3.1912 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^2}{d+e x} \, dx\)

Optimal. Leaf size=122 \[ -\frac {(b d-a e)^5 \log (d+e x)}{e^6}+\frac {b x (b d-a e)^4}{e^5}-\frac {(a+b x)^2 (b d-a e)^3}{2 e^4}+\frac {(a+b x)^3 (b d-a e)^2}{3 e^3}-\frac {(a+b x)^4 (b d-a e)}{4 e^2}+\frac {(a+b x)^5}{5 e} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac {b x (b d-a e)^4}{e^5}-\frac {(a+b x)^2 (b d-a e)^3}{2 e^4}+\frac {(a+b x)^3 (b d-a e)^2}{3 e^3}-\frac {(a+b x)^4 (b d-a e)}{4 e^2}-\frac {(b d-a e)^5 \log (d+e x)}{e^6}+\frac {(a+b x)^5}{5 e} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x),x]

[Out]

(b*(b*d - a*e)^4*x)/e^5 - ((b*d - a*e)^3*(a + b*x)^2)/(2*e^4) + ((b*d - a*e)^2*(a + b*x)^3)/(3*e^3) - ((b*d -
a*e)*(a + b*x)^4)/(4*e^2) + (a + b*x)^5/(5*e) - ((b*d - a*e)^5*Log[d + e*x])/e^6

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 167, normalized size = 1.37 \[ \frac {b e x \left (300 a^4 e^4+300 a^3 b e^3 (e x-2 d)+100 a^2 b^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+25 a b^3 e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )-60 (b d-a e)^5 \log (d+e x)}{60 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*x*(300*a^4*e^4 + 300*a^3*b*e^3*(-2*d + e*x) + 100*a^2*b^2*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 25*a*b^3*e*
(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 1
2*e^4*x^4)) - 60*(b*d - a*e)^5*Log[d + e*x])/(60*e^6)

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fricas [B]  time = 1.13, size = 259, normalized size = 2.12 \[ \frac {12 \, b^{5} e^{5} x^{5} - 15 \, {\left (b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{3} - 5 \, a b^{4} d e^{4} + 10 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \, {\left (b^{5} d^{3} e^{2} - 5 \, a b^{4} d^{2} e^{3} + 10 \, a^{2} b^{3} d e^{4} - 10 \, a^{3} b^{2} e^{5}\right )} x^{2} + 60 \, {\left (b^{5} d^{4} e - 5 \, a b^{4} d^{3} e^{2} + 10 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.16, size = 259, normalized size = 2.12 \[ -{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (12 \, b^{5} x^{5} e^{4} - 15 \, b^{5} d x^{4} e^{3} + 20 \, b^{5} d^{2} x^{3} e^{2} - 30 \, b^{5} d^{3} x^{2} e + 60 \, b^{5} d^{4} x + 75 \, a b^{4} x^{4} e^{4} - 100 \, a b^{4} d x^{3} e^{3} + 150 \, a b^{4} d^{2} x^{2} e^{2} - 300 \, a b^{4} d^{3} x e + 200 \, a^{2} b^{3} x^{3} e^{4} - 300 \, a^{2} b^{3} d x^{2} e^{3} + 600 \, a^{2} b^{3} d^{2} x e^{2} + 300 \, a^{3} b^{2} x^{2} e^{4} - 600 \, a^{3} b^{2} d x e^{3} + 300 \, a^{4} b x e^{4}\right )} e^{\left (-5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*e^(-6)*log(abs(
x*e + d)) + 1/60*(12*b^5*x^5*e^4 - 15*b^5*d*x^4*e^3 + 20*b^5*d^2*x^3*e^2 - 30*b^5*d^3*x^2*e + 60*b^5*d^4*x + 7
5*a*b^4*x^4*e^4 - 100*a*b^4*d*x^3*e^3 + 150*a*b^4*d^2*x^2*e^2 - 300*a*b^4*d^3*x*e + 200*a^2*b^3*x^3*e^4 - 300*
a^2*b^3*d*x^2*e^3 + 600*a^2*b^3*d^2*x*e^2 + 300*a^3*b^2*x^2*e^4 - 600*a^3*b^2*d*x*e^3 + 300*a^4*b*x*e^4)*e^(-5
)

