Integrand size = 44, antiderivative size = 123 \[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {4 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.21 (sec) , antiderivative size = 728, normalized size of antiderivative = 5.92, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2081, 6847, 6857, 226, 2098, 1225, 1713, 211, 6860, 1231, 1721} \[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=-\frac {4 \sqrt {x} \sqrt {a^2 x^2+b^2} \arctan \left (\frac {\sqrt {-a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {-a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {2} \sqrt {x} \sqrt {a^2 x^2+b^2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {2 \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {2 \left (\sqrt {3} \sqrt {-a^2}+a\right ) \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \left (\sqrt {3} \sqrt {-a^2}+3 a\right ) \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {a} \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {3} \sqrt {-a^2} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {-a^2} \sqrt {x} (a x+b) \sqrt {\frac {a^2 x^2+b^2}{(a x+b)^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {3} a^{3/2} \sqrt {b} \sqrt {a^2 x^3+b^2 x}} \]
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Rule 211
Rule 226
Rule 1225
Rule 1231
Rule 1713
Rule 1721
Rule 2081
Rule 2098
Rule 6847
Rule 6857
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {-b^3+a^3 x^3}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (b^3+a^3 x^3\right )} \, dx}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-b^3+a^3 x^6}{\sqrt {b^2+a^2 x^4} \left (b^3+a^3 x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {b^2+a^2 x^4}}-\frac {2 b^3}{\sqrt {b^2+a^2 x^4} \left (b^3+a^3 x^6\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4} \left (b^3+a^3 x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b^3 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{3 b^2 \left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}+\frac {2 b-a x^2}{3 b^2 \sqrt {b^2+a^2 x^4} \left (b^2-a b x^2+a^2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {2 b-a x^2}{\sqrt {b^2+a^2 x^4} \left (b^2-a b x^2+a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {b-a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {-a-\sqrt {3} \sqrt {-a^2}}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}}+\frac {-a+\sqrt {3} \sqrt {-a^2}}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}} \\ & = \frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{b+2 a b^2 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}}-\frac {\left (4 \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b^2 x+a^2 x^3}} \\ & = -\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (-a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}+\frac {\left (4 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}}-\frac {\left (8 a \left (-a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b^2 x+a^2 x^3}} \\ & = -\frac {4 \sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {-a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {-a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}+\frac {2 \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \left (3 a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (a-\sqrt {3} \sqrt {-a^2}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticPi}\left (\frac {3}{4},2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \left (3 a+\sqrt {3} \sqrt {-a^2}\right ) \sqrt {b} \sqrt {b^2 x+a^2 x^3}} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.05 \[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=-\frac {\sqrt {x} \sqrt {b^2+a^2 x^2} \left (\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{3 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \]
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Time = 0.90 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.56
method | result | size |
default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-4 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{3 \sqrt {a b}}\) | \(69\) |
pseudoelliptic | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-4 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{3 \sqrt {a b}}\) | \(69\) |
elliptic | \(\text {Expression too large to display}\) | \(871\) |
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Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (95) = 190\).
Time = 0.35 (sec) , antiderivative size = 421, normalized size of antiderivative = 3.42 \[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=\left [-\frac {\sqrt {2} a b \sqrt {\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {\frac {1}{a b}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right ) - 2 \, \sqrt {a b} \log \left (\frac {a^{4} x^{4} + 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {a b}}{a^{4} x^{4} - 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4}}\right )}{6 \, a b}, \frac {\sqrt {2} a b \sqrt {-\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a b^{3} x + b^{4} + 4 \, \sqrt {2} {\left (a^{3} b x^{2} - 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {-\frac {1}{a b}}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + b^{4}}\right ) + 8 \, \sqrt {-a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {-a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right )}{12 \, a b}\right ] \]
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\[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right )}\, dx \]
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\[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=\int { \frac {a^{3} x^{3} - b^{3}}{{\left (a^{3} x^{3} + b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
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\[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=\int { \frac {a^{3} x^{3} - b^{3}}{{\left (a^{3} x^{3} + b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x}} \,d x } \]
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Timed out. \[ \int \frac {-b^3+a^3 x^3}{\sqrt {b^2 x+a^2 x^3} \left (b^3+a^3 x^3\right )} \, dx=\text {Hanged} \]
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