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

Optimal. Leaf size=151 \[ \frac {c x^4 \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )}{4 e^3}-\frac {c^2 x^5 (c d-3 b e)}{5 e^2}+\frac {d^3 (c d-b e)^3 \log (d+e x)}{e^7}-\frac {d^2 x (c d-b e)^3}{e^6}+\frac {d x^2 (c d-b e)^3}{2 e^5}-\frac {x^3 (c d-b e)^3}{3 e^4}+\frac {c^3 x^6}{6 e} \]

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Rubi [A]  time = 0.15, antiderivative size = 151, 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 x^4 \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )}{4 e^3}-\frac {c^2 x^5 (c d-3 b e)}{5 e^2}-\frac {d^2 x (c d-b e)^3}{e^6}+\frac {d^3 (c d-b e)^3 \log (d+e x)}{e^7}-\frac {x^3 (c d-b e)^3}{3 e^4}+\frac {d x^2 (c d-b e)^3}{2 e^5}+\frac {c^3 x^6}{6 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^3/(d + e*x),x]

[Out]

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

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Mathematica [A]  time = 0.08, size = 144, normalized size = 0.95 \begin {gather*} \frac {15 c e^4 x^4 \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )-12 c^2 e^5 x^5 (c d-3 b e)+60 d^3 (c d-b e)^3 \log (d+e x)-60 d^2 e x (c d-b e)^3+20 e^3 x^3 (b e-c d)^3+30 d e^2 x^2 (c d-b e)^3+10 c^3 e^6 x^6}{60 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-60*d^2*e*(c*d - b*e)^3*x + 30*d*e^2*(c*d - b*e)^3*x^2 + 20*e^3*(-(c*d) + b*e)^3*x^3 + 15*c*e^4*(c^2*d^2 - 3*
b*c*d*e + 3*b^2*e^2)*x^4 - 12*c^2*e^5*(c*d - 3*b*e)*x^5 + 10*c^3*e^6*x^6 + 60*d^3*(c*d - b*e)^3*Log[d + e*x])/
(60*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} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 266, normalized size = 1.76 \begin {gather*} \frac {10 \, c^{3} e^{6} x^{6} - 12 \, {\left (c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 15 \, {\left (c^{3} d^{2} e^{4} - 3 \, b c^{2} d e^{5} + 3 \, b^{2} c e^{6}\right )} x^{4} - 20 \, {\left (c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 30 \, {\left (c^{3} d^{4} e^{2} - 3 \, b c^{2} d^{3} e^{3} + 3 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} - 60 \, {\left (c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} + 3 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 60 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \log \left (e x + d\right )}{60 \, 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),x, algorithm="fricas")

[Out]

1/60*(10*c^3*e^6*x^6 - 12*(c^3*d*e^5 - 3*b*c^2*e^6)*x^5 + 15*(c^3*d^2*e^4 - 3*b*c^2*d*e^5 + 3*b^2*c*e^6)*x^4 -
20*(c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*b^2*c*d*e^5 - b^3*e^6)*x^3 + 30*(c^3*d^4*e^2 - 3*b*c^2*d^3*e^3 + 3*b^2*
c*d^2*e^4 - b^3*d*e^5)*x^2 - 60*(c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*b^2*c*d^3*e^3 - b^3*d^2*e^4)*x + 60*(c^3*d^6
- 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*log(e*x + d))/e^7

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giac [A]  time = 0.17, size = 270, normalized size = 1.79 \begin {gather*} {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (10 \, c^{3} x^{6} e^{5} - 12 \, c^{3} d x^{5} e^{4} + 15 \, c^{3} d^{2} x^{4} e^{3} - 20 \, c^{3} d^{3} x^{3} e^{2} + 30 \, c^{3} d^{4} x^{2} e - 60 \, c^{3} d^{5} x + 36 \, b c^{2} x^{5} e^{5} - 45 \, b c^{2} d x^{4} e^{4} + 60 \, b c^{2} d^{2} x^{3} e^{3} - 90 \, b c^{2} d^{3} x^{2} e^{2} + 180 \, b c^{2} d^{4} x e + 45 \, b^{2} c x^{4} e^{5} - 60 \, b^{2} c d x^{3} e^{4} + 90 \, b^{2} c d^{2} x^{2} e^{3} - 180 \, b^{2} c d^{3} x e^{2} + 20 \, b^{3} x^{3} e^{5} - 30 \, b^{3} d x^{2} e^{4} + 60 \, b^{3} d^{2} x e^{3}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*e^(-7)*log(abs(x*e + d)) + 1/60*(10*c^3*x^6*e^5 - 12
*c^3*d*x^5*e^4 + 15*c^3*d^2*x^4*e^3 - 20*c^3*d^3*x^3*e^2 + 30*c^3*d^4*x^2*e - 60*c^3*d^5*x + 36*b*c^2*x^5*e^5
- 45*b*c^2*d*x^4*e^4 + 60*b*c^2*d^2*x^3*e^3 - 90*b*c^2*d^3*x^2*e^2 + 180*b*c^2*d^4*x*e + 45*b^2*c*x^4*e^5 - 60
*b^2*c*d*x^3*e^4 + 90*b^2*c*d^2*x^2*e^3 - 180*b^2*c*d^3*x*e^2 + 20*b^3*x^3*e^5 - 30*b^3*d*x^2*e^4 + 60*b^3*d^2
*x*e^3)*e^(-6)

