Optimal. Leaf size=167 \[ -\frac {4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}} \]
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Rubi [A]
time = 0.27, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {718, 839, 841,
1180, 214} \begin {gather*} -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{\sqrt {a} c^{7/4}}-\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {4 d e \sqrt {d+e x}}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 718
Rule 839
Rule 841
Rule 1180
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{a-c x^2} \, dx &=-\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\int \frac {\sqrt {d+e x} \left (-c d^2-a e^2-2 c d e x\right )}{a-c x^2} \, dx}{c}\\ &=-\frac {4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {\int \frac {c d \left (c d^2+3 a e^2\right )+c e \left (3 c d^2+a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{c^2}\\ &=-\frac {4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}+\frac {2 \text {Subst}\left (\int \frac {-c d e \left (3 c d^2+a e^2\right )+c d e \left (c d^2+3 a e^2\right )+c e \left (3 c d^2+a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{c^2}\\ &=-\frac {4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^3 \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} c}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^3 \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} c}\\ &=-\frac {4 d e \sqrt {d+e x}}{c}-\frac {2 e (d+e x)^{3/2}}{3 c}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{7/4}}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 225, normalized size = 1.35 \begin {gather*} -\frac {2 c e \sqrt {d+e x} (7 d+e x)+\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e} \tan ^{-1}\left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a}}+\frac {3 \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^3 \tan ^{-1}\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}}}{3 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 228, normalized size = 1.37
method | result | size |
derivativedivides | \(-2 e \left (\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+2 d \sqrt {e x +d}}{c}-\frac {\left (3 a d \,e^{2} c +c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-3 a d \,e^{2} c -c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(228\) |
default | \(-2 e \left (\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+2 d \sqrt {e x +d}}{c}-\frac {\left (3 a d \,e^{2} c +c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-3 a d \,e^{2} c -c^{2} d^{3}+\sqrt {a c \,e^{2}}\, a \,e^{2}+3 \sqrt {a c \,e^{2}}\, c \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(228\) |
risch | \(-\frac {2 \left (e x +7 d \right ) \sqrt {e x +d}\, e}{3 c}+\frac {3 \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) a d \,e^{3}}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {e c \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) d^{3}}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) a \,e^{3}}{c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {3 e \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) d^{2}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {3 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) a d \,e^{3}}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {e c \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) d^{3}}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) a \,e^{3}}{c \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {3 e \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right ) d^{2}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(453\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1522 vs.
\(2 (131) = 262\).
time = 1.62, size = 1522, normalized size = 9.11 \begin {gather*} -\frac {3 \, c \sqrt {\frac {c^{2} d^{5} + 10 \, a c d^{3} e^{2} + a c^{3} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}} \log \left ({\left (5 \, c^{4} d^{8} e - 14 \, a^{2} c^{2} d^{4} e^{5} + 8 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \sqrt {x e + d} + {\left (10 \, a c^{4} d^{5} e^{2} + 20 \, a^{2} c^{3} d^{3} e^{4} + 2 \, a^{3} c^{2} d e^{6} - {\left (a c^{6} d^{2} + a^{2} c^{5} e^{2}\right )} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}}\right )} \sqrt {\frac {c^{2} d^{5} + 10 \, a c d^{3} e^{2} + a c^{3} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}}\right ) - 3 \, c \sqrt {\frac {c^{2} d^{5} + 10 \, a c d^{3} e^{2} + a