Optimal. Leaf size=297 \[ -\frac {2 \left (x \left (-3 a^3 e^6-a^2 c d^2 e^4-7 a c^2 d^4 e^2+3 c^3 d^6\right )+a d e \left (c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{3/2} d^{3/2} e^{5/2}}-\frac {2 d x^2 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.29, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {849, 818, 777, 621, 206} \[ -\frac {2 \left (x \left (-a^2 c d^2 e^4-3 a^3 e^6-7 a c^2 d^4 e^2+3 c^3 d^6\right )+a d e \left (c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{3/2} d^{3/2} e^{5/2}}-\frac {2 d x^2 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 777
Rule 818
Rule 849
Rubi steps
\begin {align*} \int \frac {x^3}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\int \frac {x^3 (a e+c d x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \int \frac {x \left (2 a c d^2 e \left (c d^2-a e^2\right )+\frac {3}{2} c d \left (c d^2-a e^2\right )^2 x\right )}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \left (a d e \left (c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )+\left (3 c^3 d^6-7 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d e^2}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \left (a d e \left (c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )+\left (3 c^3 d^6-7 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \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 )}{c d e^2}\\ &=-\frac {2 d x^2 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \left (a d e \left (c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )+\left (3 c^3 d^6-7 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\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 )}{c^{3/2} d^{3/2} e^{5/2}}\\ \end {align*}
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Mathematica [C] time = 4.64, size = 1443, normalized size = 4.86 \[ \frac {a^3 e^3 (a e+c d x)^2 \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{5/2} \left (\frac {56 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^5}{a^3 e \left (c d^2-a e^2\right )^2}-\frac {280 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^4}{a^2 \left (c d^2-a e^2\right )^2}+\frac {96 \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^4}{a^3 e^2 \left (a e^2-c d^2\right )}+\frac {294 \sin ^{-1}\left (\sqrt {\frac {e (a e+c d x)}{a e^2-c d^2}}\right ) (a e+c d x)^3}{a^3 e^3 \left (\frac {e (a e+c d x)}{a e^2-c d^2}\right )^{5/2}}+\frac {392 e \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^3}{a \left (c d^2-a e^2\right )^2}-\frac {288 \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^3}{a^2 \left (a e^3-c d^2 e\right )}-\frac {1575 \sin ^{-1}\left (\sqrt {\frac {e (a e+c d x)}{a e^2-c d^2}}\right ) (a e+c d x)^2}{a^2 e^2 \left (\frac {e (a e+c d x)}{a e^2-c d^2}\right )^{5/2}}-\frac {168 e^2 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^2}{\left (c d^2-a e^2\right )^2}+\frac {288 \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)^2}{a \left (a e^2-c d^2\right )}+\frac {196 c d (d+e x) (a e+c d x)^2}{a^3 e^4 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}+\frac {3780 \sin ^{-1}\left (\sqrt {\frac {e (a e+c d x)}{a e^2-c d^2}}\right ) (a e+c d x)}{a e \left (\frac {e (a e+c d x)}{a e^2-c d^2}\right )^{5/2}}-\frac {96 e \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right ) (a e+c d x)}{a e^2-c d^2}-\frac {294 \left (c d^2-a e^2\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} (a e+c d x)}{a^3 e^5}-\frac {1050 c d (d+e x) (a e+c d x)}{a^2 e^3 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}-\frac {1995 \sin ^{-1}\left (\sqrt {\frac {e (a e+c d x)}{a e^2-c d^2}}\right )}{\left (\frac {e (a e+c d x)}{a e^2-c d^2}\right )^{5/2}}-56 \left (\frac {c d x}{a e}+1\right )^3 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+\frac {1575 \left (c d^2-a e^2\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}{a^2 e^4}+336 \left (\frac {c d x}{a e}+1\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}-504 \left (\frac {c d x}{a e}+1\right ) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+1568 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}+\frac {2520 c d (d+e x)}{a e^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}-\frac {3780 \left (c d^2-a e^2\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}{a e^3 (a e+c d x)}-\frac {1330 c d (d+e x)}{e \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} (a e+c d x)}+\frac {1995 \left (c d^2-a e^2\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}{e^2 (a e+c d x)^2}\right )}{252 c^4 d^4 ((a e+c d x) (d+e x))^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 5.33, size = 1466, normalized size = 4.94 \[ \text {result too large to display} \]
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 = 977, normalized size = 3.29 \[ -\frac {16 c^{2} d^{5} x}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e^{2}}-\frac {8 a c \,d^{4}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e}-\frac {8 c^{2} d^{6}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e^{3}}+\frac {a^{2} e^{2} x}{\left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c d}+\frac {4 a d x}{\left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}+\frac {3 c \,d^{3} x}{\left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, e^{2}}+\frac {a^{3} e^{3}}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c^{2} d^{2}}+\frac {5 a^{2} e}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c}+\frac {7 a \,d^{2}}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, e}+\frac {3 c \,d^{4}}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, e^{3}}+\frac {2 d^{3}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, e^{4}}+\frac {2 \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) d^{2}}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, e^{3}}-\frac {x}{\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c d \,e^{2}}+\frac {\ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{\sqrt {c d e}\, c d \,e^{2}}+\frac {a}{2 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c^{2} d^{2} e}+\frac {3}{2 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, c \,e^{3}} \]
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 \[ \int \frac {x^3}{\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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