3.3.36 \(\int \frac {(b x+c x^2)^3}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=234 \[ -\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac {3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac {d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac {c^3}{3 e^7 (d+e x)^3} \]

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Rubi [A]  time = 0.16, antiderivative size = 234, 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 {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac {3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac {d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}-\frac {c^3}{3 e^7 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^10,x]

[Out]

-(d^3*(c*d - b*e)^3)/(9*e^7*(d + e*x)^9) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(8*e^7*(d + e*x)^8) - (3*d*(c*d
- b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(7*e^7*(d + e*x)^7) + ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2
*e^2))/(6*e^7*(d + e*x)^6) - (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(5*e^7*(d + e*x)^5) + (3*c^2*(2*c*d - b*e
))/(4*e^7*(d + e*x)^4) - c^3/(3*e^7*(d + e*x)^3)

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

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Mathematica [A]  time = 0.07, size = 222, normalized size = 0.95 \begin {gather*} -\frac {5 b^3 e^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+12 b^2 c e^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 )+15 b c^2 e \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 )+10 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )}{2520 e^7 (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^3/(d + e*x)^10,x]

[Out]

-1/2520*(5*b^3*e^3*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 12*b^2*c*e^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) + 15*b*c^2*e*(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) + 10*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x
^5 + 84*e^6*x^6))/(e^7*(d + e*x)^9)

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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)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x + c*x^2)^3/(d + e*x)^10,x]

[Out]

IntegrateAlgebraic[(b*x + c*x^2)^3/(d + e*x)^10, x]

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fricas [A]  time = 0.40, size = 361, normalized size = 1.54 \begin {gather*} -\frac {840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3} + 630 \, {\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \, {\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, b^{2} c e^{6}\right )} x^{4} + 84 \, {\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 5 \, b^{3} e^{6}\right )} x^{3} + 36 \, {\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 5 \, b^{3} d e^{5}\right )} x^{2} + 9 \, {\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 5 \, b^{3} d^{2} e^{4}\right )} x}{2520 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} 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)^10,x, algorithm="fricas")

[Out]

-1/2520*(840*c^3*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + 5*b^3*d^3*e^3 + 630*(2*c^3*d*e^5 +
3*b*c^2*e^6)*x^5 + 126*(10*c^3*d^2*e^4 + 15*b*c^2*d*e^5 + 12*b^2*c*e^6)*x^4 + 84*(10*c^3*d^3*e^3 + 15*b*c^2*d
^2*e^4 + 12*b^2*c*d*e^5 + 5*b^3*e^6)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + 5*b^3*d*
e^5)*x^2 + 9*(10*c^3*d^5*e + 15*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 5*b^3*d^2*e^4)*x)/(e^16*x^9 + 9*d*e^15*x^8
+ 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 +
9*d^8*e^8*x + d^9*e^7)

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giac [A]  time = 0.23, size = 268, normalized size = 1.15 \begin {gather*} -\frac {{\left (840 \, c^{3} x^{6} e^{6} + 1260 \, c^{3} d x^{5} e^{5} + 1260 \, c^{3} d^{2} x^{4} e^{4} + 840 \, c^{3} d^{3} x^{3} e^{3} + 360 \, c^{3} d^{4} x^{2} e^{2} + 90 \, c^{3} d^{5} x e + 10 \, c^{3} d^{6} + 1890 \, b c^{2} x^{5} e^{6} + 1890 \, b c^{2} d x^{4} e^{5} + 1260 \, b c^{2} d^{2} x^{3} e^{4} + 540 \, b c^{2} d^{3} x^{2} e^{3} + 135 \, b c^{2} d^{4} x e^{2} + 15 \, b c^{2} d^{5} e + 1512 \, b^{2} c x^{4} e^{6} + 1008 \, b^{2} c d x^{3} e^{5} + 432 \, b^{2} c d^{2} x^{2} e^{4} + 108 \, b^{2} c d^{3} x e^{3} + 12 \, b^{2} c d^{4} e^{2} + 420 \, b^{3} x^{3} e^{6} + 180 \, b^{3} d x^{2} e^{5} + 45 \, b^{3} d^{2} x e^{4} + 5 \, b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{2520 \, {\left (x e + d\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^3/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/2520*(840*c^3*x^6*e^6 + 1260*c^3*d*x^5*e^5 + 1260*c^3*d^2*x^4*e^4 + 840*c^3*d^3*x^3*e^3 + 360*c^3*d^4*x^2*e
^2 + 90*c^3*d^5*x*e + 10*c^3*d^6 + 1890*b*c^2*x^5*e^6 + 1890*b*c^2*d*x^4*e^5 + 1260*b*c^2*d^2*x^3*e^4 + 540*b*
c^2*d^3*x^2*e^3 + 135*b*c^2*d^4*x*e^2 + 15*b*c^2*d^5*e + 1512*b^2*c*x^4*e^6 + 1008*b^2*c*d*x^3*e^5 + 432*b^2*c
*d^2*x^2*e^4 + 108*b^2*c*d^3*x*e^3 + 12*b^2*c*d^4*e^2 + 420*b^3*x^3*e^6 + 180*b^3*d*x^2*e^5 + 45*b^3*d^2*x*e^4
+ 5*b^3*d^3*e^3)*e^(-7)/(x*e + d)^9