c^{3} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}} \log \left ({\left (5 \, c^{4} d^{8} e - 14 \, a^{2} c^{2} d^{4} e^{5} + 8 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \sqrt {x e + d} - {\left (10 \, a c^{4} d^{5} e^{2} + 20 \, a^{2} c^{3} d^{3} e^{4} + 2 \, a^{3} c^{2} d e^{6} - {\left (a c^{6} d^{2} + a^{2} c^{5} e^{2}\right )} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}}\right )} \sqrt {\frac {c^{2} d^{5} + 10 \, a c d^{3} e^{2} + a c^{3} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}}\right ) + 3 \, c \sqrt {\frac {c^{2} d^{5} + 10 \, a c d^{3} e^{2} - a c^{3} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}} \log \left ({\left (5 \, c^{4} d^{8} e - 14 \, a^{2} c^{2} d^{4} e^{5} + 8 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \sqrt {x e + d} + {\left (10 \, a c^{4} d^{5} e^{2} + 20 \, a^{2} c^{3} d^{3} e^{4} + 2 \, a^{3} c^{2} d e^{6} + {\left (a c^{6} d^{2} + a^{2} c^{5} e^{2}\right )} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}}\right )} \sqrt {\frac {c^{2} d^{5} + 10 \, a c d^{3} e^{2} - a c^{3} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}}\right ) - 3 \, c \sqrt {\frac {c^{2} d^{5} + 10 \, a c d^{3} e^{2} - a c^{3} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}} \log \left ({\left (5 \, c^{4} d^{8} e - 14 \, a^{2} c^{2} d^{4} e^{5} + 8 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \sqrt {x e + d} - {\left (10 \, a c^{4} d^{5} e^{2} + 20 \, a^{2} c^{3} d^{3} e^{4} + 2 \, a^{3} c^{2} d e^{6} + {\left (a c^{6} d^{2} + a^{2} c^{5} e^{2}\right )} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}}\right )} \sqrt {\frac {c^{2} d^{5} + 10 \, a c d^{3} e^{2} - a c^{3} \sqrt {\frac {25 \, c^{4} d^{8} e^{2} + 100 \, a c^{3} d^{6} e^{4} + 110 \, a^{2} c^{2} d^{4} e^{6} + 20 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}}{a c^{7}}} + 5 \, a^{2} d e^{4}}{a c^{3}}}\right ) + 4 \, {\left (x e^{2} + 7 \, d e\right )} \sqrt {x e + d}}{6 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 498 vs.
\(2 (148) = 296\).
time = 43.62, size = 498, normalized size = 2.98 \begin {gather*} - \frac {4 a d e^{3} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left ( t \mapsto t \log {\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt {d + e x} \right )} \right )\right )}}{c} - \frac {2 a e^{3} \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log {\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )}}{c} + 4 d^{3} e \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} c e^{6} - 256 a^{2} c^{2} d^{2} e^{4}\right ) + 32 t^{2} a c d e^{2} - 1, \left ( t \mapsto t \log {\left (- 64 t^{3} a^{2} c d e^{4} + 64 t^{3} a c^{2} d^{3} e^{2} - 4 t a e^{2} - 4 t c d^{2} + \sqrt {d + e x} \right )} \right )\right )} - 4 d^{2} e \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log {\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} - 2 d^{2} e \operatorname {RootSum} {\left (256 t^{4} a^{2} c^{3} e^{4} - 32 t^{2} a c^{2} d e^{2} - a e^{2} + c d^{2}, \left ( t \mapsto t \log {\left (- 64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt {d + e x} \right )} \right )\right )} - \frac {4 d e \sqrt {d + e x}}{c} - \frac {2 e \left (d + e x\right )^{\frac {3}{2}}}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 402 vs.
\(2 (131) = 262\).
time = 2.54, size = 402, normalized size = 2.41 \begin {gather*} -\frac {{\left (\sqrt {a c} c^{4} d^{4} + 3 \, \sqrt {a c} a c^{3} d^{2} e^{2} - {\left (3 \, \sqrt {a c} a c d^{2} e^{2} + \sqrt {a c} a^{2} e^{4}\right )} c^{2} + 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{4} d + \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d - \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e}} + \frac {{\left (\sqrt {a c} c^{4} d^{4} + 3 \, \sqrt {a c} a c^{3} d^{2} e^{2} - {\left (3 \, \sqrt {a c} a c d^{2} e^{2} + \sqrt {a c} a^{2} e^{4}\right )} c^{2} - 2 \, {\left (a c^{3} d^{3} e - a^{2} c^{2} d e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{4} d - \sqrt {c^{8} d^{2} - {\left (c^{4} d^{2} - a c^{3} e^{2}\right )} c^{4}}}{c^{4}}}}\right )}{{\left (a c^{4} d + \sqrt {a c} a c^{3} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e}} - \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c^{2} e + 6 \, \sqrt {x e + d} c^{2} d e\right )}}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 2500, normalized size = 14.97 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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