Optimal. Leaf size=176 \[ \frac {15 e^2}{4 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}-\frac {15 \sqrt {c} \sqrt {d} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 44, 53, 65,
214} \begin {gather*} \frac {15 e^2}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^3}+\frac {5 e}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac {15 \sqrt {c} \sqrt {d} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 214
Rule 640
Rubi steps
\begin {align*} \int \frac {(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} \, dx &=\int \frac {1}{(a e+c d x)^3 (d+e x)^{3/2}} \, dx\\ &=-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}-\frac {(5 e) \int \frac {1}{(a e+c d x)^2 (d+e x)^{3/2}} \, dx}{4 \left (c d^2-a e^2\right )}\\ &=-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}+\frac {\left (15 e^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac {15 e^2}{4 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}+\frac {\left (15 c d e^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{8 \left (c d^2-a e^2\right )^3}\\ &=\frac {15 e^2}{4 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}+\frac {(15 c d e) \text {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 \left (c d^2-a e^2\right )^3}\\ &=\frac {15 e^2}{4 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}-\frac {15 \sqrt {c} \sqrt {d} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 157, normalized size = 0.89 \begin {gather*} \frac {1}{4} \left (\frac {8 a^2 e^4+a c d e^2 (9 d+25 e x)+c^2 d^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2 \sqrt {d+e x}}-\frac {15 \sqrt {c} \sqrt {d} e^2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\left (-c d^2+a e^2\right )^{7/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 152, normalized size = 0.86
method | result | size |
derivativedivides | \(2 e^{2} \left (-\frac {c d \left (\frac {\frac {7 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9 e^{2} a}{8}-\frac {9 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {15 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}-\frac {1}{\left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {e x +d}}\right )\) | \(152\) |
default | \(2 e^{2} \left (-\frac {c d \left (\frac {\frac {7 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9 e^{2} a}{8}-\frac {9 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {15 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{3}}-\frac {1}{\left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {e x +d}}\right )\) | \(152\) |
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 \begin {gather*} \text {Exception raised: ValueError} \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. 439 vs.
\(2 (148) = 296\).
time = 3.03, size = 893, normalized size = 5.07 \begin {gather*} \left [\frac {15 \, {\left (c^{2} d^{3} x^{2} e^{2} + a^{2} x e^{5} + {\left (2 \, a c d x^{2} + a^{2} d\right )} e^{4} + {\left (c^{2} d^{2} x^{3} + 2 \, a c d^{2} x\right )} e^{3}\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d x e + 2 \, c d^{2} - 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} - a e^{2}}{c d x + a e}\right ) + 2 \, {\left (5 \, c^{2} d^{3} x e - 2 \, c^{2} d^{4} + 25 \, a c d x e^{3} + 8 \, a^{2} e^{4} + 3 \, {\left (5 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{8 \, {\left (c^{5} d^{9} x^{2} - a^{5} x e^{9} - {\left (2 \, a^{4} c d x^{2} + a^{5} d\right )} e^{8} - {\left (a^{3} c^{2} d^{2} x^{3} - a^{4} c d^{2} x\right )} e^{7} + {\left (5 \, a^{3} c^{2} d^{3} x^{2} + 3 \, a^{4} c d^{3}\right )} e^{6} + 3 \, {\left (a^{2} c^{3} d^{4} x^{3} + a^{3} c^{2} d^{4} x\right )} e^{5} - 3 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5}\right )} e^{4} - {\left (3 \, a c^{4} d^{6} x^{3} + 5 \, a^{2} c^{3} d^{6} x\right )} e^{3} - {\left (a c^{4} d^{7} x^{2} - a^{2} c^{3} d^{7}\right )} e^{2} + {\left (c^{5} d^{8} x^{3} + 2 \, a c^{4} d^{8} x\right )} e\right )}}, -\frac {15 \, {\left (c^{2} d^{3} x^{2} e^{2} + a^{2} x e^{5} + {\left (2 \, a c d x^{2} + a^{2} d\right )} e^{4} + {\left (c^{2} d^{2} x^{3} + 2 \, a c d^{2} x\right )} e^{3}\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d x e + c d^{2}}\right ) - {\left (5 \, c^{2} d^{3} x e - 2 \, c^{2} d^{4} + 25 \, a c d x e^{3} + 8 \, a^{2} e^{4} + 3 \, {\left (5 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{4 \, {\left (c^{5} d^{9} x^{2} - a^{5} x e^{9} - {\left (2 \, a^{4} c d x^{2} + a^{5} d\right )} e^{8} - {\left (a^{3} c^{2} d^{2} x^{3} - a^{4} c d^{2} x\right )} e^{7} + {\left (5 \, a^{3} c^{2} d^{3} x^{2} + 3 \, a^{4} c d^{3}\right )} e^{6} + 3 \, {\left (a^{2} c^{3} d^{4} x^{3} + a^{3} c^{2} d^{4} x\right )} e^{5} - 3 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5}\right )} e^{4} - {\left (3 \, a c^{4} d^{6} x^{3} + 5 \, a^{2} c^{3} d^{6} x\right )} e^{3} - {\left (a c^{4} d^{7} x^{2} - a^{2} c^{3} d^{7}\right )} e^{2} + {\left (c^{5} d^{8} x^{3} + 2 \, a c^{4} d^{8} x\right )} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 258, normalized size = 1.47 \begin {gather*} \frac {15 \, c d \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right ) e^{2}}{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^{2} d^{3} + a c d e^{2}}} + \frac {2 \, e^{2}}{{\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 {x e + d}} + \frac {7 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} - 9 \, \sqrt {x e + d} c^{2} d^{3} e^{2} + 9 \, \sqrt {x e + d} a c d e^{4}}{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 ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.80, size = 251, normalized size = 1.43 \begin {gather*} -\frac {\frac {2\,e^2}{a\,e^2-c\,d^2}+\frac {25\,c\,d\,e^2\,\left (d+e\,x\right )}{4\,{\left (a\,e^2-c\,d^2\right )}^2}+\frac {15\,c^2\,d^2\,e^2\,{\left (d+e\,x\right )}^2}{4\,{\left (a\,e^2-c\,d^2\right )}^3}}{\sqrt {d+e\,x}\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )-\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}+c^2\,d^2\,{\left (d+e\,x\right )}^{5/2}}-\frac {15\,\sqrt {c}\,\sqrt {d}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^{7/2}}\right )}{4\,{\left (a\,e^2-c\,d^2\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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