Optimal. Leaf size=394 \[ \frac {\left (5 a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{5/2} d^{7/2} e^{5/2}}+\frac {2 \left (-5 a^3 e^6+c d e x \left (-5 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+9 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (-15 a^3 e^6+31 a^2 c d^2 e^4-9 a c^2 d^4 e^2+9 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x \left (c d^2-a e^2\right )^3}-\frac {2 e (a e+c d x)}{3 d x \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.59, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {851, 822, 806, 724, 206} \[ -\frac {\left (31 a^2 c d^2 e^4-15 a^3 e^6-9 a c^2 d^4 e^2+9 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x \left (c d^2-a e^2\right )^3}+\frac {2 \left (c d e x \left (-5 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+9 a^2 c d^2 e^4-5 a^3 e^6+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\left (5 a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{5/2} d^{7/2} e^{5/2}}-\frac {2 e (a e+c d x)}{3 d x \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 822
Rule 851
Rubi steps
\begin {align*} \int \frac {1}{x^2 (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 {a e+c d x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \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 {-\frac {1}{2} a e \left (3 c d^2-5 a e^2\right ) \left (c d^2-a e^2\right )+3 a c d e^2 \left (c d^2-a e^2\right ) x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 a d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 \int \frac {\frac {1}{4} a e \left (c d^2-a e^2\right ) \left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right )+\frac {1}{2} a c d e^2 \left (c d^2-a e^2\right ) \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a^2 d^2 e^2 \left (c d^2-a e^2\right )^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}-\frac {\left (3 c d^2+5 a e^2\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 a^2 d^3 e^2}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}+\frac {\left (3 c d^2+5 a e^2\right ) \operatorname {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 )}{a^2 d^3 e^2}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}+\frac {\left (3 c d^2+5 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 )}{2 a^{5/2} d^{7/2} e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 370, normalized size = 0.94 \[ \frac {(a e+c d x) \left (3 a^{3/2} d^{5/2} e^{3/2} \left (a e^2-c d^2\right )^3+\sqrt {a} d^{3/2} \sqrt {e} x \left (a e^2-c d^2\right ) \left (5 a^2 e^5-6 a c d^2 e^3+9 c^2 d^4 e\right ) (a e+c d x)+x (d+e x) \sqrt {a e+c d x} \left (\sqrt {a} \sqrt {d} \sqrt {e} \left (15 a^3 e^7-31 a^2 c d^2 e^5+9 a c^2 d^4 e^3-9 c^3 d^6 e\right ) \sqrt {a e+c d x}+3 \sqrt {d+e x} \left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )\right )+3 \sqrt {a} c d^{7/2} \sqrt {e} x \left (c d^2-a e^2\right )^2 \left (a e^2-3 c d^2\right )\right )}{3 a^{5/2} d^{7/2} e^{5/2} x \left (c d^2-a e^2\right )^3 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 30.79, size = 1812, normalized size = 4.60 \[ \text {result too large to display} \]
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 = 0.02, size = 912, normalized size = 2.31 \[ \frac {16 c^{2} e^{3} 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 )}}+\frac {8 a c \,e^{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 )}\, d}+\frac {8 c^{2} d \,e^{2}}{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 )}}+\frac {3 c^{3} d^{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}\, a^{2} e}+\frac {5 c \,e^{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}\, d^{2}}+\frac {5 a \,e^{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}\, d^{3}}+\frac {3 c^{2} d}{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}\, a}+\frac {3 c^{3} d^{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}\, a^{2} e^{2}}+\frac {5 c \,e^{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}\, d}-\frac {2 e}{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 )}\, d^{2}}+\frac {5 \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{x}\right )}{2 \sqrt {a d e}\, a \,d^{3}}+\frac {3 c \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{x}\right )}{2 \sqrt {a d e}\, a^{2} d \,e^{2}}-\frac {5}{2 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a \,d^{3}}-\frac {3 c}{2 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{2} d \,e^{2}}-\frac {1}{\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a \,d^{2} e x} \]
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 \[ \int \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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