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

Optimal. Leaf size=200 \[ \frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac {x (c d-b e) \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6}-\frac {c^2 x^3 (c d-b e)}{e^4}-\frac {d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {3 c x^2 (c d-b e) (2 c d-b e)}{2 e^5}+\frac {c^3 x^4}{4 e^3} \]

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Rubi [A]  time = 0.22, antiderivative size = 200, 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 {x (c d-b e) \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6}+\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) \log (d+e x)}{e^7}-\frac {c^2 x^3 (c d-b e)}{e^4}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}-\frac {d^3 (c d-b e)^3}{2 e^7 (d+e x)^2}+\frac {3 c x^2 (c d-b e) (2 c d-b e)}{2 e^5}+\frac {c^3 x^4}{4 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x)^3,x]

[Out]

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

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

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Mathematica [A]  time = 0.13, size = 207, normalized size = 1.04 \begin {gather*} \frac {6 c e^2 x^2 \left (b^2 e^2-3 b c d e+2 c^2 d^2\right )+4 e x \left (b^3 e^3-9 b^2 c d e^2+18 b c^2 d^2 e-10 c^3 d^3\right )+12 d \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 ) \log (d+e x)-4 c^2 e^3 x^3 (c d-b e)-\frac {2 d^3 (c d-b e)^3}{(d+e x)^2}+\frac {12 d^2 (c d-b e)^2 (2 c d-b e)}{d+e x}+c^3 e^4 x^4}{4 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*e*(-10*c^3*d^3 + 18*b*c^2*d^2*e - 9*b^2*c*d*e^2 + b^3*e^3)*x + 6*c*e^2*(2*c^2*d^2 - 3*b*c*d*e + b^2*e^2)*x^
2 - 4*c^2*e^3*(c*d - b*e)*x^3 + c^3*e^4*x^4 - (2*d^3*(c*d - b*e)^3)/(d + e*x)^2 + (12*d^2*(c*d - b*e)^2*(2*c*d
- b*e))/(d + e*x) + 12*d*(5*c^3*d^3 - 10*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*Log[d + e*x])/(4*e^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)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 428, normalized size = 2.14 \begin {gather*} \frac {c^{3} e^{6} x^{6} + 22 \, c^{3} d^{6} - 54 \, b c^{2} d^{5} e + 42 \, b^{2} c d^{4} e^{2} - 10 \, b^{3} d^{3} e^{3} - 2 \, {\left (c^{3} d e^{5} - 2 \, b c^{2} e^{6}\right )} x^{5} + {\left (5 \, c^{3} d^{2} e^{4} - 10 \, b c^{2} d e^{5} + 6 \, b^{2} c e^{6}\right )} x^{4} - 4 \, {\left (5 \, c^{3} d^{3} e^{3} - 10 \, b c^{2} d^{2} e^{4} + 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} - 2 \, {\left (34 \, c^{3} d^{4} e^{2} - 63 \, b c^{2} d^{3} e^{3} + 33 \, b^{2} c d^{2} e^{4} - 4 \, b^{3} d e^{5}\right )} x^{2} - 4 \, {\left (4 \, c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} - 3 \, b^{2} c d^{3} e^{3} + 2 \, b^{3} d^{2} e^{4}\right )} x + 12 \, {\left (5 \, c^{3} d^{6} - 10 \, b c^{2} d^{5} e + 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 2 \, {\left (5 \, c^{3} d^{5} e - 10 \, b c^{2} d^{4} e^{2} + 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/4*(c^3*e^6*x^6 + 22*c^3*d^6 - 54*b*c^2*d^5*e + 42*b^2*c*d^4*e^2 - 10*b^3*d^3*e^3 - 2*(c^3*d*e^5 - 2*b*c^2*e^
6)*x^5 + (5*c^3*d^2*e^4 - 10*b*c^2*d*e^5 + 6*b^2*c*e^6)*x^4 - 4*(5*c^3*d^3*e^3 - 10*b*c^2*d^2*e^4 + 6*b^2*c*d*
e^5 - b^3*e^6)*x^3 - 2*(34*c^3*d^4*e^2 - 63*b*c^2*d^3*e^3 + 33*b^2*c*d^2*e^4 - 4*b^3*d*e^5)*x^2 - 4*(4*c^3*d^5
*e - 3*b*c^2*d^4*e^2 - 3*b^2*c*d^3*e^3 + 2*b^3*d^2*e^4)*x + 12*(5*c^3*d^6 - 10*b*c^2*d^5*e + 6*b^2*c*d^4*e^2 -
b^3*d^3*e^3 + (5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2 + 2*(5*c^3*d^5*e - 10*b*c^
2*d^4*e^2 + 6*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

