3.15.17 \(\int \frac {(a+b x+c x^2)^{4/3}}{(b d+2 c d x)^{8/3}} \, dx\) [1417]

Optimal. Leaf size=581 \[ \frac {\sqrt [3]{d (b+2 c x)} \sqrt [3]{a+b x+c x^2}}{5 c^2 d^3}-\frac {3 \left (a+b x+c x^2\right )^{4/3}}{10 c d (d (b+2 c x))^{5/3}}-\frac {\sqrt [3]{d (b+2 c x)} \left (b^2-4 a c-(b+2 c x)^2\right ) \left (2 \sqrt [3]{c} d^{2/3}-\frac {\sqrt [3]{2} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} c^{2/3} d^{4/3}+\frac {(d (b+2 c x))^{4/3}}{\left (a+b x+c x^2\right )^{2/3}}+\frac {2^{2/3} \sqrt [3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}}{\left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac {2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1-\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}}{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{40 \sqrt [4]{3} c^{10/3} d^{11/3} \left (a+b x+c x^2\right )^{2/3} \sqrt {-\frac {(d (b+2 c x))^{2/3} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {(d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )}{\sqrt [3]{a+b x+c x^2} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}}} \]

[Out]

1/5*(d*(2*c*x+b))^(1/3)*(c*x^2+b*x+a)^(1/3)/c^2/d^3-3/10*(c*x^2+b*x+a)^(4/3)/c/d/(d*(2*c*x+b))^(5/3)-1/120*(d*
(2*c*x+b))^(1/3)*(b^2-4*a*c-(2*c*x+b)^2)*(2*c^(1/3)*d^(2/3)-2^(1/3)*(d*(2*c*x+b))^(2/3)/(c*x^2+b*x+a)^(1/3))*(
(2^(2/3)*c^(1/3)*d^(2/3)-(d*(2*c*x+b))^(2/3)*(1-3^(1/2))/(c*x^2+b*x+a)^(1/3))^2/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(2
*c*x+b))^(2/3)*(1+3^(1/2))/(c*x^2+b*x+a)^(1/3))^2)^(1/2)/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(2*c*x+b))^(2/3)*(1-3^(1/
2))/(c*x^2+b*x+a)^(1/3))*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(2*c*x+b))^(2/3)*(1+3^(1/2))/(c*x^2+b*x+a)^(1/3))*Ellipti
cF((1-(2^(2/3)*c^(1/3)*d^(2/3)-(d*(2*c*x+b))^(2/3)*(1-3^(1/2))/(c*x^2+b*x+a)^(1/3))^2/(2^(2/3)*c^(1/3)*d^(2/3)
-(d*(2*c*x+b))^(2/3)*(1+3^(1/2))/(c*x^2+b*x+a)^(1/3))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*((2*2^(1/3)*c^(2/3)*d^
(4/3)+(d*(2*c*x+b))^(4/3)/(c*x^2+b*x+a)^(2/3)+2^(2/3)*c^(1/3)*d^(2/3)*(d*(2*c*x+b))^(2/3)/(c*x^2+b*x+a)^(1/3))
/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(2*c*x+b))^(2/3)*(1+3^(1/2))/(c*x^2+b*x+a)^(1/3))^2)^(1/2)*3^(3/4)/c^(10/3)/d^(11
/3)/(c*x^2+b*x+a)^(2/3)/(-(d*(2*c*x+b))^(2/3)*(2^(2/3)*c^(1/3)*d^(2/3)-(d*(2*c*x+b))^(2/3)/(c*x^2+b*x+a)^(1/3)
)/(c*x^2+b*x+a)^(1/3)/(2^(2/3)*c^(1/3)*d^(2/3)-(d*(2*c*x+b))^(2/3)*(1+3^(1/2))/(c*x^2+b*x+a)^(1/3))^2)^(1/2)

