Integrand size = 21, antiderivative size = 555 \[ \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^5}} \, dx=-\frac {8 \left (1+\sqrt {3}\right ) b^{4/3} x^{3/2} \left (a+b x^3\right )}{7 a^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {a x^2+b x^5}}-\frac {2 \sqrt {a x^2+b x^5}}{7 a x^{9/2}}+\frac {8 b \sqrt {a x^2+b x^5}}{7 a^2 x^{3/2}}+\frac {8 \sqrt [4]{3} b^{4/3} x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}}+\frac {4 \left (1-\sqrt {3}\right ) b^{4/3} x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 \sqrt [4]{3} a^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}} \]
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Time = 0.41 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2050, 2057, 335, 314, 231, 1895} \[ \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^5}} \, dx=\frac {4 \left (1-\sqrt {3}\right ) b^{4/3} x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 \sqrt [4]{3} a^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}}+\frac {8 \sqrt [4]{3} b^{4/3} x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}}-\frac {8 \left (1+\sqrt {3}\right ) b^{4/3} x^{3/2} \left (a+b x^3\right )}{7 a^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {a x^2+b x^5}}+\frac {8 b \sqrt {a x^2+b x^5}}{7 a^2 x^{3/2}}-\frac {2 \sqrt {a x^2+b x^5}}{7 a x^{9/2}} \]
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Rule 231
Rule 314
Rule 335
Rule 1895
Rule 2050
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a x^2+b x^5}}{7 a x^{9/2}}-\frac {(4 b) \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^5}} \, dx}{7 a} \\ & = -\frac {2 \sqrt {a x^2+b x^5}}{7 a x^{9/2}}+\frac {8 b \sqrt {a x^2+b x^5}}{7 a^2 x^{3/2}}-\frac {\left (8 b^2\right ) \int \frac {x^{5/2}}{\sqrt {a x^2+b x^5}} \, dx}{7 a^2} \\ & = -\frac {2 \sqrt {a x^2+b x^5}}{7 a x^{9/2}}+\frac {8 b \sqrt {a x^2+b x^5}}{7 a^2 x^{3/2}}-\frac {\left (8 b^2 x \sqrt {a+b x^3}\right ) \int \frac {x^{3/2}}{\sqrt {a+b x^3}} \, dx}{7 a^2 \sqrt {a x^2+b x^5}} \\ & = -\frac {2 \sqrt {a x^2+b x^5}}{7 a x^{9/2}}+\frac {8 b \sqrt {a x^2+b x^5}}{7 a^2 x^{3/2}}-\frac {\left (16 b^2 x \sqrt {a+b x^3}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^6}} \, dx,x,\sqrt {x}\right )}{7 a^2 \sqrt {a x^2+b x^5}} \\ & = -\frac {2 \sqrt {a x^2+b x^5}}{7 a x^{9/2}}+\frac {8 b \sqrt {a x^2+b x^5}}{7 a^2 x^{3/2}}+\frac {\left (8 b^{4/3} x \sqrt {a+b x^3}\right ) \text {Subst}\left (\int \frac {\left (-1+\sqrt {3}\right ) a^{2/3}-2 b^{2/3} x^4}{\sqrt {a+b x^6}} \, dx,x,\sqrt {x}\right )}{7 a^2 \sqrt {a x^2+b x^5}}+\frac {\left (8 \left (1-\sqrt {3}\right ) b^{4/3} x \sqrt {a+b x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^6}} \, dx,x,\sqrt {x}\right )}{7 a^{4/3} \sqrt {a x^2+b x^5}} \\ & = -\frac {8 \left (1+\sqrt {3}\right ) b^{4/3} x^{3/2} \left (a+b x^3\right )}{7 a^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {a x^2+b x^5}}-\frac {2 \sqrt {a x^2+b x^5}}{7 a x^{9/2}}+\frac {8 b \sqrt {a x^2+b x^5}}{7 a^2 x^{3/2}}+\frac {8 \sqrt [4]{3} b^{4/3} x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}}+\frac {4 \left (1-\sqrt {3}\right ) b^{4/3} x^{3/2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 \sqrt [4]{3} a^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a x^2+b x^5}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^5}} \, dx=-\frac {2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {7}{6},\frac {1}{2},-\frac {1}{6},-\frac {b x^3}{a}\right )}{7 x^{5/2} \sqrt {x^2 \left (a+b x^3\right )}} \]
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Result contains complex when optimal does not.
Time = 2.28 (sec) , antiderivative size = 1125, normalized size of antiderivative = 2.03
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1125\) |
default | \(\text {Expression too large to display}\) | \(3048\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^5}} \, dx=-\frac {2 \, {\left (4 \, \sqrt {a} b x^{5} {\rm weierstrassZeta}\left (0, -\frac {4 \, b}{a}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right )\right ) + \sqrt {b x^{5} + a x^{2}} a \sqrt {x}\right )}}{7 \, a^{2} x^{5}} \]
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\[ \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^5}} \, dx=\int \frac {1}{x^{\frac {7}{2}} \sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \]
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\[ \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^5}} \, dx=\int { \frac {1}{\sqrt {b x^{5} + a x^{2}} x^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^5}} \, dx=\int { \frac {1}{\sqrt {b x^{5} + a x^{2}} x^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^5}} \, dx=\int \frac {1}{x^{7/2}\,\sqrt {b\,x^5+a\,x^2}} \,d x \]
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