Optimal. Leaf size=199 \[ -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 e (d+e x)^{5/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{4 e^{3/2} \left (c d^2-a e^2\right )^{3/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {676, 686, 674,
211} \begin {gather*} \frac {c^2 d^2 \text {ArcTan}\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 )}{4 e^{3/2} \left (c d^2-a e^2\right )^{3/2}}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 674
Rule 676
Rule 686
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{7/2}} \, dx &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 e (d+e x)^{5/2}}+\frac {(c d) \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}{4 e}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 e (d+e x)^{5/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {\left (c^2 d^2\right ) \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}{8 e \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}}{2 e (d+e x)^{5/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {\left (c^2 d^2\right ) \text {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 )}{4 \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}}{2 e (d+e x)^{5/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {c^2 d^2 \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 )}{4 e^{3/2} \left (c d^2-a e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 165, normalized size = 0.83 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {e} \sqrt {c d^2-a e^2} \sqrt {a e+c d x} \left (2 a e^2+c d (-d+e x)\right )+c^2 d^2 (d+e x)^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{4 e^{3/2} \left (c d^2-a e^2\right )^{3/2} \sqrt {a e+c d x} (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 282, normalized size = 1.42
method | result | size |
default | \(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (\arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{2} e^{2} x^{2}+2 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{3} e x +\arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{4}-c d e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-2 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,e^{2}+\sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c \,d^{2}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \sqrt {c d x +a e}\, \left (e^{2} a -c \,d^{2}\right ) e \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\) | \(282\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 360 vs.
\(2 (172) = 344\).
time = 2.82, size = 739, normalized size = 3.71 \begin {gather*} \left [-\frac {{\left (c^{2} d^{2} x^{3} e^{3} + 3 \, c^{2} d^{3} x^{2} e^{2} + 3 \, c^{2} d^{4} x e + c^{2} d^{5}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} - {\left (c d x^{2} + 2 \, a d\right )} e^{2} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-c d^{2} e + a e^{3}} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 2 \, {\left (c^{2} d^{3} x e^{2} - c^{2} d^{4} e - a c d x e^{4} + 3 \, a c d^{2} e^{3} - 2 \, a^{2} e^{5}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{8 \, {\left (3 \, c^{2} d^{6} x e^{3} + c^{2} d^{7} e^{2} + a^{2} x^{3} e^{9} + 3 \, a^{2} d x^{2} e^{8} - {\left (2 \, a c d^{2} x^{3} - 3 \, a^{2} d^{2} x\right )} e^{7} - {\left (6 \, a c d^{3} x^{2} - a^{2} d^{3}\right )} e^{6} + {\left (c^{2} d^{4} x^{3} - 6 \, a c d^{4} x\right )} e^{5} + {\left (3 \, c^{2} d^{5} x^{2} - 2 \, a c d^{5}\right )} e^{4}\right )}}, -\frac {{\left (c^{2} d^{2} x^{3} e^{3} + 3 \, c^{2} d^{3} x^{2} e^{2} + 3 \, c^{2} d^{4} x e + c^{2} d^{5}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {c d^{2} e - a e^{3}} \sqrt {x e + d}}{c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}}\right ) - {\left (c^{2} d^{3} x e^{2} - c^{2} d^{4} e - a c d x e^{4} + 3 \, a c d^{2} e^{3} - 2 \, a^{2} e^{5}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{4 \, {\left (3 \, c^{2} d^{6} x e^{3} + c^{2} d^{7} e^{2} + a^{2} x^{3} e^{9} + 3 \, a^{2} d x^{2} e^{8} - {\left (2 \, a c d^{2} x^{3} - 3 \, a^{2} d^{2} x\right )} e^{7} - {\left (6 \, a c d^{3} x^{2} - a^{2} d^{3}\right )} e^{6} + {\left (c^{2} d^{4} x^{3} - 6 \, a c d^{4} x\right )} e^{5} + {\left (3 \, c^{2} d^{5} x^{2} - 2 \, a c d^{5}\right )} e^{4}\right )}}\right ] \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 {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.75, size = 231, normalized size = 1.16 \begin {gather*} \frac {{\left (\frac {c^{3} d^{3} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{\sqrt {c d^{2} e - a e^{3}} {\left (c d^{2} - a e^{2}\right )}} - \frac {{\left (\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{5} e^{2} - \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{3} e^{4} - {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{3} e\right )} e^{\left (-2\right )}}{{\left (c d^{2} - a e^{2}\right )} {\left (x e + d\right )}^{2} c^{2} d^{2}}\right )} e^{\left (-2\right )}}{4 \, c d} \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 {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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