Optimal. Leaf size=404 \[ -\frac {\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac {\left (6 a d e+\left (11 c d^2+3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 d x^4}+c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )+\frac {\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 a^{3/2} d^{5/2} e^{3/2}} \]
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Rubi [A]
time = 0.29, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {863, 824, 857,
635, 212, 738} \begin {gather*} -\frac {\left (x \left (-3 a^3 e^6+11 a^2 c d^2 e^4+83 a c^2 d^4 e^2+5 c^3 d^6\right )+2 a d e \left (5 c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 a d^2 e x^2}+\frac {\left (-3 a^4 e^8+20 a^3 c d^2 e^6-90 a^2 c^2 d^4 e^4-60 a c^3 d^6 e^2+5 c^4 d^8\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{3/2} d^{5/2} e^{3/2}}+c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac {\left (x \left (3 a e^2+11 c d^2\right )+6 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 d x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 824
Rule 857
Rule 863
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (d+e x)} \, dx &=\int \frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5} \, dx\\ &=-\frac {\left (6 a d e+\left (11 c d^2+3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 d x^4}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )-8 a c^2 d^3 e^2 x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{8 a d e}\\ &=-\frac {\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac {\left (6 a d e+\left (11 c d^2+3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 d x^4}+\frac {\int \frac {-\frac {1}{4} a e \left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right )+32 a^2 c^3 d^5 e^4 x}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 a^2 d^2 e^2}\\ &=-\frac {\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac {\left (6 a d e+\left (11 c d^2+3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 d x^4}+\left (c^3 d^3 e^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx-\frac {\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 a d^2 e}\\ &=-\frac {\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac {\left (6 a d e+\left (11 c d^2+3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 d x^4}+\left (2 c^3 d^3 e^2\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )+\frac {\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right ) \text {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 a d^2 e}\\ &=-\frac {\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac {\left (6 a d e+\left (11 c d^2+3 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 d x^4}+c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )+\frac {\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 a^{3/2} d^{5/2} e^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 1.11, size = 358, normalized size = 0.89 \begin {gather*} \frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (-\sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x} \left (15 c^3 d^6 x^3+a c^2 d^4 e x^2 (118 d+337 e x)+a^2 c d^2 e^2 x \left (136 d^2+244 d e x+57 e^2 x^2\right )+3 a^3 e^3 \left (16 d^3+24 d^2 e x+2 d e^2 x^2-3 e^3 x^3\right )\right )+3 \left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right ) x^4 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+384 a^{3/2} c^{5/2} d^5 e^3 x^4 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )\right )}{192 a^{3/2} d^{5/2} e^{3/2} x^4 \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(11684\) vs.
\(2(362)=724\).
time = 0.08, size = 11685, normalized size = 28.92
method | result | size |
default | \(\text {Expression too large to display}\) | \(11685\) |
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 [A]
time = 26.42, size = 1945, normalized size = 4.81 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 1783 vs.
\(2 (353) = 706\).
time = 1.60, size = 1783, normalized size = 4.41 \begin {gather*} -\sqrt {c d} c^{2} d^{2} e^{\frac {3}{2}} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right ) - \frac {{\left (5 \, c^{4} d^{8} - 60 \, a c^{3} d^{6} e^{2} - 90 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8}\right )} \arctan \left (-\frac {\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}}{\sqrt {-a d e}}\right ) e^{\left (-1\right )}}{64 \, \sqrt {-a d e} a d^{2}} + \frac {{\left (15 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{3} c^{4} d^{11} e^{3} - 55 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} a^{2} c^{4} d^{10} e^{2} + 73 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{5} a c^{4} d^{9} e + 15 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{7} c^{4} d^{8} + 384 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{6} \sqrt {c d} a c^{3} d^{7} e^{\frac {3}{2}} - 180 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{4} c^{3} d^{9} e^{5} + 660 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} a^{3} c^{3} d^{8} e^{4} + 276 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{5} a^{2} c^{3} d^{7} e^{3} + 588 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{7} a c^{3} d^{6} e^{2} - 512 \, \sqrt {c d} a^{5} c^{2} d^{8} e^{\frac {13}{2}} + 2048 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2} \sqrt {c d} a^{4} c^{2} d^{7} e^{\frac {11}{2}} - 1152 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{4} \sqrt {c d} a^{3} c^{2} d^{6} e^{\frac {9}{2}} + 2304 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{6} \sqrt {c d} a^{2} c^{2} d^{5} e^{\frac {7}{2}} + 114 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{5} c^{2} d^{7} e^{7} + 1374 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} a^{4} c^{2} d^{6} e^{6} + 990 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{5} a^{3} c^{2} d^{5} e^{5} + 882 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{7} a^{2} c^{2} d^{4} e^{4} + 768 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2} \sqrt {c d} a^{5} c d^{5} e^{\frac {15}{2}} + 768 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{4} \sqrt {c d} a^{4} c d^{4} e^{\frac {13}{2}} + 1152 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{6} \sqrt {c d} a^{3} c d^{3} e^{\frac {11}{2}} + 60 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{6} c d^{5} e^{9} + 548 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} a^{5} c d^{4} e^{8} + 676 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{5} a^{4} c d^{3} e^{7} + 60 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{7} a^{3} c d^{2} e^{6} + 384 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{4} \sqrt {c d} a^{5} d^{2} e^{\frac {17}{2}} - 9 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{7} d^{3} e^{11} + 33 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} a^{6} d^{2} e^{10} + 33 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{5} a^{5} d e^{9} - 9 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{7} a^{4} e^{8}\right )} e^{\left (-1\right )}}{192 \, {\left (a d e - {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2}\right )}^{4} a d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^5\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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