Integrand size = 33, antiderivative size = 1366 \[ \int \frac {(a+b x)^2}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=-\frac {3 (b c-a d) (c+d x)^{2/3}}{4 d^3 \sqrt [3]{b c+a d+2 b d x}}+\frac {3 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{16 d^3}-\frac {9 (b c-a d) \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2}}{16 b^{2/3} d^5 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}+\frac {9 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b c-a d)^{5/3} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{32 b^{2/3} d^3 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}-\frac {3\ 3^{3/4} (b c-a d)^{5/3} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right ),-7-4 \sqrt {3}\right )}{8 \sqrt {2} b^{2/3} d^3 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}} \]
-3/4*(-a*d+b*c)*(d*x+c)^(2/3)/d^3/(2*b*d*x+a*d+b*c)^(1/3)+3/16*(d*x+c)^(2/ 3)*(2*b*d*x+a*d+b*c)^(2/3)/d^3-9/16*(-a*d+b*c)*((d*x+c)*(2*b*d*x+a*d+b*c)) ^(1/3)*(d^2*(4*b*d*x+a*d+3*b*c)^2)^(1/2)*((d*(a*d+3*b*c)+4*b*d^2*x)^2)^(1/ 2)/b^(2/3)/d^5/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(1/3)/(4*b*d*x+a*d+3*b*c)/( 2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))- 3/16*3^(3/4)*(-a*d+b*c)^(5/3)*((d*x+c)*(2*b*d*x+a*d+b*c))^(1/3)*((-a*d+b*c )^(2/3)+2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3))*EllipticF((2*b^(1/3)* ((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1-3^(1/2)))/(2*b^(1/3) *((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2))),I*3^(1/2) +2*I)*((d*(a*d+3*b*c)+4*b*d^2*x)^2)^(1/2)*(((-a*d+b*c)^(4/3)-2*b^(1/3)*(-a *d+b*c)^(2/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+4*b^(2/3)*((d*x+c)*(a*d+b* (2*d*x+c)))^(2/3))/(2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c) ^(2/3)*(1+3^(1/2)))^2)^(1/2)/b^(2/3)/d^3/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^( 1/3)/(4*b*d*x+a*d+3*b*c)*2^(1/2)/(d^2*(4*b*d*x+a*d+3*b*c)^2)^(1/2)/((-a*d+ b*c)^(2/3)*((-a*d+b*c)^(2/3)+2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3))/ (2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2))) ^2)^(1/2)+9/32*3^(1/4)*(-a*d+b*c)^(5/3)*((d*x+c)*(2*b*d*x+a*d+b*c))^(1/3)* ((-a*d+b*c)^(2/3)+2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3))*EllipticE(( 2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1-3^(1/2)))/ (2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2...
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.08 \[ \int \frac {(a+b x)^2}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\frac {3 (c+d x)^{2/3} \left (-3 b c+5 a d+2 b d x+3 (-b c+a d) \sqrt [3]{\frac {a d+b (c+2 d x)}{-b c+a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {2 b (c+d x)}{b c-a d}\right )\right )}{16 d^3 \sqrt [3]{a d+b (c+2 d x)}} \]
(3*(c + d*x)^(2/3)*(-3*b*c + 5*a*d + 2*b*d*x + 3*(-(b*c) + a*d)*((a*d + b* (c + 2*d*x))/(-(b*c) + a*d))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (2*b*( c + d*x))/(b*c - a*d)]))/(16*d^3*(a*d + b*(c + 2*d*x))^(1/3))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {100, 27, 90, 80, 79}
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.
| \(\displaystyle \int \frac {(a+b x)^2}{\sqrt [3]{c+d x} (a d+b c+2 b d x)^{4/3}} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {3 \int \frac {2 b^2 d^2 (b c-a d) (a+b x)}{3 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}}dx}{4 b^2 d^3 (b c-a d)}-\frac {3 (c+d x)^{2/3} (b c-a d)}{4 d^3 \sqrt [3]{a d+b c+2 b d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a+b x}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}}dx}{2 d}-\frac {3 (c+d x)^{2/3} (b c-a d)}{4 d^3 \sqrt [3]{a d+b c+2 b d x}}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\frac {3 (c+d x)^{2/3} (a d+b c+2 b d x)^{2/3}}{8 d^2}-\frac {3 (b c-a d) \int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}}dx}{4 d}}{2 d}-\frac {3 (c+d x)^{2/3} (b c-a d)}{4 d^3 \sqrt [3]{a d+b c+2 b d x}}\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle \frac {\frac {3 (c+d x)^{2/3} (a d+b c+2 b d x)^{2/3}}{8 d^2}-\frac {3 (b c-a d) \sqrt [3]{-\frac {a d+b c+2 b d x}{b c-a d}} \int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{-\frac {b c+a d}{b c-a d}-\frac {2 b d x}{b c-a d}}}dx}{4 d \sqrt [3]{a d+b c+2 b d x}}}{2 d}-\frac {3 (c+d x)^{2/3} (b c-a d)}{4 d^3 \sqrt [3]{a d+b c+2 b d x}}\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {\frac {3 (c+d x)^{2/3} (a d+b c+2 b d x)^{2/3}}{8 d^2}-\frac {9 (c+d x)^{2/3} (b c-a d) \sqrt [3]{-\frac {a d+b c+2 b d x}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {2 b (c+d x)}{b c-a d}\right )}{8 d^2 \sqrt [3]{a d+b c+2 b d x}}}{2 d}-\frac {3 (c+d x)^{2/3} (b c-a d)}{4 d^3 \sqrt [3]{a d+b c+2 b d x}}\) |
(-3*(b*c - a*d)*(c + d*x)^(2/3))/(4*d^3*(b*c + a*d + 2*b*d*x)^(1/3)) + ((3 *(c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)^(2/3))/(8*d^2) - (9*(b*c - a*d)*(c + d*x)^(2/3)*(-((b*c + a*d + 2*b*d*x)/(b*c - a*d)))^(1/3)*Hypergeometric2F 1[1/3, 2/3, 5/3, (2*b*(c + d*x))/(b*c - a*d)])/(8*d^2*(b*c + a*d + 2*b*d*x )^(1/3)))/(2*d)
3.31.30.3.1 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[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
\[\int \frac {\left (b x +a \right )^{2}}{\left (d x +c \right )^{\frac {1}{3}} \left (2 b d x +a d +b c \right )^{\frac {4}{3}}}d x\]
\[ \int \frac {(a+b x)^2}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{2}}{{\left (2 \, b d x + b c + a d\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
integral((b^2*x^2 + 2*a*b*x + a^2)*(2*b*d*x + b*c + a*d)^(2/3)*(d*x + c)^( 2/3)/(4*b^2*d^3*x^3 + b^2*c^3 + 2*a*b*c^2*d + a^2*c*d^2 + 4*(2*b^2*c*d^2 + a*b*d^3)*x^2 + (5*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x), x)
\[ \int \frac {(a+b x)^2}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int \frac {\left (a + b x\right )^{2}}{\sqrt [3]{c + d x} \left (a d + b c + 2 b d x\right )^{\frac {4}{3}}}\, dx \]
\[ \int \frac {(a+b x)^2}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{2}}{{\left (2 \, b d x + b c + a d\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {(a+b x)^2}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{2}}{{\left (2 \, b d x + b c + a d\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {(a+b x)^2}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{1/3}\,{\left (a\,d+b\,c+2\,b\,d\,x\right )}^{4/3}} \,d x \]