Optimal. Leaf size=139 \[ \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}+\frac {c d \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {672, 660, 205} \begin {gather*} \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}+\frac {c d \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 660
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {(c d) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 \left (c d^2-a e^2\right )}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {(c d e) \operatorname {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{c d^2-a e^2}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {c d \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 135, normalized size = 0.97 \begin {gather*} \frac {\sqrt {e} \sqrt {c d^2-a e^2} (a e+c d x)+c d (d+e x) \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \sqrt {d+e x} \left (c d^2-a e^2\right )^{3/2} \sqrt {(d+e x) (a e+c d x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 5.17, size = 768, normalized size = 5.53 \begin {gather*} \frac {\sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}} \left (-a^3 e^7+3 a^2 c d^2 e^5-18 a^2 c d e^5 (d+e x)-3 a c^2 d^4 e^3+36 a c^2 d^3 e^3 (d+e x)-48 a c^2 d^2 e^3 (d+e x)^2+c^3 d^6 e-18 c^3 d^5 e (d+e x)+48 c^3 d^4 e (d+e x)^2-32 c^3 d^3 e (d+e x)^3\right )+\sqrt {c d e} \left (6 a^3 e^6 (d+e x)-18 a^2 c d^2 e^4 (d+e x)+38 a^2 c d e^4 (d+e x)^2+18 a c^2 d^4 e^2 (d+e x)-76 a c^2 d^3 e^2 (d+e x)^2+64 a c^2 d^2 e^2 (d+e x)^3-6 c^3 d^6 (d+e x)+38 c^3 d^5 (d+e x)^2-64 c^3 d^4 (d+e x)^3+32 c^3 d^3 (d+e x)^4\right )}{e \sqrt {c d e} (d+e x)^{3/2} \left (a e^2-c d^2\right ) \left (-6 a^2 e^4+12 a c d^2 e^2-32 a c d e^2 (d+e x)-6 c^2 d^4+32 c^2 d^3 (d+e x)-32 c^2 d^2 (d+e x)^2\right ) \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}}+e (d+e x)^{3/2} \left (a e^2-c d^2\right ) \left (a^3 e^6-3 a^2 c d^2 e^4+18 a^2 c d e^4 (d+e x)+3 a c^2 d^4 e^2-36 a c^2 d^3 e^2 (d+e x)+48 a c^2 d^2 e^2 (d+e x)^2-c^3 d^6+18 c^3 d^5 (d+e x)-48 c^3 d^4 (d+e x)^2+32 c^3 d^3 (d+e x)^3\right )}-\frac {2 c d \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {d+e x} \sqrt {c d^2-a e^2}}{e \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}}-\sqrt {c d e} (d+e x)}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 558, normalized size = 4.01 \begin {gather*} \left [\frac {{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} e - a e^{3}\right )} \sqrt {e x + d}}{2 \, {\left (c^{2} d^{6} e - 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5} + {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x\right )}}, -\frac {{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) - \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} e - a e^{3}\right )} \sqrt {e x + d}}{c^{2} d^{6} e - 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5} + {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 172, normalized size = 1.24 \begin {gather*} \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (c d e x \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+c \,d^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )-\sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\right )}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c d x +a e}\, \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}} \end {gather*}
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*} \int \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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