Integrand size = 17, antiderivative size = 161 \[ \int \frac {x^7}{\left (a x+b x^3\right )^{3/2}} \, dx=-\frac {x^5}{b \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}+\frac {15 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{14 b^{13/4} \sqrt {a x+b x^3}} \]
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Time = 0.14 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2047, 2049, 2036, 335, 226} \[ \int \frac {x^7}{\left (a x+b x^3\right )^{3/2}} \, dx=\frac {15 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{14 b^{13/4} \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}-\frac {x^5}{b \sqrt {a x+b x^3}} \]
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Rule 226
Rule 335
Rule 2036
Rule 2047
Rule 2049
Rubi steps \begin{align*} \text {integral}& = -\frac {x^5}{b \sqrt {a x+b x^3}}+\frac {9 \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx}{2 b} \\ & = -\frac {x^5}{b \sqrt {a x+b x^3}}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}-\frac {(45 a) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx}{14 b^2} \\ & = -\frac {x^5}{b \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}+\frac {\left (15 a^2\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{14 b^3} \\ & = -\frac {x^5}{b \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}+\frac {\left (15 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{14 b^3 \sqrt {a x+b x^3}} \\ & = -\frac {x^5}{b \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}+\frac {\left (15 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{7 b^3 \sqrt {a x+b x^3}} \\ & = -\frac {x^5}{b \sqrt {a x+b x^3}}-\frac {15 a \sqrt {a x+b x^3}}{7 b^3}+\frac {9 x^2 \sqrt {a x+b x^3}}{7 b^2}+\frac {15 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{14 b^{13/4} \sqrt {a x+b x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.50 \[ \int \frac {x^7}{\left (a x+b x^3\right )^{3/2}} \, dx=\frac {x \left (-15 a^2-6 a b x^2+2 b^2 x^4+15 a^2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{7 b^3 \sqrt {x \left (a+b x^2\right )}} \]
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Time = 2.71 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.07
method | result | size |
default | \(-\frac {x \,a^{2}}{b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}+\frac {2 x^{2} \sqrt {b \,x^{3}+a x}}{7 b^{2}}-\frac {8 a \sqrt {b \,x^{3}+a x}}{7 b^{3}}+\frac {15 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{14 b^{4} \sqrt {b \,x^{3}+a x}}\) | \(172\) |
elliptic | \(-\frac {x \,a^{2}}{b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}+\frac {2 x^{2} \sqrt {b \,x^{3}+a x}}{7 b^{2}}-\frac {8 a \sqrt {b \,x^{3}+a x}}{7 b^{3}}+\frac {15 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{14 b^{4} \sqrt {b \,x^{3}+a x}}\) | \(172\) |
risch | \(-\frac {2 \left (-b \,x^{2}+4 a \right ) \left (b \,x^{2}+a \right ) x}{7 b^{3} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {a^{2} \left (\frac {11 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b \,x^{3}+a x}}-7 a \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b \,x^{3}+a x}}\right )\right )}{7 b^{3}}\) | \(287\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.50 \[ \int \frac {x^7}{\left (a x+b x^3\right )^{3/2}} \, dx=\frac {15 \, {\left (a^{2} b x^{2} + a^{3}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (2 \, b^{3} x^{4} - 6 \, a b^{2} x^{2} - 15 \, a^{2} b\right )} \sqrt {b x^{3} + a x}}{7 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \]
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\[ \int \frac {x^7}{\left (a x+b x^3\right )^{3/2}} \, dx=\int \frac {x^{7}}{\left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^7}{\left (a x+b x^3\right )^{3/2}} \, dx=\int { \frac {x^{7}}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^7}{\left (a x+b x^3\right )^{3/2}} \, dx=\int { \frac {x^{7}}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^7}{\left (a x+b x^3\right )^{3/2}} \, dx=\int \frac {x^7}{{\left (b\,x^3+a\,x\right )}^{3/2}} \,d x \]
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