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giac [A]  time = 0.17, size = 264, normalized size = 1.32 \begin {gather*} 3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{4} \, {\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 4 \, b c^{2} x^{3} e^{9} - 18 \, b c^{2} d x^{2} e^{8} + 72 \, b c^{2} d^{2} x e^{7} + 6 \, b^{2} c x^{2} e^{9} - 36 \, b^{2} c d x e^{8} + 4 \, b^{3} x e^{9}\right )} e^{\left (-12\right )} + \frac {{\left (11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} e^{\left (-7\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*e^(-7)*log(abs(x*e + d)) + 1/4*(c^3*x^4*e^9 - 4*c
^3*d*x^3*e^8 + 12*c^3*d^2*x^2*e^7 - 40*c^3*d^3*x*e^6 + 4*b*c^2*x^3*e^9 - 18*b*c^2*d*x^2*e^8 + 72*b*c^2*d^2*x*e
^7 + 6*b^2*c*x^2*e^9 - 36*b^2*c*d*x*e^8 + 4*b^3*x*e^9)*e^(-12) + 1/2*(11*c^3*d^6 - 27*b*c^2*d^5*e + 21*b^2*c*d
^4*e^2 - 5*b^3*d^3*e^3 + 6*(2*c^3*d^5*e - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x)*e^(-7)/(x*e + d)
^2

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maple [A]  time = 0.06, size = 335, normalized size = 1.68 \begin {gather*} \frac {c^{3} x^{4}}{4 e^{3}}+\frac {b \,c^{2} x^{3}}{e^{3}}-\frac {c^{3} d \,x^{3}}{e^{4}}+\frac {b^{3} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 b^{2} c \,d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {3 b^{2} c \,x^{2}}{2 e^{3}}+\frac {3 b \,c^{2} d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {9 b \,c^{2} d \,x^{2}}{2 e^{4}}-\frac {c^{3} d^{6}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {3 c^{3} d^{2} x^{2}}{e^{5}}-\frac {3 b^{3} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 b^{3} d \ln \left (e x +d \right )}{e^{4}}+\frac {b^{3} x}{e^{3}}+\frac {12 b^{2} c \,d^{3}}{\left (e x +d \right ) e^{5}}+\frac {18 b^{2} c \,d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {9 b^{2} c d x}{e^{4}}-\frac {15 b \,c^{2} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {30 b \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {18 b \,c^{2} d^{2} x}{e^{5}}+\frac {6 c^{3} d^{5}}{\left (e x +d \right ) e^{7}}+\frac {15 c^{3} d^{4} \ln \left (e x +d \right )}{e^{7}}-\frac {10 c^{3} d^{3} x}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c^3*x^4/e^3+1/e^3*x^3*b*c^2-c^3*d*x^3/e^4+3/2/e^3*x^2*b^2*c-9/2/e^4*x^2*b*c^2*d+3/e^5*x^2*c^3*d^2+1/e^3*b^
3*x-9/e^4*b^2*c*d*x+18/e^5*b*c^2*d^2*x-10/e^6*c^3*d^3*x+1/2*d^3/e^4/(e*x+d)^2*b^3-3/2*d^4/e^5/(e*x+d)^2*b^2*c+
3/2*d^5/e^6/(e*x+d)^2*b*c^2-1/2*d^6/e^7/(e*x+d)^2*c^3-3*d^2/e^4/(e*x+d)*b^3+12*d^3/e^5/(e*x+d)*b^2*c-15*d^4/e^
6/(e*x+d)*b*c^2+6*d^5/e^7/(e*x+d)*c^3-3*d/e^4*ln(e*x+d)*b^3+18*d^2/e^5*ln(e*x+d)*b^2*c-30*d^3/e^6*ln(e*x+d)*b*
c^2+15*d^4/e^7*ln(e*x+d)*c^3

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maxima [A]  time = 1.50, size = 280, normalized size = 1.40 \begin {gather*} \frac {11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{2 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac {c^{3} e^{3} x^{4} - 4 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 6 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} - 4 \, {\left (10 \, c^{3} d^{3} - 18 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{4 \, e^{6}} + \frac {3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(11*c^3*d^6 - 27*b*c^2*d^5*e + 21*b^2*c*d^4*e^2 - 5*b^3*d^3*e^3 + 6*(2*c^3*d^5*e - 5*b*c^2*d^4*e^2 + 4*b^2
*c*d^3*e^3 - b^3*d^2*e^4)*x)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(c^3*e^3*x^4 - 4*(c^3*d*e^2 - b*c^2*e^3)*x^
3 + 6*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*x^2 - 4*(10*c^3*d^3 - 18*b*c^2*d^2*e + 9*b^2*c*d*e^2 - b^3*e^3
)*x)/e^6 + 3*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*log(e*x + d)/e^7

