Optimal. Leaf size=274 \[ -\frac {3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac {3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^2 d^2 e^3}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \]
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Rubi [A] time = 0.19, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {664, 612, 621, 206} \[ \frac {3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^2 d^2 e^3}-\frac {3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 664
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}}{d+e x} \, dx &=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (2 c d^2 e-e \left (c d^2+a e^2\right )\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{2 e^2}\\ &=\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}+\frac {\left (3 \left (c d^2-a e^2\right )^3\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 c d e^2}\\ &=\frac {3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (3 \left (c d^2-a e^2\right )^5\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^2 d^2 e^3}\\ &=\frac {3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (3 \left (c d^2-a e^2\right )^5\right ) \operatorname {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 )}{128 c^2 d^2 e^3}\\ &=\frac {3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {3 \left (c d^2-a e^2\right )^5 \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 )}{256 c^{5/2} d^{5/2} e^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 384, normalized size = 1.40 \[ \frac {\sqrt {c d} \left (\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d} (d+e x) \left (-15 a^5 e^9+5 a^4 c d e^7 (14 d-e x)+2 a^3 c^2 d^2 e^5 \left (64 d^2+268 d e x+129 e^2 x^2\right )+2 a^2 c^3 d^3 e^3 \left (-35 d^3+87 d^2 e x+489 d e^2 x^2+292 e^3 x^3\right )+a c^4 d^4 e \left (15 d^4-80 d^3 e x+54 d^2 e^2 x^2+688 d e^3 x^3+464 e^4 x^4\right )+c^5 d^5 x \left (15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right )\right )-15 \left (c d^2-a e^2\right )^{11/2} \sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )\right )}{640 c^{7/2} d^{7/2} e^{7/2} \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 844, normalized size = 3.08 \[ \left [\frac {15 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (128 \, c^{5} d^{5} e^{5} x^{4} + 15 \, c^{5} d^{9} e - 70 \, a c^{4} d^{7} e^{3} + 128 \, a^{2} c^{3} d^{5} e^{5} + 70 \, a^{3} c^{2} d^{3} e^{7} - 15 \, a^{4} c d e^{9} + 16 \, {\left (11 \, c^{5} d^{6} e^{4} + 21 \, a c^{4} d^{4} e^{6}\right )} x^{3} + 8 \, {\left (c^{5} d^{7} e^{3} + 64 \, a c^{4} d^{5} e^{5} + 31 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} - 2 \, {\left (5 \, c^{5} d^{8} e^{2} - 23 \, a c^{4} d^{6} e^{4} - 233 \, a^{2} c^{3} d^{4} e^{6} - 5 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2560 \, c^{3} d^{3} e^{4}}, \frac {15 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (128 \, c^{5} d^{5} e^{5} x^{4} + 15 \, c^{5} d^{9} e - 70 \, a c^{4} d^{7} e^{3} + 128 \, a^{2} c^{3} d^{5} e^{5} + 70 \, a^{3} c^{2} d^{3} e^{7} - 15 \, a^{4} c d e^{9} + 16 \, {\left (11 \, c^{5} d^{6} e^{4} + 21 \, a c^{4} d^{4} e^{6}\right )} x^{3} + 8 \, {\left (c^{5} d^{7} e^{3} + 64 \, a c^{4} d^{5} e^{5} + 31 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} - 2 \, {\left (5 \, c^{5} d^{8} e^{2} - 23 \, a c^{4} d^{6} e^{4} - 233 \, a^{2} c^{3} d^{4} e^{6} - 5 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{1280 \, c^{3} d^{3} e^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 1123, normalized size = 4.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 915, normalized size = 3.34 \[ -\frac {3 \, c^{4} d^{9} \log \left (2 \, c d x + \frac {c d^{2}}{e} + a e + 2 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} \sqrt {\frac {c d}{e}}\right )}{256 \, \left (\frac {c d}{e}\right )^{\frac {3}{2}} e^{5}} + \frac {15 \, a c^{3} d^{7} \log \left (2 \, c d x + \frac {c d^{2}}{e} + a e + 2 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} \sqrt {\frac {c d}{e}}\right )}{256 \, \left (\frac {c d}{e}\right )^{\frac {3}{2}} e^{3}} - \frac {15 \, a^{2} c^{2} d^{5} \log \left (2 \, c d x + \frac {c d^{2}}{e} + a e + 2 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} \sqrt {\frac {c d}{e}}\right )}{128 \, \left (\frac {c d}{e}\right )^{\frac {3}{2}} e} + \frac {15 \, a^{3} c d^{3} e \log \left (2 \, c d x + \frac {c d^{2}}{e} + a e + 2 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} \sqrt {\frac {c d}{e}}\right )}{128 \, \left (\frac {c d}{e}\right )^{\frac {3}{2}}} - \frac {15 \, a^{4} d e^{3} \log \left (2 \, c d x + \frac {c d^{2}}{e} + a e + 2 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} \sqrt {\frac {c d}{e}}\right )}{256 \, \left (\frac {c d}{e}\right )^{\frac {3}{2}}} + \frac {3 \, a^{5} e^{5} \log \left (2 \, c d x + \frac {c d^{2}}{e} + a e + 2 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} \sqrt {\frac {c d}{e}}\right )}{256 \, c d \left (\frac {c d}{e}\right )^{\frac {3}{2}}} - \frac {9}{64} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} a c d^{3} x + \frac {3 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} c^{2} d^{5} x}{64 \, e^{2}} + \frac {9}{64} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} a^{2} d e^{2} x - \frac {3 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} a^{3} e^{4} x}{64 \, c d} + \frac {3 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} c^{2} d^{6}}{128 \, e^{3}} - \frac {3 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} a c d^{4}}{64 \, e} + \frac {3 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} a^{3} e^{3}}{64 \, c} - \frac {3 \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} a^{4} e^{5}}{128 \, c^{2} d^{2}} - \frac {{\left (c d e x^{2} + c d^{2} x + a e^{2} x + a d e\right )}^{\frac {3}{2}} c d^{2} x}{8 \, e} + \frac {1}{8} \, {\left (c d e x^{2} + c d^{2} x + a e^{2} x + a d e\right )}^{\frac {3}{2}} a e x - \frac {{\left (c d e x^{2} + c d^{2} x + a e^{2} x + a d e\right )}^{\frac {3}{2}} c d^{3}}{16 \, e^{2}} + \frac {{\left (c d e x^{2} + c d^{2} x + a e^{2} x + a d e\right )}^{\frac {3}{2}} a^{2} e^{2}}{16 \, c d} + \frac {{\left (c d e x^{2} + c d^{2} x + a e^{2} x + a d e\right )}^{\frac {5}{2}}}{5 \, e} \]
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 \[ \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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