Integrand size = 19, antiderivative size = 301 \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {1768 a^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {884 a^{27/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{100947 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \]
[Out]
Time = 0.30 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2043, 2045, 2050, 2036, 335, 226} \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx=-\frac {884 a^{27/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{100947 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {1768 a^6 \sqrt {a x+b \sqrt [3]{x}}}{100947 b^5 x^{2/3}}+\frac {1768 a^5 \sqrt {a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac {1768 a^4 \sqrt {a x+b \sqrt [3]{x}}}{216315 b^3 x^2}+\frac {136 a^3 \sqrt {a x+b \sqrt [3]{x}}}{19665 b^2 x^{8/3}}-\frac {8 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1311 b x^{10/3}}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{9 x^5}-\frac {4 a \sqrt {a x+b \sqrt [3]{x}}}{69 x^4} \]
[In]
[Out]
Rule 226
Rule 335
Rule 2036
Rule 2043
Rule 2045
Rule 2050
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {\left (b x+a x^3\right )^{3/2}}{x^{16}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac {1}{3} (2 a) \text {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x^{13}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac {1}{69} \left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{x^{10} \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (68 a^3\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1311 b} \\ & = -\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac {\left (884 a^4\right ) \text {Subst}\left (\int \frac {1}{x^6 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{19665 b^2} \\ & = -\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (884 a^5\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{24035 b^3} \\ & = -\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}+\frac {\left (884 a^6\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{33649 b^4} \\ & = -\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {1768 a^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (884 a^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{100947 b^5} \\ & = -\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {1768 a^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (884 a^7 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{100947 b^5 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {1768 a^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {\left (1768 a^7 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{100947 b^5 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{69 x^4}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1311 b x^{10/3}}+\frac {136 a^3 \sqrt {b \sqrt [3]{x}+a x}}{19665 b^2 x^{8/3}}-\frac {1768 a^4 \sqrt {b \sqrt [3]{x}+a x}}{216315 b^3 x^2}+\frac {1768 a^5 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {1768 a^6 \sqrt {b \sqrt [3]{x}+a x}}{100947 b^5 x^{2/3}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{9 x^5}-\frac {884 a^{27/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{100947 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.21 \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx=-\frac {2 b \sqrt {b \sqrt [3]{x}+a x} \operatorname {Hypergeometric2F1}\left (-\frac {27}{4},-\frac {3}{2},-\frac {23}{4},-\frac {a x^{2/3}}{b}\right )}{9 \sqrt {1+\frac {a x^{2/3}}{b}} x^{14/3}} \]
[In]
[Out]
Time = 5.86 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.67
method | result | size |
default | \(-\frac {2 \left (6630 a^{6} \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) x^{\frac {26}{3}}-1768 x^{\frac {23}{3}} a^{5} b^{2}+5304 x^{\frac {25}{3}} a^{6} b +952 a^{4} b^{3} x^{7}+216755 x^{\frac {17}{3}} a^{2} b^{5}-616 x^{\frac {19}{3}} a^{3} b^{4}+380380 a \,b^{6} x^{5}+13260 a^{7} x^{9}+168245 x^{\frac {13}{3}} b^{7}\right )}{1514205 b^{5} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, x^{\frac {26}{3}}}\) | \(201\) |
derivativedivides | \(-\frac {2 b \sqrt {b \,x^{\frac {1}{3}}+a x}}{9 x^{\frac {14}{3}}}-\frac {58 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{207 x^{4}}-\frac {8 a^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1311 b \,x^{\frac {10}{3}}}+\frac {136 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{19665 b^{2} x^{\frac {8}{3}}}-\frac {1768 a^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{216315 b^{3} x^{2}}+\frac {1768 a^{5} \sqrt {b \,x^{\frac {1}{3}}+a x}}{168245 b^{4} x^{\frac {4}{3}}}-\frac {1768 a^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}{100947 b^{5} x^{\frac {2}{3}}}-\frac {884 a^{6} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{100947 b^{5} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(265\) |
[In]
[Out]
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (a\,x+b\,x^{1/3}\right )}^{3/2}}{x^6} \,d x \]
[In]
[Out]