Integrand size = 20, antiderivative size = 149 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 e \sqrt {d+e x}}{c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}} \]
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Time = 0.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {718, 841, 1180, 214} \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{5/4}}-\frac {2 e \sqrt {d+e x}}{c} \]
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Rule 214
Rule 718
Rule 841
Rule 1180
Rubi steps \begin{align*} \text {integral}& = -\frac {2 e \sqrt {d+e x}}{c}-\frac {\int \frac {-c d^2-a e^2-2 c d e x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{c} \\ & = -\frac {2 e \sqrt {d+e x}}{c}-\frac {2 \text {Subst}\left (\int \frac {2 c d^2 e+e \left (-c d^2-a e^2\right )-2 c d e x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{c} \\ & = -\frac {2 e \sqrt {d+e x}}{c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \text {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {a} \sqrt {c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^2 \text {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {a} \sqrt {c}} \\ & = -\frac {2 e \sqrt {d+e x}}{c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.40 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\frac {-2 e \sqrt {d+e x}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{c} \]
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Time = 2.05 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(-2 e \left (\frac {\sqrt {e x +d}}{c}+\frac {\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(168\) |
pseudoelliptic | \(-e \left (\frac {2 \sqrt {e x +d}}{c}+\frac {\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(168\) |
default | \(2 e \left (-\frac {\sqrt {e x +d}}{c}+\frac {\left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-e^{2} a -c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(169\) |
risch | \(-\frac {2 e \sqrt {e x +d}}{c}-2 e \left (\frac {\left (-e^{2} a -c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 974 vs. \(2 (107) = 214\).
Time = 0.31 (sec) , antiderivative size = 974, normalized size of antiderivative = 6.54 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\frac {c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} - a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} - a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} + a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) + c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} + {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} + a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - c \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}} \log \left (-{\left (3 \, c^{2} d^{4} e - 2 \, a c d^{2} e^{3} - a^{2} e^{5}\right )} \sqrt {e x + d} - {\left (3 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4} + a c^{4} d \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}\right )} \sqrt {\frac {c d^{3} + 3 \, a d e^{2} - a c^{2} \sqrt {\frac {9 \, c^{2} d^{4} e^{2} + 6 \, a c d^{2} e^{4} + a^{2} e^{6}}{a c^{5}}}}{a c^{2}}}\right ) - 4 \, \sqrt {e x + d} e}{2 \, c} \]
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\[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=- \int \frac {d \sqrt {d + e x}}{- a + c x^{2}}\, dx - \int \frac {e x \sqrt {d + e x}}{- a + c x^{2}}\, dx \]
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\[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\int { -\frac {{\left (e x + d\right )}^{\frac {3}{2}}}{c x^{2} - a} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (107) = 214\).
Time = 0.32 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.11 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=-\frac {2 \, \sqrt {e x + d} e}{c} - \frac {{\left (\sqrt {a c} c^{3} d^{3} e - \sqrt {a c} a c^{2} d e^{3} + {\left (a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d + \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d - \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {a c} c^{3} d^{3} e - \sqrt {a c} a c^{2} d e^{3} - {\left (a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d - \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d + \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} \]
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Time = 9.63 (sec) , antiderivative size = 1581, normalized size of antiderivative = 10.61 \[ \int \frac {(d+e x)^{3/2}}{a-c x^2} \, dx=\text {Too large to display} \]
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