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maple [B]  time = 0.13, size = 302, normalized size = 2.00 \begin {gather*} \frac {c^{3} x^{6}}{6 e}+\frac {3 b \,c^{2} x^{5}}{5 e}-\frac {c^{3} d \,x^{5}}{5 e^{2}}+\frac {3 b^{2} c \,x^{4}}{4 e}-\frac {3 b \,c^{2} d \,x^{4}}{4 e^{2}}+\frac {c^{3} d^{2} x^{4}}{4 e^{3}}+\frac {b^{3} x^{3}}{3 e}-\frac {b^{2} c d \,x^{3}}{e^{2}}+\frac {b \,c^{2} d^{2} x^{3}}{e^{3}}-\frac {c^{3} d^{3} x^{3}}{3 e^{4}}-\frac {b^{3} d \,x^{2}}{2 e^{2}}+\frac {3 b^{2} c \,d^{2} x^{2}}{2 e^{3}}-\frac {3 b \,c^{2} d^{3} x^{2}}{2 e^{4}}+\frac {c^{3} d^{4} x^{2}}{2 e^{5}}-\frac {b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {b^{3} d^{2} x}{e^{3}}+\frac {3 b^{2} c \,d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {3 b^{2} c \,d^{3} x}{e^{4}}-\frac {3 b \,c^{2} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {3 b \,c^{2} d^{4} x}{e^{5}}+\frac {c^{3} d^{6} \ln \left (e x +d \right )}{e^{7}}-\frac {c^{3} d^{5} 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),x)

[Out]

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

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maxima [A]  time = 1.40, size = 264, normalized size = 1.75 \begin {gather*} \frac {10 \, c^{3} e^{5} x^{6} - 12 \, {\left (c^{3} d e^{4} - 3 \, b c^{2} e^{5}\right )} x^{5} + 15 \, {\left (c^{3} d^{2} e^{3} - 3 \, b c^{2} d e^{4} + 3 \, b^{2} c e^{5}\right )} x^{4} - 20 \, {\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 30 \, {\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} - 60 \, {\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e + 3 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x}{60 \, e^{6}} + \frac {{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} 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),x, algorithm="maxima")

[Out]

1/60*(10*c^3*e^5*x^6 - 12*(c^3*d*e^4 - 3*b*c^2*e^5)*x^5 + 15*(c^3*d^2*e^3 - 3*b*c^2*d*e^4 + 3*b^2*c*e^5)*x^4 -
20*(c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*b^2*c*d*e^4 - b^3*e^5)*x^3 + 30*(c^3*d^4*e - 3*b*c^2*d^3*e^2 + 3*b^2*c*
d^2*e^3 - b^3*d*e^4)*x^2 - 60*(c^3*d^5 - 3*b*c^2*d^4*e + 3*b^2*c*d^3*e^2 - b^3*d^2*e^3)*x)/e^6 + (c^3*d^6 - 3*
b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*log(e*x + d)/e^7

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mupad [B]  time = 0.18, size = 294, normalized size = 1.95 \begin {gather*} x^3\,\left (\frac {b^3}{3\,e}-\frac {d\,\left (\frac {3\,b^2\,c}{e}-\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{e}\right )}{3\,e}\right )+x^5\,\left (\frac {3\,b\,c^2}{5\,e}-\frac {c^3\,d}{5\,e^2}\right )+x^4\,\left (\frac {3\,b^2\,c}{4\,e}-\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{4\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6\right )}{e^7}+\frac {c^3\,x^6}{6\,e}-\frac {d\,x^2\,\left (\frac {b^3}{e}-\frac {d\,\left (\frac {3\,b^2\,c}{e}-\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{e}\right )}{e}\right )}{2\,e}+\frac {d^2\,x\,\left (\frac {b^3}{e}-\frac {d\,\left (\frac {3\,b^2\,c}{e}-\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{e}\right )}{e}\right )}{e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3/(d + e*x),x)

[Out]

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

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sympy [A]  time = 0.48, size = 243, normalized size = 1.61 \begin {gather*} \frac {c^{3} x^{6}}{6 e} - \frac {d^{3} \left (b e - c d\right )^{3} \log {\left (d + e x \right )}}{e^{7}} + x^{5} \left (\frac {3 b c^{2}}{5 e} - \frac {c^{3} d}{5 e^{2}}\right ) + x^{4} \left (\frac {3 b^{2} c}{4 e} - \frac {3 b c^{2} d}{4 e^{2}} + \frac {c^{3} d^{2}}{4 e^{3}}\right ) + x^{3} \left (\frac {b^{3}}{3 e} - \frac {b^{2} c d}{e^{2}} + \frac {b c^{2} d^{2}}{e^{3}} - \frac {c^{3} d^{3}}{3 e^{4}}\right ) + x^{2} \left (- \frac {b^{3} d}{2 e^{2}} + \frac {3 b^{2} c d^{2}}{2 e^{3}} - \frac {3 b c^{2} d^{3}}{2 e^{4}} + \frac {c^{3} d^{4}}{2 e^{5}}\right ) + x \left (\frac {b^{3} d^{2}}{e^{3}} - \frac {3 b^{2} c d^{3}}{e^{4}} + \frac {3 b c^{2} d^{4}}{e^{5}} - \frac {c^{3} d^{5}}{e^{6}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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