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Rubi [A]
time = 1.42, antiderivative size = 581, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {708, 283, 285, 335, 247, 231} \begin {gather*} -\frac {\left (-4 a c+b^2-(b+2 c x)^2\right ) \sqrt [3]{d (b+2 c x)} \left (2 \sqrt [3]{c} d^{2/3}-\frac {\sqrt [3]{2} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right ) \sqrt {\frac {\frac {2^{2/3} \sqrt [3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}+\frac {(d (b+2 c x))^{4/3}}{\left (a+b x+c x^2\right )^{2/3}}+2 \sqrt [3]{2} c^{2/3} d^{4/3}}{\left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}} F\left (\text {ArcCos}\left (\frac {2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1-\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{c x^2+b x+a}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{40 \sqrt [4]{3} c^{10/3} d^{11/3} \left (a+b x+c x^2\right )^{2/3} \sqrt {-\frac {(d (b+2 c x))^{2/3} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {(d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )}{\sqrt [3]{a+b x+c x^2} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}}}+\frac {\sqrt [3]{a+b x+c x^2} \sqrt [3]{d (b+2 c x)}}{5 c^2 d^3}-\frac {3 \left (a+b x+c x^2\right )^{4/3}}{10 c d (d (b+2 c x))^{5/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(8/3),x]

[Out]

((d*(b + 2*c*x))^(1/3)*(a + b*x + c*x^2)^(1/3))/(5*c^2*d^3) - (3*(a + b*x + c*x^2)^(4/3))/(10*c*d*(d*(b + 2*c*
x))^(5/3)) - ((d*(b + 2*c*x))^(1/3)*(b^2 - 4*a*c - (b + 2*c*x)^2)*(2*c^(1/3)*d^(2/3) - (2^(1/3)*(d*(b + 2*c*x)
)^(2/3))/(a + b*x + c*x^2)^(1/3))*Sqrt[(2*2^(1/3)*c^(2/3)*d^(4/3) + (d*(b + 2*c*x))^(4/3)/(a + b*x + c*x^2)^(2
/3) + (2^(2/3)*c^(1/3)*d^(2/3)*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3) - ((1
+ Sqrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ArcCos[(2^(2/3)*c^(1/3)*d^(2/3) - ((1
- Sqrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))/(2^(2/3)*c^(1/3)*d^(2/3) - ((1 + Sqrt[3])*(d*(b + 2
*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3))], (2 + Sqrt[3])/4])/(40*3^(1/4)*c^(10/3)*d^(11/3)*(a + b*x + c*x^2)^(2/
3)*Sqrt[-(((d*(b + 2*c*x))^(2/3)*(2^(2/3)*c^(1/3)*d^(2/3) - (d*(b + 2*c*x))^(2/3)/(a + b*x + c*x^2)^(1/3)))/((
a + b*x + c*x^2)^(1/3)*(2^(2/3)*c^(1/3)*d^(2/3) - ((1 + Sqrt[3])*(d*(b + 2*c*x))^(2/3))/(a + b*x + c*x^2)^(1/3
))^2))])

Rule 231

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s +
r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*(
(s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^
2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x]

Rule 247

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a/(a + b*x^n))^(p + 1/n)*(a + b*x^n)^(p + 1/n), Subst[In
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p,
0] && NeQ[p, -2^(-1)] && LtQ[Denominator[p + 1/n], Denominator[p]]