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mupad [B]  time = 0.22, size = 352, normalized size = 1.76 \begin {gather*} x^3\,\left (\frac {b\,c^2}{e^3}-\frac {c^3\,d}{e^4}\right )-x^2\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{2\,e}-\frac {3\,b^2\,c}{2\,e^3}+\frac {3\,c^3\,d^2}{2\,e^5}\right )+\frac {x\,\left (-3\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-15\,b\,c^2\,d^4\,e+6\,c^3\,d^5\right )+\frac {-5\,b^3\,d^3\,e^3+21\,b^2\,c\,d^4\,e^2-27\,b\,c^2\,d^5\,e+11\,c^3\,d^6}{2\,e}}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x\,\left (\frac {b^3}{e^3}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e}-\frac {3\,b^2\,c}{e^3}+\frac {3\,c^3\,d^2}{e^5}\right )}{e}-\frac {c^3\,d^3}{e^6}-\frac {3\,d^2\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e^2}\right )+\frac {c^3\,x^4}{4\,e^3}+\frac {\ln \left (d+e\,x\right )\,\left (-3\,b^3\,d\,e^3+18\,b^2\,c\,d^2\,e^2-30\,b\,c^2\,d^3\,e+15\,c^3\,d^4\right )}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3/(d + e*x)^3,x)

[Out]

x^3*((b*c^2)/e^3 - (c^3*d)/e^4) - x^2*((3*d*((3*b*c^2)/e^3 - (3*c^3*d)/e^4))/(2*e) - (3*b^2*c)/(2*e^3) + (3*c^
3*d^2)/(2*e^5)) + (x*(6*c^3*d^5 - 3*b^3*d^2*e^3 + 12*b^2*c*d^3*e^2 - 15*b*c^2*d^4*e) + (11*c^3*d^6 - 5*b^3*d^3
*e^3 + 21*b^2*c*d^4*e^2 - 27*b*c^2*d^5*e)/(2*e))/(d^2*e^6 + e^8*x^2 + 2*d*e^7*x) + x*(b^3/e^3 + (3*d*((3*d*((3
*b*c^2)/e^3 - (3*c^3*d)/e^4))/e - (3*b^2*c)/e^3 + (3*c^3*d^2)/e^5))/e - (c^3*d^3)/e^6 - (3*d^2*((3*b*c^2)/e^3
- (3*c^3*d)/e^4))/e^2) + (c^3*x^4)/(4*e^3) + (log(d + e*x)*(15*c^3*d^4 - 3*b^3*d*e^3 + 18*b^2*c*d^2*e^2 - 30*b
*c^2*d^3*e))/e^7

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sympy [A]  time = 1.61, size = 284, normalized size = 1.42 \begin {gather*} \frac {c^{3} x^{4}}{4 e^{3}} - \frac {3 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + x^{3} \left (\frac {b c^{2}}{e^{3}} - \frac {c^{3} d}{e^{4}}\right ) + x^{2} \left (\frac {3 b^{2} c}{2 e^{3}} - \frac {9 b c^{2} d}{2 e^{4}} + \frac {3 c^{3} d^{2}}{e^{5}}\right ) + x \left (\frac {b^{3}}{e^{3}} - \frac {9 b^{2} c d}{e^{4}} + \frac {18 b c^{2} d^{2}}{e^{5}} - \frac {10 c^{3} d^{3}}{e^{6}}\right ) + \frac {- 5 b^{3} d^{3} e^{3} + 21 b^{2} c d^{4} e^{2} - 27 b c^{2} d^{5} e + 11 c^{3} d^{6} + x \left (- 6 b^{3} d^{2} e^{4} + 24 b^{2} c d^{3} e^{3} - 30 b c^{2} d^{4} e^{2} + 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*x**4/(4*e**3) - 3*d*(b*e - c*d)*(b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(d + e*x)/e**7 + x**3*(b*c**2/e*
*3 - c**3*d/e**4) + x**2*(3*b**2*c/(2*e**3) - 9*b*c**2*d/(2*e**4) + 3*c**3*d**2/e**5) + x*(b**3/e**3 - 9*b**2*
c*d/e**4 + 18*b*c**2*d**2/e**5 - 10*c**3*d**3/e**6) + (-5*b**3*d**3*e**3 + 21*b**2*c*d**4*e**2 - 27*b*c**2*d**
5*e + 11*c**3*d**6 + x*(-6*b**3*d**2*e**4 + 24*b**2*c*d**3*e**3 - 30*b*c**2*d**4*e**2 + 12*c**3*d**5*e))/(2*d*
*2*e**7 + 4*d*e**8*x + 2*e**9*x**2)

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