Optimal. Leaf size=269 \[ \frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {15 c^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {15 c^2 d^2 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {686, 680, 674,
211} \begin {gather*} -\frac {15 c^2 d^2 \sqrt {e} \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 \left (c d^2-a e^2\right )^{7/2}}-\frac {15 c^2 d^2 \sqrt {d+e x}}{4 \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 {5 c d}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 674
Rule 680
Rule 686
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(5 c d) \int \frac {1}{\sqrt {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}{4 \left (c d^2-a e^2\right )}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (15 c^2 d^2\right ) \int \frac {\sqrt {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}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {15 c^2 d^2 \sqrt {d+e x}}{4 \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 {\left (15 c^2 d^2 e\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 \left (c d^2-a e^2\right )^3}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {15 c^2 d^2 \sqrt {d+e x}}{4 \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 {\left (15 c^2 d^2 e^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 )^3}\\ &=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {15 c^2 d^2 \sqrt {d+e x}}{4 \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 {15 c^2 d^2 \sqrt {e} \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 \left (c d^2-a e^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 183, normalized size = 0.68 \begin {gather*} \frac {-\sqrt {c d^2-a e^2} \left (-2 a^2 e^4+a c d e^2 (9 d+5 e x)+c^2 d^2 \left (8 d^2+25 d e x+15 e^2 x^2\right )\right )-15 c^2 d^2 \sqrt {e} \sqrt {a e+c d x} (d+e x)^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2} (d+e x)^{3/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 374, normalized size = 1.39
method | result | size |
default | \(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{2} e^{3} x^{2}+30 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{3} e^{2} x +15 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{4} e -15 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{2} e^{2} x^{2}-5 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c d \,e^{3} x -25 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{3} e x +2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}-9 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}-8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\) | \(374\) |
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. 540 vs.
\(2 (238) = 476\).
time = 2.64, size = 1118, normalized size = 4.16 \begin {gather*} \left [\frac {15 \, {\left (c^{3} d^{6} x + a c^{2} d^{2} x^{3} e^{4} + {\left (c^{3} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2}\right )} e^{3} + 3 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{2} + {\left (3 \, c^{3} d^{5} x^{2} + a c^{2} d^{5}\right )} e\right )} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} - {\left (c d x^{2} + 2 \, a d\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - 2 \, {\left (25 \, c^{2} d^{3} x e + 8 \, c^{2} d^{4} + 5 \, a c d x e^{3} - 2 \, a^{2} e^{4} + 3 \, {\left (5 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\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^{4} d^{8} x^{3} e^{2} + c^{4} d^{10} x - a^{4} x^{3} e^{10} - 3 \, a^{4} d^{2} x e^{8} - {\left (a^{3} c d x^{4} + 3 \, a^{4} d x^{2}\right )} e^{9} + {\left (3 \, a^{2} c^{2} d^{3} x^{4} + 6 \, a^{3} c d^{3} x^{2} - a^{4} d^{3}\right )} e^{7} + 2 \, {\left (3 \, a^{2} c^{2} d^{4} x^{3} + 4 \, a^{3} c d^{4} x\right )} e^{6} - 3 \, {\left (a c^{3} d^{5} x^{4} - a^{3} c d^{5}\right )} e^{5} - 2 \, {\left (4 \, a c^{3} d^{6} x^{3} + 3 \, a^{2} c^{2} d^{6} x\right )} e^{4} + {\left (c^{4} d^{7} x^{4} - 6 \, a c^{3} d^{7} x^{2} - 3 \, a^{2} c^{2} d^{7}\right )} e^{3} + {\left (3 \, c^{4} d^{9} x^{2} + a c^{3} d^{9}\right )} e\right )}}, -\frac {\frac {15 \, {\left (c^{3} d^{6} x + a c^{2} d^{2} x^{3} e^{4} + {\left (c^{3} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2}\right )} e^{3} + 3 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{2} + {\left (3 \, c^{3} d^{5} x^{2} + a c^{2} d^{5}\right )} e\right )} \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} - a e^{2}} \sqrt {x e + d} e^{\frac {1}{2}}}{c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}}\right ) e^{\frac {1}{2}}}{\sqrt {c d^{2} - a e^{2}}} + {\left (25 \, c^{2} d^{3} x e + 8 \, c^{2} d^{4} + 5 \, a c d x e^{3} - 2 \, a^{2} e^{4} + 3 \, {\left (5 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\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^{4} d^{8} x^{3} e^{2} + c^{4} d^{10} x - a^{4} x^{3} e^{10} - 3 \, a^{4} d^{2} x e^{8} - {\left (a^{3} c d x^{4} + 3 \, a^{4} d x^{2}\right )} e^{9} + {\left (3 \, a^{2} c^{2} d^{3} x^{4} + 6 \, a^{3} c d^{3} x^{2} - a^{4} d^{3}\right )} e^{7} + 2 \, {\left (3 \, a^{2} c^{2} d^{4} x^{3} + 4 \, a^{3} c d^{4} x\right )} e^{6} - 3 \, {\left (a c^{3} d^{5} x^{4} - a^{3} c d^{5}\right )} e^{5} - 2 \, {\left (4 \, a c^{3} d^{6} x^{3} + 3 \, a^{2} c^{2} d^{6} x\right )} e^{4} + {\left (c^{4} d^{7} x^{4} - 6 \, a c^{3} d^{7} x^{2} - 3 \, a^{2} c^{2} d^{7}\right )} e^{3} + {\left (3 \, c^{4} d^{9} x^{2} + a c^{3} d^{9}\right )} e\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 {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \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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Giac [A]
time = 1.19, size = 350, normalized size = 1.30 \begin {gather*} -\frac {15 \, c^{2} d^{2} \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}{4 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {2 \, c^{2} d^{2} e}{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}} - \frac {{\left (9 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{4} e^{2} - 9 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e^{4} + 7 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e\right )} e^{\left (-2\right )}}{4 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (x e + d\right )}^{2} c^{2} d^{2}} \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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