Rule 283

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

Rule 285

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 708

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{8/3}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^{4/3}}{x^{8/3}} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=-\frac {3 \left (a+b x+c x^2\right )^{4/3}}{10 c d (d (b+2 c x))^{5/3}}+\frac {\text {Subst}\left (\int \frac {\sqrt [3]{a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}}}{x^{2/3}} \, dx,x,b d+2 c d x\right )}{5 c^2 d^3}\\ &=\frac {\sqrt [3]{d (b+2 c x)} \sqrt [3]{a+b x+c x^2}}{5 c^2 d^3}-\frac {3 \left (a+b x+c x^2\right )^{4/3}}{10 c d (d (b+2 c x))^{5/3}}-\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{x^{2/3} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )^{2/3}} \, dx,x,b d+2 c d x\right )}{30 c^3 d^3}\\ &=\frac {\sqrt [3]{d (b+2 c x)} \sqrt [3]{a+b x+c x^2}}{5 c^2 d^3}-\frac {3 \left (a+b x+c x^2\right )^{4/3}}{10 c d (d (b+2 c x))^{5/3}}-\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{\left (a-\frac {b^2}{4 c}+\frac {x^6}{4 c d^2}\right )^{2/3}} \, dx,x,\sqrt [3]{d (b+2 c x)}\right )}{10 c^3 d^3}\\ &=\frac {\sqrt [3]{d (b+2 c x)} \sqrt [3]{a+b x+c x^2}}{5 c^2 d^3}-\frac {3 \left (a+b x+c x^2\right )^{4/3}}{10 c d (d (b+2 c x))^{5/3}}-\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^6}{4 c d^2}}} \, dx,x,\frac {\sqrt [3]{d (b+2 c x)}}{\sqrt [6]{a+x (b+c x)}}\right )}{10 c^3 d^3 \sqrt {\frac {a-\frac {b^2}{4 c}}{a+x (b+c x)}} \sqrt {a+x (b+c x)}}\\ &=\frac {\sqrt [3]{d (b+2 c x)} \sqrt [3]{a+b x+c x^2}}{5 c^2 d^3}-\frac {3 \left (a+b x+c x^2\right )^{4/3}}{10 c d (d (b+2 c x))^{5/3}}-\frac {\left (b^2-4 a c\right ) \sqrt [3]{d (b+2 c x)} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {(d (b+2 c x))^{2/3}}{\sqrt [3]{a+x (b+c x)}}\right ) \sqrt {\frac {2 \sqrt [3]{2} c^{2/3} d^{4/3}+\frac {(d (b+2 c x))^{4/3}}{(a+x (b+c x))^{2/3}}+\frac {2^{2/3} \sqrt [3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+x (b+c x)}}}{\left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+x (b+c x)}}\right )^2}} F\left (\cos ^{-1}\left (\frac {2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1-\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+x (b+c x)}}}{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+x (b+c x)}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5\ 2^{2/3} \sqrt [4]{3} c^{10/3} d^{11/3} \sqrt {\frac {4 a-\frac {b^2}{c}}{a+x (b+c x)}} (a+x (b+c x))^{2/3} \sqrt {4-\frac {(b+2 c x)^2}{c (a+x (b+c x))}} \sqrt {-\frac {(d (b+2 c x))^{2/3} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {(d (b+2 c x))^{2/3}}{\sqrt [3]{a+x (b+c x)}}\right )}{\sqrt [3]{a+x (b+c x)} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+x (b+c x)}}\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 = 10.06, size = 104, normalized size = 0.18 \begin {gather*} \frac {3 \left (b^2-4 a c\right ) \sqrt [3]{a+x (b+c x)} \, _2F_1\left (-\frac {4}{3},-\frac {5}{6};\frac {1}{6};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{40\ 2^{2/3} c^2 d (d (b+2 c x))^{5/3} \sqrt [3]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(8/3),x]

[Out]

(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-4/3, -5/6, 1/6, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(40*
2^(2/3)*c^2*d*(d*(b + 2*c*x))^(5/3)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {4}{3}}}{\left (2 c d x +b d \right )^{\frac {8}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(8/3),x)

[Out]

int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(8/3),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((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(8/3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(8/3), 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((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(8/3),x, algorithm="fricas")

[Out]

integral((2*c*d*x + b*d)^(1/3)*(c*x^2 + b*x + a)^(4/3)/(8*c^3*d^3*x^3 + 12*b*c^2*d^3*x^2 + 6*b^2*c*d^3*x + b^3
*d^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {4}{3}}}{\left (d \left (b + 2 c x\right )\right )^{\frac {8}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(4/3)/(2*c*d*x+b*d)**(8/3),x)

[Out]

Integral((a + b*x + c*x**2)**(4/3)/(d*(b + 2*c*x))**(8/3), 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((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(8/3),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(8/3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{4/3}}{{\left (b\,d+2\,c\,d\,x\right )}^{8/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(8/3),x)

[Out]

int((a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(8/3), x)

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