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maple [B]  time = 0.05, size = 302, normalized size = 2.48 \[ \frac {b^{5} x^{5}}{5 e}+\frac {5 a \,b^{4} x^{4}}{4 e}-\frac {b^{5} d \,x^{4}}{4 e^{2}}+\frac {10 a^{2} b^{3} x^{3}}{3 e}-\frac {5 a \,b^{4} d \,x^{3}}{3 e^{2}}+\frac {b^{5} d^{2} x^{3}}{3 e^{3}}+\frac {5 a^{3} b^{2} x^{2}}{e}-\frac {5 a^{2} b^{3} d \,x^{2}}{e^{2}}+\frac {5 a \,b^{4} d^{2} x^{2}}{2 e^{3}}-\frac {b^{5} d^{3} x^{2}}{2 e^{4}}+\frac {a^{5} \ln \left (e x +d \right )}{e}-\frac {5 a^{4} b d \ln \left (e x +d \right )}{e^{2}}+\frac {5 a^{4} b x}{e}+\frac {10 a^{3} b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {10 a^{3} b^{2} d x}{e^{2}}-\frac {10 a^{2} b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {10 a^{2} b^{3} d^{2} x}{e^{3}}+\frac {5 a \,b^{4} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {5 a \,b^{4} d^{3} x}{e^{4}}-\frac {b^{5} d^{5} \ln \left (e x +d \right )}{e^{6}}+\frac {b^{5} d^{4} x}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^5/e*x^5+5/4*b^4/e*x^4*a-1/4*b^5/e^2*x^4*d+10/3*b^3/e*x^3*a^2-5/3*b^4/e^2*x^3*a*d+1/3*b^5/e^3*x^3*d^2+5*b
^2/e*x^2*a^3-5*b^3/e^2*x^2*a^2*d+5/2*b^4/e^3*x^2*a*d^2-1/2*b^5/e^4*x^2*d^3+5*b/e*a^4*x-10*b^2/e^2*a^3*d*x+10*b
^3/e^3*a^2*d^2*x-5*b^4/e^4*a*d^3*x+b^5/e^5*d^4*x+1/e*ln(e*x+d)*a^5-5/e^2*ln(e*x+d)*a^4*b*d+10/e^3*ln(e*x+d)*a^
3*b^2*d^2-10/e^4*ln(e*x+d)*a^2*b^3*d^3+5/e^5*ln(e*x+d)*a*b^4*d^4-1/e^6*ln(e*x+d)*b^5*d^5

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maxima [B]  time = 0.55, size = 258, normalized size = 2.11 \[ \frac {12 \, b^{5} e^{4} x^{5} - 15 \, {\left (b^{5} d e^{3} - 5 \, a b^{4} e^{4}\right )} x^{4} + 20 \, {\left (b^{5} d^{2} e^{2} - 5 \, a b^{4} d e^{3} + 10 \, a^{2} b^{3} e^{4}\right )} x^{3} - 30 \, {\left (b^{5} d^{3} e - 5 \, a b^{4} d^{2} e^{2} + 10 \, a^{2} b^{3} d e^{3} - 10 \, a^{3} b^{2} e^{4}\right )} x^{2} + 60 \, {\left (b^{5} d^{4} - 5 \, a b^{4} d^{3} e + 10 \, a^{2} b^{3} d^{2} e^{2} - 10 \, a^{3} b^{2} d e^{3} + 5 \, a^{4} b e^{4}\right )} x}{60 \, e^{5}} - \frac {{\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.99, size = 280, normalized size = 2.30 \[ x\,\left (\frac {5\,a^4\,b}{e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{e}-\frac {b^5\,d}{e^2}\right )}{e}-\frac {10\,a^2\,b^3}{e}\right )}{e}+\frac {10\,a^3\,b^2}{e}\right )}{e}\right )+x^4\,\left (\frac {5\,a\,b^4}{4\,e}-\frac {b^5\,d}{4\,e^2}\right )+x^2\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{e}-\frac {b^5\,d}{e^2}\right )}{e}-\frac {10\,a^2\,b^3}{e}\right )}{2\,e}+\frac {5\,a^3\,b^2}{e}\right )-x^3\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{e}-\frac {b^5\,d}{e^2}\right )}{3\,e}-\frac {10\,a^2\,b^3}{3\,e}\right )+\frac {b^5\,x^5}{5\,e}+\frac {\ln \left (d+e\,x\right )\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{e^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.51, size = 209, normalized size = 1.71 \[ \frac {b^{5} x^{5}}{5 e} + x^{4} \left (\frac {5 a b^{4}}{4 e} - \frac {b^{5} d}{4 e^{2}}\right ) + x^{3} \left (\frac {10 a^{2} b^{3}}{3 e} - \frac {5 a b^{4} d}{3 e^{2}} + \frac {b^{5} d^{2}}{3 e^{3}}\right ) + x^{2} \left (\frac {5 a^{3} b^{2}}{e} - \frac {5 a^{2} b^{3} d}{e^{2}} + \frac {5 a b^{4} d^{2}}{2 e^{3}} - \frac {b^{5} d^{3}}{2 e^{4}}\right ) + x \left (\frac {5 a^{4} b}{e} - \frac {10 a^{3} b^{2} d}{e^{2}} + \frac {10 a^{2} b^{3} d^{2}}{e^{3}} - \frac {5 a b^{4} d^{3}}{e^{4}} + \frac {b^{5} d^{4}}{e^{5}}\right ) + \frac {\left (a e - b d\right )^{5} \log {\left (d + e x \right )}}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**5*x**5/(5*e) + x**4*(5*a*b**4/(4*e) - b**5*d/(4*e**2)) + x**3*(10*a**2*b**3/(3*e) - 5*a*b**4*d/(3*e**2) + b
**5*d**2/(3*e**3)) + x**2*(5*a**3*b**2/e - 5*a**2*b**3*d/e**2 + 5*a*b**4*d**2/(2*e**3) - b**5*d**3/(2*e**4)) +
x*(5*a**4*b/e - 10*a**3*b**2*d/e**2 + 10*a**2*b**3*d**2/e**3 - 5*a*b**4*d**3/e**4 + b**5*d**4/e**5) + (a*e -
b*d)**5*log(d + e*x)/e**6

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