} + 6 \, B a c e^{5}\right )} x^{3} + 28 \, {\left (5 \, B c^{2} d^{3} e^{2} + 3 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} + 10 \, A a c e^{5}\right )} x^{2} + 8 \, {\left (5 \, B c^{2} d^{4} e + 3 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 10 \, A a c d e^{4} + 15 \, B a^{2} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} 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)^9,x, algorithm="maxima")

[Out]

-1/840*(280*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 3*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 + 10*A*a*c*d^2*e^3 + 15*B*a^2*d*e^4
+ 105*A*a^2*e^5 + 70*(5*B*c^2*d*e^4 + 3*A*c^2*e^5)*x^4 + 56*(5*B*c^2*d^2*e^3 + 3*A*c^2*d*e^4 + 6*B*a*c*e^5)*x^
3 + 28*(5*B*c^2*d^3*e^2 + 3*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 + 10*A*a*c*e^5)*x^2 + 8*(5*B*c^2*d^4*e + 3*A*c^2*d^3
*e^2 + 6*B*a*c*d^2*e^3 + 10*A*a*c*d*e^4 + 15*B*a^2*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7 + 28*d^2*e^12*x^6 + 56*d^3
*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)

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mupad [B]  time = 0.12, size = 310, normalized size = 1.50 \begin {gather*} -\frac {\frac {15\,B\,a^2\,d\,e^4+105\,A\,a^2\,e^5+6\,B\,a\,c\,d^3\,e^2+10\,A\,a\,c\,d^2\,e^3+5\,B\,c^2\,d^5+3\,A\,c^2\,d^4\,e}{840\,e^6}+\frac {x\,\left (15\,B\,a^2\,e^4+6\,B\,a\,c\,d^2\,e^2+10\,A\,a\,c\,d\,e^3+5\,B\,c^2\,d^4+3\,A\,c^2\,d^3\,e\right )}{105\,e^5}+\frac {c\,x^3\,\left (5\,B\,c\,d^2+3\,A\,c\,d\,e+6\,B\,a\,e^2\right )}{15\,e^3}+\frac {c^2\,x^4\,\left (3\,A\,e+5\,B\,d\right )}{12\,e^2}+\frac {c\,x^2\,\left (5\,B\,c\,d^3+3\,A\,c\,d^2\,e+6\,B\,a\,d\,e^2+10\,A\,a\,e^3\right )}{30\,e^4}+\frac {B\,c^2\,x^5}{3\,e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \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)^9,x)

[Out]

-((105*A*a^2*e^5 + 5*B*c^2*d^5 + 15*B*a^2*d*e^4 + 3*A*c^2*d^4*e + 10*A*a*c*d^2*e^3 + 6*B*a*c*d^3*e^2)/(840*e^6
) + (x*(15*B*a^2*e^4 + 5*B*c^2*d^4 + 3*A*c^2*d^3*e + 10*A*a*c*d*e^3 + 6*B*a*c*d^2*e^2))/(105*e^5) + (c*x^3*(6*
B*a*e^2 + 5*B*c*d^2 + 3*A*c*d*e))/(15*e^3) + (c^2*x^4*(3*A*e + 5*B*d))/(12*e^2) + (c*x^2*(10*A*a*e^3 + 5*B*c*d
^3 + 6*B*a*d*e^2 + 3*A*c*d^2*e))/(30*e^4) + (B*c^2*x^5)/(3*e))/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 +
56*d^5*e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*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)**9,x)

[Out]

Timed out

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