3.3.34 \(\int \frac {(b x+c x^2)^3}{(d+e x)^8} \, dx\)

Optimal. Leaf size=230 \[ -\frac {c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {d^3 (c d-b e)^3}{7 e^7 (d+e x)^7}+\frac {d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^6}-\frac {c^3}{e^7 (d+e x)} \]

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Rubi [A]  time = 0.16, antiderivative size = 230, 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} \begin {gather*} -\frac {c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}+\frac {d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^6}-\frac {d^3 (c d-b e)^3}{7 e^7 (d+e x)^7}-\frac {c^3}{e^7 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^8,x]

[Out]

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

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

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Mathematica [A]  time = 0.07, size = 221, normalized size = 0.96 \begin {gather*} -\frac {b^3 e^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 b^2 c e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+10 b c^2 e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+20 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )}{140 e^7 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^8,x]

[Out]

-1/140*(b^3*e^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*b^2*c*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2
+ 35*d*e^3*x^3 + 35*e^4*x^4) + 10*b*c^2*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 +
21*e^5*x^5) + 20*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^
6*x^6))/(e^7*(d + e*x)^7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x + c*x^2)^3/(d + e*x)^8,x]

[Out]

IntegrateAlgebraic[(b*x + c*x^2)^3/(d + e*x)^8, x]

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fricas [A]  time = 0.39, size = 334, normalized size = 1.45 \begin {gather*} -\frac {140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 210 \, {\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, b^{2} c e^{6}\right )} x^{4} + 35 \, {\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 21 \, {\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 7 \, {\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{140 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/140*(140*c^3*e^6*x^6 + 20*c^3*d^6 + 10*b*c^2*d^5*e + 4*b^2*c*d^4*e^2 + b^3*d^3*e^3 + 210*(2*c^3*d*e^5 + b*c
^2*e^6)*x^5 + 70*(10*c^3*d^2*e^4 + 5*b*c^2*d*e^5 + 2*b^2*c*e^6)*x^4 + 35*(20*c^3*d^3*e^3 + 10*b*c^2*d^2*e^4 +
4*b^2*c*d*e^5 + b^3*e^6)*x^3 + 21*(20*c^3*d^4*e^2 + 10*b*c^2*d^3*e^3 + 4*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 7*(2
0*c^3*d^5*e + 10*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5
+ 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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giac [A]  time = 0.17, size = 267, normalized size = 1.16 \begin {gather*} -\frac {{\left (140 \, c^{3} x^{6} e^{6} + 420 \, c^{3} d x^{5} e^{5} + 700 \, c^{3} d^{2} x^{4} e^{4} + 700 \, c^{3} d^{3} x^{3} e^{3} + 420 \, c^{3} d^{4} x^{2} e^{2} + 140 \, c^{3} d^{5} x e + 20 \, c^{3} d^{6} + 210 \, b c^{2} x^{5} e^{6} + 350 \, b c^{2} d x^{4} e^{5} + 350 \, b c^{2} d^{2} x^{3} e^{4} + 210 \, b c^{2} d^{3} x^{2} e^{3} + 70 \, b c^{2} d^{4} x e^{2} + 10 \, b c^{2} d^{5} e + 140 \, b^{2} c x^{4} e^{6} + 140 \, b^{2} c d x^{3} e^{5} + 84 \, b^{2} c d^{2} x^{2} e^{4} + 28 \, b^{2} c d^{3} x e^{3} + 4 \, b^{2} c d^{4} e^{2} + 35 \, b^{3} x^{3} e^{6} + 21 \, b^{3} d x^{2} e^{5} + 7 \, b^{3} d^{2} x e^{4} + b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{140 \, {\left (x e + d\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/140*(140*c^3*x^6*e^6 + 420*c^3*d*x^5*e^5 + 700*c^3*d^2*x^4*e^4 + 700*c^3*d^3*x^3*e^3 + 420*c^3*d^4*x^2*e^2
+ 140*c^3*d^5*x*e + 20*c^3*d^6 + 210*b*c^2*x^5*e^6 + 350*b*c^2*d*x^4*e^5 + 350*b*c^2*d^2*x^3*e^4 + 210*b*c^2*d
^3*x^2*e^3 + 70*b*c^2*d^4*x*e^2 + 10*b*c^2*d^5*e + 140*b^2*c*x^4*e^6 + 140*b^2*c*d*x^3*e^5 + 84*b^2*c*d^2*x^2*
e^4 + 28*b^2*c*d^3*x*e^3 + 4*b^2*c*d^4*e^2 + 35*b^3*x^3*e^6 + 21*b^3*d*x^2*e^5 + 7*b^3*d^2*x*e^4 + b^3*d^3*e^3
)*e^(-7)/(x*e + d)^7

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maple [A]  time = 0.05, size = 274, normalized size = 1.19 \begin {gather*} -\frac {c^{3}}{\left (e x +d \right ) 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}}{7 \left (e x +d \right )^{7} e^{7}}-\frac {3 \left (b e -2 c d \right ) c^{2}}{2 \left (e x +d \right )^{2} e^{7}}-\frac {\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}}{2 \left (e x +d \right )^{6} e^{7}}-\frac {\left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) c}{\left (e x +d \right )^{3} e^{7}}+\frac {3 \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}{5 \left (e x +d \right )^{5} 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}}{4 \left (e x +d \right )^{4} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^3/(e*x+d)^8,x)

[Out]