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maple [A]  time = 0.05, size = 274, normalized size = 1.17 \begin {gather*} -\frac {c^{3}}{3 \left (e x +d \right )^{3} 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}}{9 \left (e x +d \right )^{9} e^{7}}-\frac {3 \left (b e -2 c d \right ) c^{2}}{4 \left (e x +d \right )^{4} 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}}{8 \left (e x +d \right )^{8} e^{7}}-\frac {3 \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) c}{5 \left (e x +d \right )^{5} 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}{7 \left (e x +d \right )^{7} 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}}{6 \left (e x +d \right )^{6} 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)^10,x)

[Out]

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

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maxima [A]  time = 1.62, size = 361, normalized size = 1.54 \begin {gather*} -\frac {840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3} + 630 \, {\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \, {\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, b^{2} c e^{6}\right )} x^{4} + 84 \, {\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 5 \, b^{3} e^{6}\right )} x^{3} + 36 \, {\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 5 \, b^{3} d e^{5}\right )} x^{2} + 9 \, {\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 5 \, b^{3} d^{2} e^{4}\right )} x}{2520 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} 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)^10,x, algorithm="maxima")

[Out]

-1/2520*(840*c^3*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + 5*b^3*d^3*e^3 + 630*(2*c^3*d*e^5 +
3*b*c^2*e^6)*x^5 + 126*(10*c^3*d^2*e^4 + 15*b*c^2*d*e^5 + 12*b^2*c*e^6)*x^4 + 84*(10*c^3*d^3*e^3 + 15*b*c^2*d
^2*e^4 + 12*b^2*c*d*e^5 + 5*b^3*e^6)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + 5*b^3*d*
e^5)*x^2 + 9*(10*c^3*d^5*e + 15*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 5*b^3*d^2*e^4)*x)/(e^16*x^9 + 9*d*e^15*x^8
+ 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 +
9*d^8*e^8*x + d^9*e^7)

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mupad [B]  time = 0.31, size = 343, normalized size = 1.47 \begin {gather*} -\frac {\frac {d^3\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+10\,c^3\,d^3\right )}{2520\,e^7}+\frac {x^3\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+10\,c^3\,d^3\right )}{30\,e^4}+\frac {c^3\,x^6}{3\,e}+\frac {c^2\,x^5\,\left (3\,b\,e+2\,c\,d\right )}{4\,e^2}+\frac {c\,x^4\,\left (12\,b^2\,e^2+15\,b\,c\,d\,e+10\,c^2\,d^2\right )}{20\,e^3}+\frac {d\,x^2\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+10\,c^3\,d^3\right )}{70\,e^5}+\frac {d^2\,x\,\left (5\,b^3\,e^3+12\,b^2\,c\,d\,e^2+15\,b\,c^2\,d^2\,e+10\,c^3\,d^3\right )}{280\,e^6}}{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((b*x + c*x^2)^3/(d + e*x)^10,x)

[Out]

-((d^3*(5*b^3*e^3 + 10*c^3*d^3 + 15*b*c^2*d^2*e + 12*b^2*c*d*e^2))/(2520*e^7) + (x^3*(5*b^3*e^3 + 10*c^3*d^3 +
15*b*c^2*d^2*e + 12*b^2*c*d*e^2))/(30*e^4) + (c^3*x^6)/(3*e) + (c^2*x^5*(3*b*e + 2*c*d))/(4*e^2) + (c*x^4*(12
*b^2*e^2 + 10*c^2*d^2 + 15*b*c*d*e))/(20*e^3) + (d*x^2*(5*b^3*e^3 + 10*c^3*d^3 + 15*b*c^2*d^2*e + 12*b^2*c*d*e
^2))/(70*e^5) + (d^2*x*(5*b^3*e^3 + 10*c^3*d^3 + 15*b*c^2*d^2*e + 12*b^2*c*d*e^2))/(280*e^6))/(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((c*x**2+b*x)**3/(e*x+d)**10,x)

[Out]

Timed out

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