3.360 \(\int \frac{c+d x+e x^2}{(a+b x^3)^4} \, dx\)

Optimal. Leaf size=250 \[ -\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3} \]

[Out]

(x*(8*c + 7*d*x))/(54*a^2*(a + b*x^3)^2) + (2*x*(10*c + 7*d*x))/(81*a^3*(a + b*x^3)) - (a*e - b*x*(c + d*x))/(
9*a*b*(a + b*x^3)^3) - (2*(20*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*
Sqrt[3]*a^(11/3)*b^(2/3)) + (2*(20*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(11/3)*b^(2/3)) -
((20*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(243*a^(11/3)*b^(2/3))

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Rubi [A]  time = 0.222237, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1854, 1855, 1860, 31, 634, 617, 204, 628} \[ -\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2)/(a + b*x^3)^4,x]

[Out]

(x*(8*c + 7*d*x))/(54*a^2*(a + b*x^3)^2) + (2*x*(10*c + 7*d*x))/(81*a^3*(a + b*x^3)) - (a*e - b*x*(c + d*x))/(
9*a*b*(a + b*x^3)^3) - (2*(20*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*
Sqrt[3]*a^(11/3)*b^(2/3)) + (2*(20*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(11/3)*b^(2/3)) -
((20*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(243*a^(11/3)*b^(2/3))

Rule 1854

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[((a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*(a + b*x^n)^(p + 1))/(a*b*n*(p + 1)), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1855

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Di
st[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b},
x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{c+d x+e x^2}{\left (a+b x^3\right )^4} \, dx &=-\frac{a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}-\frac{\int \frac{-8 c-7 d x}{\left (a+b x^3\right )^3} \, dx}{9 a}\\ &=\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}-\frac{a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}+\frac{\int \frac{40 c+28 d x}{\left (a+b x^3\right )^2} \, dx}{54 a^2}\\ &=\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}-\frac{\int \frac{-80 c-28 d x}{a+b x^3} \, dx}{162 a^3}\\ &=\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}-\frac{\int \frac{\sqrt [3]{a} \left (-160 \sqrt [3]{b} c-28 \sqrt [3]{a} d\right )+\sqrt [3]{b} \left (80 \sqrt [3]{b} c-28 \sqrt [3]{a} d\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{486 a^{11/3} \sqrt [3]{b}}+\frac{\left (2 \left (20 c-\frac{7 \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{11/3}}\\ &=\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{243 a^{11/3} b^{2/3}}+\frac{\left (20 c+\frac{7 \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{81 a^{10/3}}\\ &=\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{\left (2 \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{81 a^{11/3} b^{2/3}}\\ &=\frac{x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac{2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac{a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}-\frac{2 \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.231645, size = 239, normalized size = 0.96 \[ \frac{\frac{2 \left (7 a^{2/3} d-20 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{4 \left (20 \sqrt [3]{a} \sqrt [3]{b} c-7 a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac{54 a^3 (a e-b x (c+d x))}{b \left (a+b x^3\right )^3}+\frac{9 a^2 x (8 c+7 d x)}{\left (a+b x^3\right )^2}-\frac{4 \sqrt{3} \sqrt [3]{a} \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{12 a x (10 c+7 d x)}{a+b x^3}}{486 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2)/(a + b*x^3)^4,x]

[Out]

((9*a^2*x*(8*c + 7*d*x))/(a + b*x^3)^2 + (12*a*x*(10*c + 7*d*x))/(a + b*x^3) - (54*a^3*(a*e - b*x*(c + d*x)))/
(b*(a + b*x^3)^3) - (4*Sqrt[3]*a^(1/3)*(20*b^(1/3)*c + 7*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]
])/b^(2/3) + (4*(20*a^(1/3)*b^(1/3)*c - 7*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (2*(-20*a^(1/3)*b^(1/
3)*c + 7*a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(486*a^4)