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

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maxima [A]  time = 1.60, size = 334, normalized size = 1.45 \begin {gather*} -\frac {140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 210 \, {\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, b^{2} c e^{6}\right )} x^{4} + 35 \, {\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 21 \, {\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 7 \, {\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{140 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/140*(140*c^3*e^6*x^6 + 20*c^3*d^6 + 10*b*c^2*d^5*e + 4*b^2*c*d^4*e^2 + b^3*d^3*e^3 + 210*(2*c^3*d*e^5 + b*c
^2*e^6)*x^5 + 70*(10*c^3*d^2*e^4 + 5*b*c^2*d*e^5 + 2*b^2*c*e^6)*x^4 + 35*(20*c^3*d^3*e^3 + 10*b*c^2*d^2*e^4 +
4*b^2*c*d*e^5 + b^3*e^6)*x^3 + 21*(20*c^3*d^4*e^2 + 10*b*c^2*d^3*e^3 + 4*b^2*c*d^2*e^4 + b^3*d*e^5)*x^2 + 7*(2
0*c^3*d^5*e + 10*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 + b^3*d^2*e^4)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5
+ 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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mupad [B]  time = 0.22, size = 315, normalized size = 1.37 \begin {gather*} -\frac {\frac {d^3\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+20\,c^3\,d^3\right )}{140\,e^7}+\frac {x^3\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+20\,c^3\,d^3\right )}{4\,e^4}+\frac {c^3\,x^6}{e}+\frac {3\,c^2\,x^5\,\left (b\,e+2\,c\,d\right )}{2\,e^2}+\frac {c\,x^4\,\left (2\,b^2\,e^2+5\,b\,c\,d\,e+10\,c^2\,d^2\right )}{2\,e^3}+\frac {3\,d\,x^2\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+20\,c^3\,d^3\right )}{20\,e^5}+\frac {d^2\,x\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+20\,c^3\,d^3\right )}{20\,e^6}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3/(d + e*x)^8,x)

[Out]

-((d^3*(b^3*e^3 + 20*c^3*d^3 + 10*b*c^2*d^2*e + 4*b^2*c*d*e^2))/(140*e^7) + (x^3*(b^3*e^3 + 20*c^3*d^3 + 10*b*
c^2*d^2*e + 4*b^2*c*d*e^2))/(4*e^4) + (c^3*x^6)/e + (3*c^2*x^5*(b*e + 2*c*d))/(2*e^2) + (c*x^4*(2*b^2*e^2 + 10
*c^2*d^2 + 5*b*c*d*e))/(2*e^3) + (3*d*x^2*(b^3*e^3 + 20*c^3*d^3 + 10*b*c^2*d^2*e + 4*b^2*c*d*e^2))/(20*e^5) +
(d^2*x*(b^3*e^3 + 20*c^3*d^3 + 10*b*c^2*d^2*e + 4*b^2*c*d*e^2))/(20*e^6))/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^
5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)

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sympy [A]  time = 86.18, size = 362, normalized size = 1.57 \begin {gather*} \frac {- b^{3} d^{3} e^{3} - 4 b^{2} c d^{4} e^{2} - 10 b c^{2} d^{5} e - 20 c^{3} d^{6} - 140 c^{3} e^{6} x^{6} + x^{5} \left (- 210 b c^{2} e^{6} - 420 c^{3} d e^{5}\right ) + x^{4} \left (- 140 b^{2} c e^{6} - 350 b c^{2} d e^{5} - 700 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 35 b^{3} e^{6} - 140 b^{2} c d e^{5} - 350 b c^{2} d^{2} e^{4} - 700 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 21 b^{3} d e^{5} - 84 b^{2} c d^{2} e^{4} - 210 b c^{2} d^{3} e^{3} - 420 c^{3} d^{4} e^{2}\right ) + x \left (- 7 b^{3} d^{2} e^{4} - 28 b^{2} c d^{3} e^{3} - 70 b c^{2} d^{4} e^{2} - 140 c^{3} d^{5} e\right )}{140 d^{7} e^{7} + 980 d^{6} e^{8} x + 2940 d^{5} e^{9} x^{2} + 4900 d^{4} e^{10} x^{3} + 4900 d^{3} e^{11} x^{4} + 2940 d^{2} e^{12} x^{5} + 980 d e^{13} x^{6} + 140 e^{14} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**3/(e*x+d)**8,x)

[Out]

(-b**3*d**3*e**3 - 4*b**2*c*d**4*e**2 - 10*b*c**2*d**5*e - 20*c**3*d**6 - 140*c**3*e**6*x**6 + x**5*(-210*b*c*
*2*e**6 - 420*c**3*d*e**5) + x**4*(-140*b**2*c*e**6 - 350*b*c**2*d*e**5 - 700*c**3*d**2*e**4) + x**3*(-35*b**3
*e**6 - 140*b**2*c*d*e**5 - 350*b*c**2*d**2*e**4 - 700*c**3*d**3*e**3) + x**2*(-21*b**3*d*e**5 - 84*b**2*c*d**
2*e**4 - 210*b*c**2*d**3*e**3 - 420*c**3*d**4*e**2) + x*(-7*b**3*d**2*e**4 - 28*b**2*c*d**3*e**3 - 70*b*c**2*d
**4*e**2 - 140*c**3*d**5*e))/(140*d**7*e**7 + 980*d**6*e**8*x + 2940*d**5*e**9*x**2 + 4900*d**4*e**10*x**3 + 4
900*d**3*e**11*x**4 + 2940*d**2*e**12*x**5 + 980*d*e**13*x**6 + 140*e**14*x**7)

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