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

Optimal. Leaf size=130 \[ -\frac {5 b^4 (d+e x)^3 (b d-a e)}{3 e^6}+\frac {5 b^3 (d+e x)^2 (b d-a e)^2}{e^6}-\frac {10 b^2 x (b d-a e)^3}{e^5}+\frac {(b d-a e)^5}{e^6 (d+e x)}+\frac {5 b (b d-a e)^4 \log (d+e x)}{e^6}+\frac {b^5 (d+e x)^4}{4 e^6} \]

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Rubi [A]  time = 0.15, antiderivative size = 130, 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} \begin {gather*} -\frac {5 b^4 (d+e x)^3 (b d-a e)}{3 e^6}+\frac {5 b^3 (d+e x)^2 (b d-a e)^2}{e^6}-\frac {10 b^2 x (b d-a e)^3}{e^5}+\frac {(b d-a e)^5}{e^6 (d+e x)}+\frac {5 b (b d-a e)^4 \log (d+e x)}{e^6}+\frac {b^5 (d+e x)^4}{4 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.07, size = 228, normalized size = 1.75 \begin {gather*} \frac {-12 a^5 e^5+60 a^4 b d e^4+120 a^3 b^2 e^3 \left (-d^2+d e x+e^2 x^2\right )+60 a^2 b^3 e^2 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+20 a b^4 e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+60 b (d+e x) (b d-a e)^4 \log (d+e x)+b^5 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )}{12 e^6 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(60*a^4*b*d*e^4 - 12*a^5*e^5 + 120*a^3*b^2*e^3*(-d^2 + d*e*x + e^2*x^2) + 60*a^2*b^3*e^2*(2*d^3 - 4*d^2*e*x -
3*d*e^2*x^2 + e^3*x^3) + 20*a*b^4*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b^5*(12*d^5
- 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + 60*b*(b*d - a*e)^4*(d + e*x)*Log[
d + e*x])/(12*e^6*(d + e*x))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 373, normalized size = 2.87 \begin {gather*} \frac {3 \, b^{5} e^{5} x^{5} + 12 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e + 120 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 12 \, a^{5} e^{5} - 5 \, {\left (b^{5} d e^{4} - 4 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} - 30 \, {\left (b^{5} d^{3} e^{2} - 4 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 4 \, a^{3} b^{2} e^{5}\right )} x^{2} - 12 \, {\left (4 \, b^{5} d^{4} e - 15 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} - 10 \, a^{3} b^{2} d e^{4}\right )} x + 60 \, {\left (b^{5} d^{5} - 4 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} - 4 \, a^{3} b^{2} d^{2} e^{3} + a^{4} b d e^{4} + {\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{7} x + d e^{6}\right )}} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

1/12*(3*b^5*e^5*x^5 + 12*b^5*d^5 - 60*a*b^4*d^4*e + 120*a^2*b^3*d^3*e^2 - 120*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4
- 12*a^5*e^5 - 5*(b^5*d*e^4 - 4*a*b^4*e^5)*x^4 + 10*(b^5*d^2*e^3 - 4*a*b^4*d*e^4 + 6*a^2*b^3*e^5)*x^3 - 30*(b
^5*d^3*e^2 - 4*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 4*a^3*b^2*e^5)*x^2 - 12*(4*b^5*d^4*e - 15*a*b^4*d^3*e^2 + 20*
a^2*b^3*d^2*e^3 - 10*a^3*b^2*d*e^4)*x + 60*(b^5*d^5 - 4*a*b^4*d^4*e + 6*a^2*b^3*d^3*e^2 - 4*a^3*b^2*d^2*e^3 +
a^4*b*d*e^4 + (b^5*d^4*e - 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(e*x + d))
/(e^7*x + d*e^6)

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giac [B]  time = 0.19, size = 328, normalized size = 2.52 \begin {gather*} \frac {1}{12} \, {\left (3 \, b^{5} - \frac {20 \, {\left (b^{5} d e - a b^{4} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {60 \, {\left (b^{5} d^{2} e^{2} - 2 \, a b^{4} d e^{3} + a^{2} b^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {120 \, {\left (b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )} {\left (x e + d\right )}^{4} e^{\left (-6\right )} - 5 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (\frac {b^{5} d^{5} e^{4}}{x e + d} - \frac {5 \, a b^{4} d^{4} e^{5}}{x e + d} + \frac {10 \, a^{2} b^{3} d^{3} e^{6}}{x e + d} - \frac {10 \, a^{3} b^{2} d^{2} e^{7}}{x e + d} + \frac {5 \, a^{4} b d e^{8}}{x e + d} - \frac {a^{5} e^{9}}{x e + d}\right )} e^{\left (-10\right )} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