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Maple [A]  time = 0.004, size = 360, normalized size = 1.4 \begin{align*}{\frac{cx}{9\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{4\,cx}{27\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{20\,cx}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) }}+{\frac{40\,c}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,c}{243\,{a}^{3}b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{40\,c\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d{x}^{2}}{9\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{7\,d{x}^{2}}{54\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{14\,d{x}^{2}}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{14\,d}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,d}{243\,{a}^{3}b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{14\,d\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e{x}^{3}}{9\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{e{x}^{3}}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{e}{9\,b{a}^{2} \left ( b{x}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d*x+c)/(b*x^3+a)^4,x)

[Out]

1/9*c/a*x/(b*x^3+a)^3+4/27*c/a^2*x/(b*x^3+a)^2+20/81*c/a^3*x/(b*x^3+a)+40/243*c/a^3/b/(1/b*a)^(2/3)*ln(x+(1/b*
a)^(1/3))-20/243*c/a^3/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+40/243*c/a^3/b/(1/b*a)^(2/3)*3^(1
/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+1/9*d/a*x^2/(b*x^3+a)^3+7/54*d/a^2*x^2/(b*x^3+a)^2+14/81*d/a^3*x
^2/(b*x^3+a)-14/243*d/a^3/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+7/243*d/a^3/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)
*x+(1/b*a)^(2/3))+14/243*d/a^3*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+1/9*e/a*x^3/(
b*x^3+a)^3+1/9*e/a^2*x^3/(b*x^3+a)^2-1/9*e/a^2/b/(b*x^3+a)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)/(b*x^3+a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 6.22442, size = 6020, normalized size = 24.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)/(b*x^3+a)^4,x, algorithm="fricas")

[Out]

1/972*(168*b^3*d*x^8 + 240*b^3*c*x^7 + 462*a*b^2*d*x^5 + 624*a*b^2*c*x^4 + 402*a^2*b*d*x^2 + 492*a^2*b*c*x - 1
08*a^3*e - 2*(a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343
*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((80
00*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*log(7/4*(4^(1/3)*(I*sqrt(3) +
1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqr
t(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^8*b*d
- 400*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1
/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/
(a^11*b^2))^(1/3)))*a^4*b*c^2 + 7840*a*c*d^2 + 4*(8000*b*c^3 + 343*a*d^3)*x) + ((a^3*b^4*x^9 + 3*a^4*b^3*x^6 +
3*a^5*b^2*x^3 + a^6*b)*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^
3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*
c^3 - 343*a*d^3)/(a^11*b^2))^(1/3))) + 3*sqrt(1/3)*(a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)*sqrt(
-((4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) -
140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11
*b^2))^(1/3)))^2*a^7*b + 8960*c*d)/(a^7*b)))*log(-7/4*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11
*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 34
3*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^8*b*d + 400*(4^(1/3)*(I*sqrt(3) + 1)*((
8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3)
+ 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*a^4*b*c^2 - 78
40*a*c*d^2 + 8*(8000*b*c^3 + 343*a*d^3)*x + 3/4*sqrt(1/3)*(7*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3
)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c
^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*a^8*b*d + 1600*a^4*b*c^2)*sqrt(-((4^
(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*
4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2)
)^(1/3)))^2*a^7*b + 8960*c*d)/(a^7*b))) + ((a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)*(4^(1/3)*(I*s
qrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*
d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))
- 3*sqrt(1/3)*(a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)*sqrt(-((4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c
^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^
7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^7*b + 8960*c*d)/(a
^7*b)))*log(-7/4*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^1
1*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 3
43*a*d^3)/(a^11*b^2))^(1/3)))^2*a^8*b*d + 400*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) +
(8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)
/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*a^4*b*c^2 - 7840*a*c*d^2 + 8*(8000*b*c^3 + 343*a*d^
3)*x - 3/4*sqrt(1/3)*(7*(4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^
3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*((8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*
c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))*a^8*b*d + 1600*a^4*b*c^2)*sqrt(-((4^(1/3)*(I*sqrt(3) + 1)*((8000*b*c^3 +
343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3) - 140*4^(2/3)*c*d*(-I*sqrt(3) + 1)/(a^7*b*(
(8000*b*c^3 + 343*a*d^3)/(a^11*b^2) + (8000*b*c^3 - 343*a*d^3)/(a^11*b^2))^(1/3)))^2*a^7*b + 8960*c*d)/(a^7*b)
)))/(a^3*b^4*x^9 + 3*a^4*b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)

