Integrand size = 19, antiderivative size = 655 \[ \int (a+b x)^{4/3} \sqrt [3]{c+d x} \, dx=-\frac {3 (b c-a d)^2 \sqrt [3]{a+b x} \sqrt [3]{c+d x}}{20 b d^2}+\frac {3 (b c-a d) (a+b x)^{4/3} \sqrt [3]{c+d x}}{40 b d}+\frac {3 (a+b x)^{7/3} \sqrt [3]{c+d x}}{8 b}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} (b c-a d)^3 ((a+b x) (c+d x))^{2/3} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{10\ 2^{2/3} b^{4/3} d^{7/3} (a+b x)^{2/3} (c+d x)^{2/3} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}} \]
-3/20*(-a*d+b*c)^2*(b*x+a)^(1/3)*(d*x+c)^(1/3)/b/d^2+3/40*(-a*d+b*c)*(b*x+ a)^(4/3)*(d*x+c)^(1/3)/b/d+3/8*(b*x+a)^(7/3)*(d*x+c)^(1/3)/b+1/20*3^(3/4)* (-a*d+b*c)^3*((b*x+a)*(d*x+c))^(2/3)*((-a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^( 1/3)*((b*x+a)*(d*x+c))^(1/3))*EllipticF((2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)* (d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1-3^(1/2)))/(2^(2/3)*b^(1/3)*d^(1/3)*((b* x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*((2*b*d*x +a*d+b*c)^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*(((-a*d+b*c)^(4/3)-2^(2/3)*b^ (1/3)*d^(1/3)*(-a*d+b*c)^(2/3)*((b*x+a)*(d*x+c))^(1/3)+2*2^(1/3)*b^(2/3)*d ^(2/3)*((b*x+a)*(d*x+c))^(2/3))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c)) ^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))^2)^(1/2)*2^(1/3)/b^(4/3)/d^(7/3)/(b*x +a)^(2/3)/(d*x+c)^(2/3)/(2*b*d*x+a*d+b*c)/((a*d+b*(2*d*x+c))^2)^(1/2)/((-a *d+b*c)^(2/3)*((-a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^ (1/3))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*( 1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.11 \[ \int (a+b x)^{4/3} \sqrt [3]{c+d x} \, dx=\frac {3 (a+b x)^{7/3} \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {7}{3},\frac {10}{3},\frac {d (a+b x)}{-b c+a d}\right )}{7 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \]
(3*(a + b*x)^(7/3)*(c + d*x)^(1/3)*Hypergeometric2F1[-1/3, 7/3, 10/3, (d*( a + b*x))/(-(b*c) + a*d)])/(7*b*((b*(c + d*x))/(b*c - a*d))^(1/3))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {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.
\(\displaystyle \int (a+b x)^{4/3} \sqrt [3]{c+d x} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {\sqrt [3]{c+d x} \int (a+b x)^{4/3} \sqrt [3]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}dx}{\sqrt [3]{\frac {b (c+d x)}{b c-a d}}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {3 (a+b x)^{7/3} \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {7}{3},\frac {10}{3},-\frac {d (a+b x)}{b c-a d}\right )}{7 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}}\) |
(3*(a + b*x)^(7/3)*(c + d*x)^(1/3)*Hypergeometric2F1[-1/3, 7/3, 10/3, -((d *(a + b*x))/(b*c - a*d))])/(7*b*((b*(c + d*x))/(b*c - a*d))^(1/3))
3.16.82.3.1 Defintions of rubi rules used
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])
\[\int \left (b x +a \right )^{\frac {4}{3}} \left (d x +c \right )^{\frac {1}{3}}d x\]
\[ \int (a+b x)^{4/3} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \]
\[ \int (a+b x)^{4/3} \sqrt [3]{c+d x} \, dx=\int \left (a + b x\right )^{\frac {4}{3}} \sqrt [3]{c + d x}\, dx \]
\[ \int (a+b x)^{4/3} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \]
\[ \int (a+b x)^{4/3} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \]
Timed out. \[ \int (a+b x)^{4/3} \sqrt [3]{c+d x} \, dx=\int {\left (a+b\,x\right )}^{4/3}\,{\left (c+d\,x\right )}^{1/3} \,d x \]
\[ \int (a+b x)^{4/3} \sqrt [3]{c+d x} \, dx=\frac {39 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} a^{2} c d +27 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} a^{2} d^{2} x -9 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} a b \,c^{2}+30 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} a b c d x +15 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} a b \,d^{2} x^{2}+3 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} b^{2} c^{2} x +15 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} b^{2} c d \,x^{2}+4 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a^{4} d^{4}-8 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a^{3} b c \,d^{3}+8 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a \,b^{3} c^{3} d -4 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) b^{4} c^{4}}{40 d \left (a d +b c \right )} \]
(39*(c + d*x)**(1/3)*(a + b*x)**(1/3)*a**2*c*d + 27*(c + d*x)**(1/3)*(a + b*x)**(1/3)*a**2*d**2*x - 9*(c + d*x)**(1/3)*(a + b*x)**(1/3)*a*b*c**2 + 3 0*(c + d*x)**(1/3)*(a + b*x)**(1/3)*a*b*c*d*x + 15*(c + d*x)**(1/3)*(a + b *x)**(1/3)*a*b*d**2*x**2 + 3*(c + d*x)**(1/3)*(a + b*x)**(1/3)*b**2*c**2*x + 15*(c + d*x)**(1/3)*(a + b*x)**(1/3)*b**2*c*d*x**2 + 4*int(((c + d*x)** (1/3)*(a + b*x)**(1/3)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 + 2*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x**2),x)*a**4*d**4 - 8*int(((c + d*x)**(1/3)*(a + b*x)**(1/3)*x)/(a**2*c*d + a**2*d**2*x + a*b*c**2 + 2*a* b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x**2),x)*a**3*b*c*d**3 + 8*int(((c + d*x)**(1/3)*(a + b*x)**(1/3)*x)/(a**2*c*d + a**2*d**2*x + a*b* c**2 + 2*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x**2),x)*a*b** 3*c**3*d - 4*int(((c + d*x)**(1/3)*(a + b*x)**(1/3)*x)/(a**2*c*d + a**2*d* *2*x + a*b*c**2 + 2*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x + b**2*c*d*x** 2),x)*b**4*c**4)/(40*d*(a*d + b*c))