Integrand size = 19, antiderivative size = 457 \[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=-\frac {108 (b c-a d)^2 \sqrt {a+b x} \sqrt [3]{c+d x}}{935 b d^2}+\frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}-\frac {108\ 3^{3/4} \sqrt {2-\sqrt {3}} (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{935 b^{4/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
[Out]
Time = 0.44 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {52, 65, 225} \[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=-\frac {108\ 3^{3/4} \sqrt {2-\sqrt {3}} (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{935 b^{4/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {108 \sqrt {a+b x} \sqrt [3]{c+d x} (b c-a d)^2}{935 b d^2}+\frac {12 (a+b x)^{3/2} \sqrt [3]{c+d x} (b c-a d)}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b} \]
[In]
[Out]
Rule 52
Rule 65
Rule 225
Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}+\frac {(2 (b c-a d)) \int \frac {(a+b x)^{3/2}}{(c+d x)^{2/3}} \, dx}{17 b} \\ & = \frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}-\frac {\left (18 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{2/3}} \, dx}{187 b d} \\ & = -\frac {108 (b c-a d)^2 \sqrt {a+b x} \sqrt [3]{c+d x}}{935 b d^2}+\frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}+\frac {\left (54 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{2/3}} \, dx}{935 b d^2} \\ & = -\frac {108 (b c-a d)^2 \sqrt {a+b x} \sqrt [3]{c+d x}}{935 b d^2}+\frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}+\frac {\left (162 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^3}{d}}} \, dx,x,\sqrt [3]{c+d x}\right )}{935 b d^3} \\ & = -\frac {108 (b c-a d)^2 \sqrt {a+b x} \sqrt [3]{c+d x}}{935 b d^2}+\frac {12 (b c-a d) (a+b x)^{3/2} \sqrt [3]{c+d x}}{187 b d}+\frac {6 (a+b x)^{5/2} \sqrt [3]{c+d x}}{17 b}-\frac {108\ 3^{3/4} \sqrt {2-\sqrt {3}} (b c-a d)^3 \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right )|-7+4 \sqrt {3}\right )}{935 b^{4/3} d^3 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.16 \[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\frac {2 (a+b x)^{5/2} \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {5}{2},\frac {7}{2},\frac {d (a+b x)}{-b c+a d}\right )}{5 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \]
[In]
[Out]
\[\int \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {1}{3}}d x\]
[In]
[Out]
\[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \]
[In]
[Out]
\[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int \left (a + b x\right )^{\frac {3}{2}} \sqrt [3]{c + d x}\, dx \]
[In]
[Out]
\[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \]
[In]
[Out]
\[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int (a+b x)^{3/2} \sqrt [3]{c+d x} \, dx=\int {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{1/3} \,d x \]
[In]
[Out]