3.257 \(\int \frac {(b x+c x^2)^3}{(d+e x)^9} \, dx\)

Optimal. Leaf size=231 \[ -\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7 (d+e x)^5}-\frac {d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^6}+\frac {c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac {d^3 (c d-b e)^3}{8 e^7 (d+e x)^8}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{7 e^7 (d+e x)^7}-\frac {c^3}{2 e^7 (d+e x)^2} \]

[Out]

-1/8*d^3*(-b*e+c*d)^3/e^7/(e*x+d)^8+3/7*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)/e^7/(e*x+d)^7-1/2*d*(-b*e+c*d)*(b^2*e^2-
5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^6+1/5*(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)/e^7/(e*x+d)^5-3/4*c*(b^2*e
^2-5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^4+c^2*(-b*e+2*c*d)/e^7/(e*x+d)^3-1/2*c^3/e^7/(e*x+d)^2

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Rubi [A]  time = 0.16, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ -\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7 (d+e x)^5}-\frac {d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^6}+\frac {c^2 (2 c d-b e)}{e^7 (d+e x)^3}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{7 e^7 (d+e x)^7}-\frac {d^3 (c d-b e)^3}{8 e^7 (d+e x)^8}-\frac {c^3}{2 e^7 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^9,x]

[Out]

-(d^3*(c*d - b*e)^3)/(8*e^7*(d + e*x)^8) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(7*e^7*(d + e*x)^7) - (d*(c*d -
b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(2*e^7*(d + e*x)^6) + ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e
^2))/(5*e^7*(d + e*x)^5) - (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(4*e^7*(d + e*x)^4) + (c^2*(2*c*d - b*e))/(
e^7*(d + e*x)^3) - c^3/(2*e^7*(d + e*x)^2)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 221, normalized size = 0.96 \[ -\frac {b^3 e^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 b^2 c e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b c^2 e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )}{280 e^7 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^9,x]

[Out]

-1/280*(b^3*e^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*b^2*c*e^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) + 5*b*c^2*e*(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 + 5
6*e^5*x^5) + 5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6
*x^6))/(e^7*(d + e*x)^8)

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fricas [A]  time = 1.01, size = 344, normalized size = 1.49 \[ -\frac {140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 280 \, {\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} + 56 \, {\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 28 \, {\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 8 \, {\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 3 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{280 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/280*(140*c^3*e^6*x^6 + 5*c^3*d^6 + 5*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + b^3*d^3*e^3 + 280*(c^3*d*e^5 + b*c^2*e
^6)*x^5 + 70*(5*c^3*d^2*e^4 + 5*b*c^2*d*e^5 + 3*b^2*c*e^6)*x^4 + 56*(5*c^3*d^3*e^3 + 5*b*c^2*d^2*e^4 + 3*b^2*c
*d*e^5 + b^3*e^6)*x^3 + 28*(5*c^3*d^4*e^2 + 5*b*c^2*d^3*e^3 + 3*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 8*(5*c^3*d^5*
e + 5*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^
12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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giac [A]  time = 0.16, size = 267, normalized size = 1.16 \[ -\frac {{\left (140 \, c^{3} x^{6} e^{6} + 280 \, c^{3} d x^{5} e^{5} + 350 \, c^{3} d^{2} x^{4} e^{4} + 280 \, c^{3} d^{3} x^{3} e^{3} + 140 \, c^{3} d^{4} x^{2} e^{2} + 40 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 280 \, b c^{2} x^{5} e^{6} + 350 \, b c^{2} d x^{4} e^{5} + 280 \, b c^{2} d^{2} x^{3} e^{4} + 140 \, b c^{2} d^{3} x^{2} e^{3} + 40 \, b c^{2} d^{4} x e^{2} + 5 \, b c^{2} d^{5} e + 210 \, b^{2} c x^{4} e^{6} + 168 \, b^{2} c d x^{3} e^{5} + 84 \, b^{2} c d^{2} x^{2} e^{4} + 24 \, b^{2} c d^{3} x e^{3} + 3 \, b^{2} c d^{4} e^{2} + 56 \, b^{3} x^{3} e^{6} + 28 \, b^{3} d x^{2} e^{5} + 8 \, b^{3} d^{2} x e^{4} + b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{280 \, {\left (x e + d\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/280*(140*c^3*x^6*e^6 + 280*c^3*d*x^5*e^5 + 350*c^3*d^2*x^4*e^4 + 280*c^3*d^3*x^3*e^3 + 140*c^3*d^4*x^2*e^2
+ 40*c^3*d^5*x*e + 5*c^3*d^6 + 280*b*c^2*x^5*e^6 + 350*b*c^2*d*x^4*e^5 + 280*b*c^2*d^2*x^3*e^4 + 140*b*c^2*d^3
*x^2*e^3 + 40*b*c^2*d^4*x*e^2 + 5*b*c^2*d^5*e + 210*b^2*c*x^4*e^6 + 168*b^2*c*d*x^3*e^5 + 84*b^2*c*d^2*x^2*e^4
+ 24*b^2*c*d^3*x*e^3 + 3*b^2*c*d^4*e^2 + 56*b^3*x^3*e^6 + 28*b^3*d*x^2*e^5 + 8*b^3*d^2*x*e^4 + b^3*d^3*e^3)*e
^(-7)/(x*e + d)^8

