Integrand size = 28, antiderivative size = 100 \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx=\frac {3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},-\frac {1}{6},\frac {5}{6},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \sqrt [3]{b d+2 c d x} \sqrt [3]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}} \] Output:
3/8*(-4*a*c+b^2)*(c*x^2+b*x+a)^(1/3)*hypergeom([-4/3, -1/6],[5/6],(2*c*x+b )^2/(-4*a*c+b^2))/c^2/d/(2*c*d*x+b*d)^(1/3)/(1-(2*c*x+b)^2/(-4*a*c+b^2))^( 1/3)
Time = 10.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx=\frac {3 \left (b^2-4 a c\right ) \sqrt [3]{a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},-\frac {1}{6},\frac {5}{6},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{8\ 2^{2/3} c^2 d \sqrt [3]{d (b+2 c x)} \sqrt [3]{\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \] Input:
Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(4/3),x]
Output:
(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/3)*Hypergeometric2F1[-4/3, -1/6, 5/6 , (b + 2*c*x)^2/(b^2 - 4*a*c)])/(8*2^(2/3)*c^2*d*(d*(b + 2*c*x))^(1/3)*((c *(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3))
Time = 0.25 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.41, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1118, 27, 279, 278}
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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx\) |
\(\Big \downarrow \) 1118 |
\(\displaystyle \frac {\int \frac {\left (-\frac {b^2}{c}+\frac {(b d+2 c x d)^2}{c d^2}+4 a\right )^{4/3}}{4\ 2^{2/3} (b d+2 c x d)^{4/3}}d(b d+2 c x d)}{2 c d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\left (-\frac {b^2}{c}+\frac {(b d+2 c x d)^2}{c d^2}+4 a\right )^{4/3}}{(b d+2 c x d)^{4/3}}d(b d+2 c x d)}{8\ 2^{2/3} c d}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {\left (4 a-\frac {b^2}{c}\right ) \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}} \int \frac {\left (1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}\right )^{4/3}}{(b d+2 c x d)^{4/3}}d(b d+2 c x d)}{8\ 2^{2/3} c d \sqrt [3]{1-\frac {(b d+2 c d x)^2}{d^2 \left (b^2-4 a c\right )}}}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {3 \left (4 a-\frac {b^2}{c}\right ) \sqrt [3]{4 a-\frac {b^2}{c}+\frac {(b d+2 c d x)^2}{c d^2}} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},-\frac {1}{6},\frac {5}{6},\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}\right )}{8\ 2^{2/3} c d \sqrt [3]{b d+2 c d x} \sqrt [3]{1-\frac {(b d+2 c d x)^2}{d^2 \left (b^2-4 a c\right )}}}\) |
Input:
Int[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(4/3),x]
Output:
(-3*(4*a - b^2/c)*(4*a - b^2/c + (b*d + 2*c*d*x)^2/(c*d^2))^(1/3)*Hypergeo metric2F1[-4/3, -1/6, 5/6, (b*d + 2*c*d*x)^2/((b^2 - 4*a*c)*d^2)])/(8*2^(2 /3)*c*d*(b*d + 2*c*d*x)^(1/3)*(1 - (b*d + 2*c*d*x)^2/((b^2 - 4*a*c)*d^2))^ (1/3))
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[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
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]
\[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {4}{3}}}{\left (2 c d x +b d \right )^{\frac {4}{3}}}d x\]
Input:
int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(4/3),x)
Output:
int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(4/3),x)
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac {4}{3}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(4/3),x, algorithm="fricas")
Output:
integral((2*c*d*x + b*d)^(2/3)*(c*x^2 + b*x + a)^(4/3)/(4*c^2*d^2*x^2 + 4* b*c*d^2*x + b^2*d^2), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {4}{3}}}{\left (d \left (b + 2 c x\right )\right )^{\frac {4}{3}}}\, dx \] Input:
integrate((c*x**2+b*x+a)**(4/3)/(2*c*d*x+b*d)**(4/3),x)
Output:
Integral((a + b*x + c*x**2)**(4/3)/(d*(b + 2*c*x))**(4/3), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac {4}{3}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(4/3),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(4/3), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac {4}{3}}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(4/3),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(4/3), x)
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{4/3}}{{\left (b\,d+2\,c\,d\,x\right )}^{4/3}} \,d x \] Input:
int((a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(4/3),x)
Output:
int((a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(4/3), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx=\frac {\left (\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}}}{\left (2 c x +b \right )^{\frac {1}{3}} b +2 \left (2 c x +b \right )^{\frac {1}{3}} c x}d x \right ) a +\left (\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} x^{2}}{\left (2 c x +b \right )^{\frac {1}{3}} b +2 \left (2 c x +b \right )^{\frac {1}{3}} c x}d x \right ) c +\left (\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} x}{\left (2 c x +b \right )^{\frac {1}{3}} b +2 \left (2 c x +b \right )^{\frac {1}{3}} c x}d x \right ) b}{d^{\frac {4}{3}}} \] Input:
int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(4/3),x)
Output:
(int((a + b*x + c*x**2)**(1/3)/((b + 2*c*x)**(1/3)*b + 2*(b + 2*c*x)**(1/3 )*c*x),x)*a + int(((a + b*x + c*x**2)**(1/3)*x**2)/((b + 2*c*x)**(1/3)*b + 2*(b + 2*c*x)**(1/3)*c*x),x)*c + int(((a + b*x + c*x**2)**(1/3)*x)/((b + 2*c*x)**(1/3)*b + 2*(b + 2*c*x)**(1/3)*c*x),x)*b)/(d**(1/3)*d)