Optimal. Leaf size=381 \[ \frac {\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}}-\frac {\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^4}+\frac {\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^3}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]
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Rubi [A] time = 0.39, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {794, 664, 612, 621, 206} \[ -\frac {\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^4}+\frac {\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^3}+\frac {\left (5 a e^2+7 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{9/2}}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 664
Rule 794
Rubi steps
\begin {align*} \int \frac {x \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 )^{7/2}}{6 c d e (d+e x)}+\frac {1}{12} \left (-\frac {7 d}{e}-\frac {5 a e}{c d}\right ) \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 {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (\left (\frac {7 d}{e}+\frac {5 a e}{c d}\right ) \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}{24 e}\\ &=\frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a 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}}{192 c^2 d^2 e^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right )\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 c^2 d^2 e^3}\\ &=-\frac {\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right ) \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}}{512 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a 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}}{192 c^2 d^2 e^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^3 d^3 e^4}\\ &=-\frac {\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right ) \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}}{512 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a 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}}{192 c^2 d^2 e^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\right )\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 )}{512 c^3 d^3 e^4}\\ &=-\frac {\left (c d^2-a e^2\right )^3 \left (7 c d^2+5 a e^2\right ) \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}}{512 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (7 c d^2+5 a 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}}{192 c^2 d^2 e^3}-\frac {1}{60} \left (\frac {5 a}{c d}+\frac {7 d}{e^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6 c d e (d+e x)}+\frac {\left (c d^2-a e^2\right )^5 \left (7 c d^2+5 a e^2\right ) \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 )}{1024 c^{7/2} d^{7/2} e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 2.74, size = 506, normalized size = 1.33 \[ \frac {(a e+c d x) ((d+e x) (a e+c d x))^{5/2} \left (7-\frac {7 \sqrt {c d} \sqrt {c d^2-a e^2} \left (5 a e^2+7 c d^2\right ) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{3/2} \left (15 \sqrt {e} \sqrt {c d} \left (c d^2-a e^2\right )^{11/2} (a e+c d x) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}-15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^6 \sqrt {a e+c d x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )-10 e^{3/2} \sqrt {c d} \left (c d^2-a e^2\right )^{9/2} (a e+c d x)^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+8 e^{5/2} \sqrt {c d} \left (c d^2-a e^2\right )^{7/2} (a e+c d x)^3 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+16 e^{7/2} \sqrt {c d} \left (c d^2-a e^2\right )^{3/2} (a e+c d x)^4 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \left (c d (11 d+8 e x)-3 a e^2\right )\right )}{1280 c^5 d^5 e^{7/2} (d+e x)^4 (a e+c d x)^4}\right )}{42 c d e} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 1046, normalized size = 2.75 \[ \left [-\frac {15 \, {\left (7 \, c^{6} d^{12} - 30 \, a c^{5} d^{10} e^{2} + 45 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} e^{8} + 18 \, a^{5} c d^{2} e^{10} - 5 \, a^{6} e^{12}\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 (1280 \, c^{6} d^{6} e^{6} x^{5} - 105 \, c^{6} d^{11} e + 415 \, a c^{5} d^{9} e^{3} - 546 \, a^{2} c^{4} d^{7} e^{5} + 150 \, a^{3} c^{3} d^{5} e^{7} - 245 \, a^{4} c^{2} d^{3} e^{9} + 75 \, a^{5} c d e^{11} + 128 \, {\left (13 \, c^{6} d^{7} e^{5} + 25 \, a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (3 \, c^{6} d^{8} e^{4} + 278 \, a c^{5} d^{6} e^{6} + 135 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} - 8 \, {\left (7 \, c^{6} d^{9} e^{3} - 27 \, a c^{5} d^{7} e^{5} - 423 \, a^{2} c^{4} d^{5} e^{7} - 5 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (35 \, c^{6} d^{10} e^{2} - 136 \, a c^{5} d^{8} e^{4} + 174 \, a^{2} c^{4} d^{6} e^{6} + 80 \, a^{3} c^{3} d^{4} e^{8} - 25 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{30720 \, c^{4} d^{4} e^{5}}, -\frac {15 \, {\left (7 \, c^{6} d^{12} - 30 \, a c^{5} d^{10} e^{2} + 45 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} e^{8} + 18 \, a^{5} c d^{2} e^{10} - 5 \, a^{6} e^{12}\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 (1280 \, c^{6} d^{6} e^{6} x^{5} - 105 \, c^{6} d^{11} e + 415 \, a c^{5} d^{9} e^{3} - 546 \, a^{2} c^{4} d^{7} e^{5} + 150 \, a^{3} c^{3} d^{5} e^{7} - 245 \, a^{4} c^{2} d^{3} e^{9} + 75 \, a^{5} c d e^{11} + 128 \, {\left (13 \, c^{6} d^{7} e^{5} + 25 \, a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (3 \, c^{6} d^{8} e^{4} + 278 \, a c^{5} d^{6} e^{6} + 135 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} - 8 \, {\left (7 \, c^{6} d^{9} e^{3} - 27 \, a c^{5} d^{7} e^{5} - 423 \, a^{2} c^{4} d^{5} e^{7} - 5 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (35 \, c^{6} d^{10} e^{2} - 136 \, a c^{5} d^{8} e^{4} + 174 \, a^{2} c^{4} d^{6} e^{6} + 80 \, a^{3} c^{3} d^{4} e^{8} - 25 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15360 \, c^{4} d^{4} e^{5}}\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 = 2411, normalized size = 6.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{d+e\,x} \,d x \]
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 \[ \int \frac {x \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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