Integrand size = 28, antiderivative size = 637 \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{20/3}} \, dx=-\frac {3 \sqrt [3]{a+b x+c x^2}}{187 c^2 d^3 (d (b+2 c x))^{11/3}}+\frac {6 \sqrt [3]{a+b x+c x^2}}{935 c^2 \left (b^2-4 a c\right ) d^5 (d (b+2 c x))^{5/3}}-\frac {3 \left (a+b x+c x^2\right )^{4/3}}{34 c d (d (b+2 c x))^{17/3}}+\frac {3\ 3^{3/4} \sqrt [3]{d (b+2 c x)} \left (b^2-4 a c-(b+2 c x)^2\right ) \left (2 \sqrt [3]{c} d^{2/3}-\frac {\sqrt [3]{2} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right ) \sqrt {\frac {2 \sqrt [3]{2} c^{2/3} d^{4/3}+\frac {(d (b+2 c x))^{4/3}}{\left (a+b x+c x^2\right )^{2/3}}+\frac {2^{2/3} \sqrt [3]{c} d^{2/3} (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}}{\left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1-\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}}{2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7480 c^{10/3} \left (b^2-4 a c\right )^2 d^{23/3} \left (a+b x+c x^2\right )^{2/3} \sqrt {-\frac {(d (b+2 c x))^{2/3} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {(d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )}{\sqrt [3]{a+b x+c x^2} \left (2^{2/3} \sqrt [3]{c} d^{2/3}-\frac {\left (1+\sqrt {3}\right ) (d (b+2 c x))^{2/3}}{\sqrt [3]{a+b x+c x^2}}\right )^2}}} \]
-3/187*(c*x^2+b*x+a)^(1/3)/c^2/d^3/(d*(2*c*x+b))^(11/3)+6/935*(c*x^2+b*x+a )^(1/3)/c^2/(-4*a*c+b^2)/d^5/(d*(2*c*x+b))^(5/3)-3/34*(c*x^2+b*x+a)^(4/3)/ c/d/(d*(2*c*x+b))^(17/3)+3/7480*3^(3/4)*(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))*EllipticF((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)/c^(10/3)/(-4*a*c+b^2)^2/d^(23/3)/(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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.99 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.18 \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{20/3}} \, dx=\frac {3 \left (b^2-4 a c\right ) \sqrt [3]{d (b+2 c x)} \sqrt [3]{a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {17}{6},-\frac {4}{3},-\frac {11}{6},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{136\ 2^{2/3} c^2 d^7 (b+2 c x)^6 \sqrt [3]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]
(3*(b^2 - 4*a*c)*(d*(b + 2*c*x))^(1/3)*(a + x*(b + c*x))^(1/3)*Hypergeomet ric2F1[-17/6, -4/3, -11/6, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(136*2^(2/3)*c^2* d^7*(b + 2*c*x)^6*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))
Time = 0.51 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1118, 27, 247, 247, 264, 266, 771, 766}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{20/3}} \, dx\) |
\(\Big \downarrow \) 1118 |
\(\displaystyle \frac {\int \frac {\left (-\frac {b^2}{c}+\frac {(b d+2 c x d)^2}{c d^2}+4 a\right )^{4/3}}{4\ 2^{2/3} (b d+2 c x d)^{20/3}}d(b d+2 c x d)}{2 c d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\left (-\frac {b^2}{c}+\frac {(b d+2 c x d)^2}{c d^2}+4 a\right )^{4/3}}{(b d+2 c x d)^{20/3}}d(b d+2 c x d)}{8\ 2^{2/3} c d}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {\frac {8 \int \frac {\sqrt [3]{-\frac {b^2}{c}+\frac {(b d+2 c x d)^2}{c d^2}+4 a}}{(b d+2 c x d)^{14/3}}d(b d+2 c x d)}{17 c d^2}-\frac {3 \left (4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}\right )^{4/3}}{17 (b d+2 c d x)^{17/3}}}{8\ 2^{2/3} c d}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {\frac {8 \left (\frac {2 \int \frac {1}{(b d+2 c x d)^{8/3} \left (-\frac {b^2}{c}+\frac {(b d+2 c x d)^2}{c d^2}+4 a\right )^{2/3}}d(b d+2 c x d)}{11 c d^2}-\frac {3 \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}{11 (b d+2 c d x)^{11/3}}\right )}{17 c d^2}-\frac {3 \left (4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}\right )^{4/3}}{17 (b d+2 c d x)^{17/3}}}{8\ 2^{2/3} c d}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {\frac {8 \left (\frac {2 \left (\frac {3 \int \frac {1}{(b d+2 c x d)^{2/3} \left (-\frac {b^2}{c}+\frac {(b d+2 c x d)^2}{c d^2}+4 a\right )^{2/3}}d(b d+2 c x d)}{5 d^2 \left (b^2-4 a c\right )}+\frac {3 c \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}{5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/3}}\right )}{11 c d^2}-\frac {3 \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}{11 (b d+2 c d x)^{11/3}}\right )}{17 c d^2}-\frac {3 \left (4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}\right )^{4/3}}{17 (b d+2 c d x)^{17/3}}}{8\ 2^{2/3} c d}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {8 \left (\frac {2 \left (\frac {9 \int \frac {1}{\left (-\frac {b^2}{c}+\frac {(b d+2 c x d)^2}{c d^2}+4 a\right )^{2/3}}d\sqrt [3]{b d+2 c x d}}{5 d^2 \left (b^2-4 a c\right )}+\frac {3 c \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}{5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/3}}\right )}{11 c d^2}-\frac {3 \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}{11 (b d+2 c d x)^{11/3}}\right )}{17 c d^2}-\frac {3 \left (4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}\right )^{4/3}}{17 (b d+2 c d x)^{17/3}}}{8\ 2^{2/3} c d}\) |
\(\Big \downarrow \) 771 |
\(\displaystyle \frac {\frac {8 \left (\frac {2 \left (\frac {9 \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{c d^2}}}d\frac {\sqrt [3]{b d+2 c x d}}{\sqrt [6]{-\frac {b^2}{c}+\frac {(b d+2 c x d)^2}{c d^2}+4 a}}}{5 d^2 \left (b^2-4 a c\right ) \sqrt {\frac {c d^2 \left (4 a-\frac {b^2}{c}\right )}{c d^2 \left (4 a-\frac {b^2}{c}\right )+(b d+2 c d x)^2}} \sqrt {4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}+\frac {3 c \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}{5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/3}}\right )}{11 c d^2}-\frac {3 \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}{11 (b d+2 c d x)^{11/3}}\right )}{17 c d^2}-\frac {3 \left (4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}\right )^{4/3}}{17 (b d+2 c d x)^{17/3}}}{8\ 2^{2/3} c d}\) |
\(\Big \downarrow \) 766 |
\(\displaystyle \frac {\frac {8 \left (\frac {2 \left (\frac {3\ 3^{3/4} \sqrt [3]{b d+2 c d x} \left (\sqrt [3]{c} d^{2/3}-(b d+2 c d x)^{2/3}\right ) \sqrt {\frac {\sqrt [3]{c} d^{2/3} (b d+2 c d x)^{2/3}+(b d+2 c d x)^{4/3}+c^{2/3} d^{4/3}}{\left (\sqrt [3]{c} d^{2/3}-\left (1+\sqrt {3}\right ) (b d+2 c d x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{c} d^{2/3}-\left (1-\sqrt {3}\right ) (b d+2 c x d)^{2/3}}{\sqrt [3]{c} d^{2/3}-\left (1+\sqrt {3}\right ) (b d+2 c x d)^{2/3}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{10 \sqrt [3]{c} d^{8/3} \left (b^2-4 