1/12*(3*b^5 - 20*(b^5*d*e - a*b^4*e^2)*e^(-1)/(x*e + d) + 60*(b^5*d^2*e^2 - 2*a*b^4*d*e^3 + a^2*b^3*e^4)*e^(-2
)/(x*e + d)^2 - 120*(b^5*d^3*e^3 - 3*a*b^4*d^2*e^4 + 3*a^2*b^3*d*e^5 - a^3*b^2*e^6)*e^(-3)/(x*e + d)^3)*(x*e +
d)^4*e^(-6) - 5*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*e^(-6)*log(abs(x*
e + d)*e^(-1)/(x*e + d)^2) + (b^5*d^5*e^4/(x*e + d) - 5*a*b^4*d^4*e^5/(x*e + d) + 10*a^2*b^3*d^3*e^6/(x*e + d)
- 10*a^3*b^2*d^2*e^7/(x*e + d) + 5*a^4*b*d*e^8/(x*e + d) - a^5*e^9/(x*e + d))*e^(-10)

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maple [B]  time = 0.05, size = 326, normalized size = 2.51 \begin {gather*} \frac {b^{5} x^{4}}{4 e^{2}}+\frac {5 a \,b^{4} x^{3}}{3 e^{2}}-\frac {2 b^{5} d \,x^{3}}{3 e^{3}}+\frac {5 a^{2} b^{3} x^{2}}{e^{2}}-\frac {5 a \,b^{4} d \,x^{2}}{e^{3}}+\frac {3 b^{5} d^{2} x^{2}}{2 e^{4}}-\frac {a^{5}}{\left (e x +d \right ) e}+\frac {5 a^{4} b d}{\left (e x +d \right ) e^{2}}+\frac {5 a^{4} b \ln \left (e x +d \right )}{e^{2}}-\frac {10 a^{3} b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {20 a^{3} b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {10 a^{3} b^{2} x}{e^{2}}+\frac {10 a^{2} b^{3} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {30 a^{2} b^{3} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {20 a^{2} b^{3} d x}{e^{3}}-\frac {5 a \,b^{4} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {20 a \,b^{4} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {15 a \,b^{4} d^{2} x}{e^{4}}+\frac {b^{5} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {5 b^{5} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {4 b^{5} d^{3} x}{e^{5}} \end {gather*}

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)^2,x)

[Out]

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

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maxima [B]  time = 0.51, size = 264, normalized size = 2.03 \begin {gather*} \frac {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^{7} x + d e^{6}} + \frac {3 \, b^{5} e^{3} x^{4} - 4 \, {\left (2 \, b^{5} d e^{2} - 5 \, a b^{4} e^{3}\right )} x^{3} + 6 \, {\left (3 \, b^{5} d^{2} e - 10 \, a b^{4} d e^{2} + 10 \, a^{2} b^{3} e^{3}\right )} x^{2} - 12 \, {\left (4 \, b^{5} d^{3} - 15 \, a b^{4} d^{2} e + 20 \, a^{2} b^{3} d e^{2} - 10 \, a^{3} b^{2} e^{3}\right )} x}{12 \, e^{5}} + \frac {5 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \end {gather*}

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)^2,x, algorithm="maxima")

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

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mupad [B]  time = 2.02, size = 327, normalized size = 2.52 \begin {gather*} x^3\,\left (\frac {5\,a\,b^4}{3\,e^2}-\frac {2\,b^5\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{e^2}-\frac {2\,b^5\,d}{e^3}\right )}{e}-\frac {5\,a^2\,b^3}{e^2}+\frac {b^5\,d^2}{2\,e^4}\right )+x\,\left (\frac {10\,a^3\,b^2}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {5\,a\,b^4}{e^2}-\frac {2\,b^5\,d}{e^3}\right )}{e}-\frac {10\,a^2\,b^3}{e^2}+\frac {b^5\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {5\,a\,b^4}{e^2}-\frac {2\,b^5\,d}{e^3}\right )}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )}{e^6}-\frac {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}{e\,\left (x\,e^6+d\,e^5\right )}+\frac {b^5\,x^4}{4\,e^2} \end {gather*}

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)^2,x)

[Out]

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

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sympy [A]  time = 0.90, size = 231, normalized size = 1.78 \begin {gather*} \frac {b^{5} x^{4}}{4 e^{2}} + \frac {5 b \left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{6}} + x^{3} \left (\frac {5 a b^{4}}{3 e^{2}} - \frac {2 b^{5} d}{3 e^{3}}\right ) + x^{2} \left (\frac {5 a^{2} b^{3}}{e^{2}} - \frac {5 a b^{4} d}{e^{3}} + \frac {3 b^{5} d^{2}}{2 e^{4}}\right ) + x \left (\frac {10 a^{3} b^{2}}{e^{2}} - \frac {20 a^{2} b^{3} d}{e^{3}} + \frac {15 a b^{4} d^{2}}{e^{4}} - \frac {4 b^{5} d^{3}}{e^{5}}\right ) + \frac {- 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}}{d e^{6} + e^{7} x} \end {gather*}

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)**2,x)

[Out]

b**5*x**4/(4*e**2) + 5*b*(a*e - b*d)**4*log(d + e*x)/e**6 + x**3*(5*a*b**4/(3*e**2) - 2*b**5*d/(3*e**3)) + x**
2*(5*a**2*b**3/e**2 - 5*a*b**4*d/e**3 + 3*b**5*d**2/(2*e**4)) + x*(10*a**3*b**2/e**2 - 20*a**2*b**3*d/e**3 + 1
5*a*b**4*d**2/e**4 - 4*b**5*d**3/e**5) + (-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)/(d*e**6 + e**7*x)

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