3.3.49 \(\int \frac {(b x+c x^2)^3}{d+e x} \, dx\) [249]

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

[Out]

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

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Rubi [A]
time = 0.11, 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 = {712} \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^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} \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 712

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 = 0.06, size = 144, normalized size = 0.95 \begin {gather*} \frac {-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 \left (c^2 d^2-3 b c d e+3 b^2 e^2\right ) 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} \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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(309\) vs. \(2(141)=282\).
time = 0.46, size = 310, normalized size = 2.05

method	result	size
norman	\(\frac {d^{2} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) x}{e^{6}}+\frac {c^{3} x^{6}}{6 e}+\frac {\left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) x^{3}}{3 e^{4}}+\frac {c \left (3 b^{2} e^{2}-3 b c d e +d^{2} c^{2}\right ) x^{4}}{4 e^{3}}-\frac {d \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) x^{2}}{2 e^{5}}+\frac {c^{2} \left (3 b e -c d \right ) x^{5}}{5 e^{2}}-\frac {d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(247\)
risch	\(\frac {c^{3} x^{6}}{6 e}+\frac {3 x^{5} c^{2} b}{5 e}-\frac {c^{3} d \,x^{5}}{5 e^{2}}+\frac {3 x^{4} b^{2} c}{4 e}-\frac {3 x^{4} b \,c^{2} d}{4 e^{2}}+\frac {x^{4} c^{3} d^{2}}{4 e^{3}}+\frac {x^{3} b^{3}}{3 e}-\frac {x^{3} b^{2} c d}{e^{2}}+\frac {x^{3} b \,c^{2} d^{2}}{e^{3}}-\frac {x^{3} c^{3} d^{3}}{3 e^{4}}-\frac {x^{2} b^{3} d}{2 e^{2}}+\frac {3 x^{2} b^{2} c \,d^{2}}{2 e^{3}}-\frac {3 x^{2} b \,c^{2} d^{3}}{2 e^{4}}+\frac {x^{2} c^{3} d^{4}}{2 e^{5}}+\frac {b^{3} d^{2} x}{e^{3}}-\frac {3 b^{2} c \,d^{3} x}{e^{4}}+\frac {3 b \,c^{2} d^{4} x}{e^{5}}-\frac {c^{3} d^{5} x}{e^{6}}-\frac {d^{3} \ln \left (e x +d \right ) b^{3}}{e^{4}}+\frac {3 d^{4} \ln \left (e x +d \right ) b^{2} c}{e^{5}}-\frac {3 d^{5} \ln \left (e x +d \right ) b \,c^{2}}{e^{6}}+\frac {d^{6} \ln \left (e x +d \right ) c^{3}}{e^{7}}\)	\(302\)
default	\(\frac {\frac {c^{3} x^{6} e^{5}}{6}+\frac {\left (\left (b e -c d \right ) e^{4} c^{2}+2 c^{2} e^{5} b \right ) x^{5}}{5}+\frac {\left (2 \left (b e -c d \right ) e^{4} b c +c e \left (e^{4} b^{2}-d \,e^{3} b c +d^{2} e^{2} c^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (b e -c d \right ) \left (e^{4} b^{2}-d \,e^{3} b c +d^{2} e^{2} c^{2}\right )+c e \left (-d \,e^{3} b^{2}+d^{2} e^{2} b c \right )\right ) x^{3}}{3}+\frac {\left (\left (b e -c d \right ) \left (-d \,e^{3} b^{2}+d^{2} e^{2} b c \right )+c e \left (d^{2} e^{2} b^{2}-2 d^{3} e b c +c^{2} d^{4}\right )\right ) x^{2}}{2}+\left (b e -c d \right ) \left (d^{2} e^{2} b^{2}-2 d^{3} e b c +c^{2} d^{4}\right ) x}{e^{6}}-\frac {d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(310\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 249, normalized size = 1.65 \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 (x e + d\right ) + \frac {1}{60} \, {\left (10 \, c^{3} x^{6} e^{5} - 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\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="maxima")

[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(x*e + d) + 1/60*(10*c^3*x^6*e^5 - 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)

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Fricas [A]
time = 2.57, size = 258, normalized size = 1.71 \begin {gather*} -\frac {1}{60} \, {\left (60 \, c^{3} d^{5} x e - {\left (10 \, c^{3} x^{6} + 36 \, b c^{2} x^{5} + 45 \, b^{2} c x^{4} + 20 \, b^{3} x^{3}\right )} e^{6} + 3 \, {\left (4 \, c^{3} d x^{5} + 15 \, b c^{2} d x^{4} + 20 \, b^{2} c d x^{3} + 10 \, b^{3} d x^{2}\right )} e^{5} - 15 \, {\left (c^{3} d^{2} x^{4} + 4 \, b c^{2} d^{2} x^{3} + 6 \, b^{2} c d^{2} x^{2} + 4 \, b^{3} d^{2} x\right )} e^{4} + 10 \, {\left (2 \, c^{3} d^{3} x^{3} + 9 \, b c^{2} d^{3} x^{2} + 18 \, b^{2} c d^{3} x\right )} e^{3} - 30 \, {\left (c^{3} d^{4} x^{2} + 6 \, b c^{2} d^{4} x\right )} e^{2} - 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 (x e + d\right )\right )} e^{\left (-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),x, algorithm="fricas")

[Out]

-1/60*(60*c^3*d^5*x*e - (10*c^3*x^6 + 36*b*c^2*x^5 + 45*b^2*c*x^4 + 20*b^3*x^3)*e^6 + 3*(4*c^3*d*x^5 + 15*b*c^
2*d*x^4 + 20*b^2*c*d*x^3 + 10*b^3*d*x^2)*e^5 - 15*(c^3*d^2*x^4 + 4*b*c^2*d^2*x^3 + 6*b^2*c*d^2*x^2 + 4*b^3*d^2
*x)*e^4 + 10*(2*c^3*d^3*x^3 + 9*b*c^2*d^3*x^2 + 18*b^2*c*d^3*x)*e^3 - 30*(c^3*d^4*x^2 + 6*b*c^2*d^4*x)*e^2 - 6
0*(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(x*e + d))*e^(-7)

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Sympy [A]
time = 0.24, 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} \cdot \left (\frac {3 b c^{2}}{5 e} - \frac {c^{3} d}{5 e^{2}}\right ) + x^{4} \cdot \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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Giac [A]
time = 0.62, 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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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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