a c\right ) \sqrt {-\frac {(b d+2 c d x)^{2/3} \left (\sqrt [3]{c} d^{2/3}-(b d+2 c d x)^{2/3}\right )}{\left (\sqrt [3]{c} d^{2/3}-\left (1+\sqrt {3}\right ) (b d+2 c d x)^{2/3}\right )^2}} \sqrt {1-\frac {(b d+2 c d x)^2}{c d^2}} \sqrt {\frac {c d^2 \left (4 a-\frac {b^2}{c}\right )}{c d^2 \left (4 a-\frac {b^2}{c}\right )+(b d+2 c d x)^2}} \left (4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}\right )^{2/3}}+\frac {3 c \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}{5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/3}}\right )}{11 c d^2}-\frac {3 \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}}}{11 (b d+2 c d x)^{11/3}}\right )}{17 c d^2}-\frac {3 \left (4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}\right )^{4/3}}{17 (b d+2 c d x)^{17/3}}}{8\ 2^{2/3} c d}\) |
((-3*(4*a - b^2/c + (b*d + 2*c*d*x)^2/(c*d^2))^(4/3))/(17*(b*d + 2*c*d*x)^ (17/3)) + (8*((-3*(4*a - b^2/c + (b*d + 2*c*d*x)^2/(c*d^2))^(1/3))/(11*(b* d + 2*c*d*x)^(11/3)) + (2*((3*c*(4*a - b^2/c + (b*d + 2*c*d*x)^2/(c*d^2))^ (1/3))/(5*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(5/3)) + (3*3^(3/4)*(b*d + 2*c*d*x )^(1/3)*(c^(1/3)*d^(2/3) - (b*d + 2*c*d*x)^(2/3))*Sqrt[(c^(2/3)*d^(4/3) + c^(1/3)*d^(2/3)*(b*d + 2*c*d*x)^(2/3) + (b*d + 2*c*d*x)^(4/3))/(c^(1/3)*d^ (2/3) - (1 + Sqrt[3])*(b*d + 2*c*d*x)^(2/3))^2]*EllipticF[ArcCos[(c^(1/3)* d^(2/3) - (1 - Sqrt[3])*(b*d + 2*c*d*x)^(2/3))/(c^(1/3)*d^(2/3) - (1 + Sqr t[3])*(b*d + 2*c*d*x)^(2/3))], (2 + Sqrt[3])/4])/(10*c^(1/3)*(b^2 - 4*a*c) *d^(8/3)*Sqrt[-(((b*d + 2*c*d*x)^(2/3)*(c^(1/3)*d^(2/3) - (b*d + 2*c*d*x)^ (2/3)))/(c^(1/3)*d^(2/3) - (1 + Sqrt[3])*(b*d + 2*c*d*x)^(2/3))^2)]*Sqrt[( (4*a - b^2/c)*c*d^2)/((4*a - b^2/c)*c*d^2 + (b*d + 2*c*d*x)^2)]*Sqrt[1 - ( b*d + 2*c*d*x)^2/(c*d^2)]*(4*a - b^2/c + (b*d + 2*c*d*x)^2/(c*d^2))^(2/3)) ))/(11*c*d^2)))/(17*c*d^2))/(8*2^(2/3)*c*d)
3.15.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
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 ]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a/(a + b*x^n))^(p + 1 /n)*(a + b*x^n)^(p + 1/n) Subst[Int[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]]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[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] && EqQ[2*c*d - b*e, 0]
\[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {4}{3}}}{\left (2 c d x +b d \right )^{\frac {20}{3}}}d x\]
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{20/3}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac {20}{3}}} \,d x } \]
integral((2*c*d*x + b*d)^(1/3)*(c*x^2 + b*x + a)^(4/3)/(128*c^7*d^7*x^7 + 448*b*c^6*d^7*x^6 + 672*b^2*c^5*d^7*x^5 + 560*b^3*c^4*d^7*x^4 + 280*b^4*c^ 3*d^7*x^3 + 84*b^5*c^2*d^7*x^2 + 14*b^6*c*d^7*x + b^7*d^7), x)
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{20/3}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{20/3}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac {20}{3}}} \,d x } \]
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{20/3}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac {20}{3}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{20/3}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{4/3}}{{\left (b\,d+2\,c\,d\,x\right )}^{20/3}} \,d x \]