Optimal. Leaf size=386 \[ -\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac {\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}} \]
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Rubi [A]
time = 0.32, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {863, 848, 820,
734, 738, 212} \begin {gather*} \frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^5 \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 )}{1024 a^{7/2} d^{9/2} e^{7/2}}-\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 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}}{192 a^2 d^3 e^2 x^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 734
Rule 738
Rule 820
Rule 848
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^7 (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^7} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (5 c d^2-7 a e^2\right )+a c d e^2 x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6} \, dx}{6 a d e}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}-\frac {\left (\frac {5 c^2 d^2}{a}+2 c e^2-\frac {7 a e^4}{d^2}\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5} \, dx}{24 e}\\ &=\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+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}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{128 a^2 d^3 e^2}\\ &=-\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+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}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 a^3 d^4 e^3}\\ &=-\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+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}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right )\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 )}{512 a^3 d^4 e^3}\\ &=-\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+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}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac {\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\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 )}{1024 a^{7/2} d^{9/2} e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.97, size = 404, normalized size = 1.05 \begin {gather*} \frac {\left (-c d^2+a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (75 c^5 d^{10} x^5-5 a c^4 d^8 e x^4 (10 d+49 e x)+10 a^2 c^3 d^6 e^2 x^3 \left (4 d^2+16 d e x+15 e^2 x^2\right )+6 a^3 c^2 d^4 e^3 x^2 \left (360 d^3+564 d^2 e x+58 d e^2 x^2-91 e^3 x^3\right )+a^4 c d^2 e^4 x \left (3200 d^4+4448 d^3 e x+216 d^2 e^2 x^2-272 d e^3 x^3+415 e^4 x^4\right )+a^5 e^5 \left (1280 d^5+1664 d^4 e x+48 d^3 e^2 x^2-56 d^2 e^3 x^3+70 d e^4 x^4-105 e^5 x^5\right )\right )}{\left (c d^2-a e^2\right )^5 x^6 (a e+c d x) (d+e x)}-\frac {15 \left (5 c d^2+7 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{7680 a^{7/2} d^{9/2} e^{7/2}} \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. \(32290\) vs.
\(2(352)=704\).
time = 0.07, size = 32291, normalized size = 83.66
method | result | size |
default | \(\text {Expression too large to display}\) | \(32291\) |
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 = 85.94, size = 1115, normalized size = 2.89 \begin {gather*} \left [-\frac {{\left (15 \, {\left (5 \, c^{6} d^{12} x^{6} - 18 \, a c^{5} d^{10} x^{6} e^{2} + 15 \, a^{2} c^{4} d^{8} x^{6} e^{4} + 20 \, a^{3} c^{3} d^{6} x^{6} e^{6} - 45 \, a^{4} c^{2} d^{4} x^{6} e^{8} + 30 \, a^{5} c d^{2} x^{6} e^{10} - 7 \, a^{6} x^{6} e^{12}\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} - 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) + 4 \, {\left (75 \, a c^{5} d^{11} x^{5} e - 50 \, a^{2} c^{4} d^{10} x^{4} e^{2} - 105 \, a^{6} d x^{5} e^{11} + 70 \, a^{6} d^{2} x^{4} e^{10} + {\left (415 \, a^{5} c d^{3} x^{5} - 56 \, a^{6} d^{3} x^{3}\right )} e^{9} - 16 \, {\left (17 \, a^{5} c d^{4} x^{4} - 3 \, a^{6} d^{4} x^{2}\right )} e^{8} - 2 \, {\left (273 \, a^{4} c^{2} d^{5} x^{5} - 108 \, a^{5} c d^{5} x^{3} - 832 \, a^{6} d^{5} x\right )} e^{7} + 4 \, {\left (87 \, a^{4} c^{2} d^{6} x^{4} + 1112 \, a^{5} c d^{6} x^{2} + 320 \, a^{6} d^{6}\right )} e^{6} + 2 \, {\left (75 \, a^{3} c^{3} d^{7} x^{5} + 1692 \, a^{4} c^{2} d^{7} x^{3} + 1600 \, a^{5} c d^{7} x\right )} e^{5} + 80 \, {\left (2 \, a^{3} c^{3} d^{8} x^{4} + 27 \, a^{4} c^{2} d^{8} x^{2}\right )} e^{4} - 5 \, {\left (49 \, a^{2} c^{4} d^{9} x^{5} - 8 \, a^{3} c^{3} d^{9} x^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{30720 \, a^{4} d^{5} x^{6}}, -\frac {{\left (15 \, {\left (5 \, c^{6} d^{12} x^{6} - 18 \, a c^{5} d^{10} x^{6} e^{2} + 15 \, a^{2} c^{4} d^{8} x^{6} e^{4} + 20 \, a^{3} c^{3} d^{6} x^{6} e^{6} - 45 \, a^{4} c^{2} d^{4} x^{6} e^{8} + 30 \, a^{5} c d^{2} x^{6} e^{10} - 7 \, a^{6} x^{6} e^{12}\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (75 \, a c^{5} d^{11} x^{5} e - 50 \, a^{2} c^{4} d^{10} x^{4} e^{2} - 105 \, a^{6} d x^{5} e^{11} + 70 \, a^{6} d^{2} x^{4} e^{10} + {\left (415 \, a^{5} c d^{3} x^{5} - 56 \, a^{6} d^{3} x^{3}\right )} e^{9} - 16 \, {\left (17 \, a^{5} c d^{4} x^{4} - 3 \, a^{6} d^{4} x^{2}\right )} e^{8} - 2 \, {\left (273 \, a^{4} c^{2} d^{5} x^{5} - 108 \, a^{5} c d^{5} x^{3} - 832 \, a^{6} d^{5} x\right )} e^{7} + 4 \, {\left (87 \, a^{4} c^{2} d^{6} x^{4} + 1112 \, a^{5} c d^{6} x^{2} + 320 \, a^{6} d^{6}\right )} e^{6} + 2 \, {\left (75 \, a^{3} c^{3} d^{7} x^{5} + 1692 \, a^{4} c^{2} d^{7} x^{3} + 1600 \, a^{5} c d^{7} x\right )} e^{5} + 80 \, {\left (2 \, a^{3} c^{3} d^{8} x^{4} + 27 \, a^{4} c^{2} d^{8} x^{2}\right )} e^{4} - 5 \, {\left (49 \, a^{2} c^{4} d^{9} x^{5} - 8 \, a^{3} c^{3} d^{9} x^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{15360 \, a^{4} d^{5} x^{6}}\right ] \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. 3430 vs.
\(2 (347) = 694\).
time = 1.91, size = 3430, normalized size = 8.89 \begin {gather*} \text {Too large to display} \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^7\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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