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Sympy [A]  time = 4.04936, size = 202, normalized size = 0.81 \begin{align*} \operatorname{RootSum}{\left (14348907 t^{3} a^{11} b^{2} + 408240 t a^{4} b c d + 2744 a d^{3} - 64000 b c^{3}, \left ( t \mapsto t \log{\left (x + \frac{413343 t^{2} a^{8} b d + 194400 t a^{4} b c^{2} + 7840 a c d^{2}}{1372 a d^{3} + 32000 b c^{3}} \right )} \right )\right )} + \frac{- 18 a^{3} e + 82 a^{2} b c x + 67 a^{2} b d x^{2} + 104 a b^{2} c x^{4} + 77 a b^{2} d x^{5} + 40 b^{3} c x^{7} + 28 b^{3} d x^{8}}{162 a^{6} b + 486 a^{5} b^{2} x^{3} + 486 a^{4} b^{3} x^{6} + 162 a^{3} b^{4} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d*x+c)/(b*x**3+a)**4,x)

[Out]

RootSum(14348907*_t**3*a**11*b**2 + 408240*_t*a**4*b*c*d + 2744*a*d**3 - 64000*b*c**3, Lambda(_t, _t*log(x + (
413343*_t**2*a**8*b*d + 194400*_t*a**4*b*c**2 + 7840*a*c*d**2)/(1372*a*d**3 + 32000*b*c**3)))) + (-18*a**3*e +
82*a**2*b*c*x + 67*a**2*b*d*x**2 + 104*a*b**2*c*x**4 + 77*a*b**2*d*x**5 + 40*b**3*c*x**7 + 28*b**3*d*x**8)/(1
62*a**6*b + 486*a**5*b**2*x**3 + 486*a**4*b**3*x**6 + 162*a**3*b**4*x**9)

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Giac [A]  time = 1.08396, size = 333, normalized size = 1.33 \begin{align*} -\frac{2 \,{\left (7 \, d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 20 \, c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{4}} + \frac{2 \, \sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{243 \, a^{4} b^{2}} + \frac{28 \, b^{3} d x^{8} + 40 \, b^{3} c x^{7} + 77 \, a b^{2} d x^{5} + 104 \, a b^{2} c x^{4} + 67 \, a^{2} b d x^{2} + 82 \, a^{2} b c x - 18 \, a^{3} e}{162 \,{\left (b x^{3} + a\right )}^{3} a^{3} b} + \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{243 \, a^{5} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)/(b*x^3+a)^4,x, algorithm="giac")

[Out]

-2/243*(7*d*(-a/b)^(1/3) + 20*c)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^4 + 2/243*sqrt(3)*(20*(-a*b^2)^(1/3
)*b*c - 7*(-a*b^2)^(2/3)*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^2) + 1/162*(28*b^3*d*
x^8 + 40*b^3*c*x^7 + 77*a*b^2*d*x^5 + 104*a*b^2*c*x^4 + 67*a^2*b*d*x^2 + 82*a^2*b*c*x - 18*a^3*e)/((b*x^3 + a)
^3*a^3*b) + 1/243*(20*(-a*b^2)^(1/3)*a*b^3*c + 7*(-a*b^2)^(2/3)*a*b^2*d)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/
3))/(a^5*b^4)