3.25.84 \(\int (d+e x) (a+b x+c x^2)^{4/3} \, dx\) [2484]

Optimal. Leaf size=539 \[ -\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{110 c^3}+\frac {3 (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{44 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (b^2-4 a c\right )^2 (2 c d-b e) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt {3}\right )}{55\ 2^{2/3} c^{10/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}} \]

[Out]

-3/110*(-4*a*c+b^2)*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(1/3)/c^3+3/44*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(
4/3)/c^2+3/14*e*(c*x^2+b*x+a)^(7/3)/c+1/110*3^(3/4)*(-4*a*c+b^2)^2*(-b*e+2*c*d)*((-4*a*c+b^2)^(1/3)+2^(2/3)*c^
(1/3)*(c*x^2+b*x+a)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1-3^(1/2)))/(2^(
2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)+1/2*2^(1/2))*(((-
4*a*c+b^2)^(2/3)-2^(2/3)*c^(1/3)*(-4*a*c+b^2)^(1/3)*(c*x^2+b*x+a)^(1/3)+2*2^(1/3)*c^(2/3)*(c*x^2+b*x+a)^(2/3))
/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))^2)^(1/2)*2^(1/3)/c^(10/3)/(2*c*x+b)/((-4
*a*c+b^2)^(1/3)*((-4*a*c+b^2)^(1/3)+2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+
(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.39, antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 637, 327, 224} \begin {gather*} \frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} (2 c d-b e) F\left (\text {ArcSin}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt {3}\right )}{55\ 2^{2/3} c^{10/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2} (2 c d-b e)}{110 c^3}+\frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3} (2 c d-b e)}{44 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/3))/(110*c^3) + (3*(2*c*d - b*e)*(b + 2*c*x)*
(a + b*x + c*x^2)^(4/3))/(44*c^2) + (3*e*(a + b*x + c*x^2)^(7/3))/(14*c) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4
*a*c)^2*(2*c*d - b*e)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3
) - 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/(
(1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3]
)*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*
c^(1/3)*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/(55*2^(2/3)*c^(10/3)*(b + 2*c*x)*Sqrt[((b^2 - 4*a*c)^(1/3)
*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)
*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2])

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 637

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[d*(Sqrt[(b + 2*c*x)
^2]/(b + 2*c*x)), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 654

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

Rubi steps

\begin {align*} \int (d+e x) \left (a+b x+c x^2\right )^{4/3} \, dx &=\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}+\frac {(2 c d-b e) \int \left (a+b x+c x^2\right )^{4/3} \, dx}{2 c}\\ &=\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}+\frac {\left (3 (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2 c (b+2 c x)}\\ &=\frac {3 (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{44 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}-\frac {\left (3 \left (b^2-4 a c\right ) (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{11 c^2 (b+2 c x)}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{110 c^3}+\frac {3 (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{44 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}+\frac {\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{110 c^3 (b+2 c x)}\\ &=-\frac {3 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{110 c^3}+\frac {3 (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{44 c^2}+\frac {3 e \left (a+b x+c x^2\right )^{7/3}}{14 c}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} \left (b^2-4 a c\right )^2 (2 c d-b e) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt {3}\right )}{55\ 2^{2/3} c^{10/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.15, size = 113, normalized size = 0.21 \begin {gather*} \frac {(a+x (b+c x))^{4/3} \left (48 c^2 e (a+x (b+c x))-\frac {7 \sqrt [3]{2} c (-2 c d+b e) (b+2 c x) \, _2F_1\left (-\frac {4}{3},\frac {1}{2};\frac {3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (-\frac {c (a+x (b+c x))}{b^2-4 a c}\right )^{4/3}}\right )}{224 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + x*(b + c*x))^(4/3)*(48*c^2*e*(a + x*(b + c*x)) - (7*2^(1/3)*c*(-2*c*d + b*e)*(b + 2*c*x)*Hypergeometric2
F1[-4/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(-((c*(a + x*(b + c*x)))/(b^2 - 4*a*c)))^(4/3)))/(224*c^3)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (e x +d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {4}{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*d*x^2 + b*d*x + a*d + (c*x^3 + b*x^2 + a*x)*e)*(c*x^2 + b*x + a)^(1/3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {4}{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{4/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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