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maple [A]  time = 0.05, size = 274, normalized size = 1.19 \[ -\frac {c^{3}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {\left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) d^{3}}{8 \left (e x +d \right )^{8} e^{7}}-\frac {\left (b e -2 c d \right ) c^{2}}{\left (e x +d \right )^{3} e^{7}}-\frac {3 \left (b^{3} e^{3}-4 b^{2} c d \,e^{2}+5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right ) d^{2}}{7 \left (e x +d \right )^{7} e^{7}}-\frac {3 \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) c}{4 \left (e x +d \right )^{4} e^{7}}+\frac {\left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right ) d}{2 \left (e x +d \right )^{6} e^{7}}-\frac {b^{3} e^{3}-12 b^{2} c d \,e^{2}+30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{5 \left (e x +d \right )^{5} e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d)^9,x)

[Out]

-c^2*(b*e-2*c*d)/e^7/(e*x+d)^3-1/5*(b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/e^7/(e*x+d)^5-1/2*c^3/e^
7/(e*x+d)^2+1/8*d^3*(b^3*e^3-3*b^2*c*d*e^2+3*b*c^2*d^2*e-c^3*d^3)/e^7/(e*x+d)^8-3/4*c*(b^2*e^2-5*b*c*d*e+5*c^2
*d^2)/e^7/(e*x+d)^4+1/2*d*(b^3*e^3-6*b^2*c*d*e^2+10*b*c^2*d^2*e-5*c^3*d^3)/e^7/(e*x+d)^6-3/7*d^2*(b^3*e^3-4*b^
2*c*d*e^2+5*b*c^2*d^2*e-2*c^3*d^3)/e^7/(e*x+d)^7

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maxima [A]  time = 1.59, size = 344, normalized size = 1.49 \[ -\frac {140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 280 \, {\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} + 56 \, {\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 28 \, {\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 8 \, {\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 3 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{280 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^9,x, algorithm="maxima")

[Out]

-1/280*(140*c^3*e^6*x^6 + 5*c^3*d^6 + 5*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + b^3*d^3*e^3 + 280*(c^3*d*e^5 + b*c^2*e
^6)*x^5 + 70*(5*c^3*d^2*e^4 + 5*b*c^2*d*e^5 + 3*b^2*c*e^6)*x^4 + 56*(5*c^3*d^3*e^3 + 5*b*c^2*d^2*e^4 + 3*b^2*c
*d*e^5 + b^3*e^6)*x^3 + 28*(5*c^3*d^4*e^2 + 5*b*c^2*d^3*e^3 + 3*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 8*(5*c^3*d^5*
e + 5*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^
12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

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mupad [B]  time = 0.24, size = 325, normalized size = 1.41 \[ -\frac {\frac {d^3\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+5\,c^3\,d^3\right )}{280\,e^7}+\frac {x^3\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+5\,c^3\,d^3\right )}{5\,e^4}+\frac {c^3\,x^6}{2\,e}+\frac {c^2\,x^5\,\left (b\,e+c\,d\right )}{e^2}+\frac {c\,x^4\,\left (3\,b^2\,e^2+5\,b\,c\,d\,e+5\,c^2\,d^2\right )}{4\,e^3}+\frac {d\,x^2\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+5\,c^3\,d^3\right )}{10\,e^5}+\frac {d^2\,x\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+5\,c^3\,d^3\right )}{35\,e^6}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3/(d + e*x)^9,x)

[Out]

-((d^3*(b^3*e^3 + 5*c^3*d^3 + 5*b*c^2*d^2*e + 3*b^2*c*d*e^2))/(280*e^7) + (x^3*(b^3*e^3 + 5*c^3*d^3 + 5*b*c^2*
d^2*e + 3*b^2*c*d*e^2))/(5*e^4) + (c^3*x^6)/(2*e) + (c^2*x^5*(b*e + c*d))/e^2 + (c*x^4*(3*b^2*e^2 + 5*c^2*d^2
+ 5*b*c*d*e))/(4*e^3) + (d*x^2*(b^3*e^3 + 5*c^3*d^3 + 5*b*c^2*d^2*e + 3*b^2*c*d*e^2))/(10*e^5) + (d^2*x*(b^3*e
^3 + 5*c^3*d^3 + 5*b*c^2*d^2*e + 3*b^2*c*d*e^2))/(35*e^6))/(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 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**9,x)

[Out]

Timed out

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