Optimal. Leaf size=181 \[ \frac {2 \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {2 \left (c d^2-a e^2\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{e^{5/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {678, 674, 211}
\begin {gather*} \frac {2 \left (c d^2-a e^2\right )^{3/2} \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 )}{e^{5/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 674
Rule 678
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {\left (2 c d^2 e-e \left (c d^2+a e^2\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{3/2}} \, dx}{e^2}\\ &=\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {\left (c d^2-a e^2\right )^2 \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}{e^2}\\ &=\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {\left (2 \left (c d^2-a e^2\right )^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 )}{e}\\ &=\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {2 \left (c d^2-a e^2\right )^{3/2} \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 )}{e^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 137, normalized size = 0.76 \begin {gather*} \frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {e} \sqrt {a e+c d x} \left (4 a e^2+c d (-3 d+e x)\right )+3 \left (c d^2-a e^2\right )^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{3 e^{5/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 265, normalized size = 1.46
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a^{2} e^{4}-6 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a c \,d^{2} e^{2}+3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{2} d^{4}-c d e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-4 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,e^{2}+3 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c \,d^{2}\right )}{3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, e^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\) | \(265\) |
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 [A]
time = 2.80, size = 382, normalized size = 2.11 \begin {gather*} \left [\frac {3 \, {\left (c d^{2} x e + c d^{3} - a x e^{3} - a d e^{2}\right )} \sqrt {-{\left (c d^{2} - a e^{2}\right )} e^{\left (-1\right )}} \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} \sqrt {x e + d} \sqrt {-{\left (c d^{2} - a e^{2}\right )} e^{\left (-1\right )}} e - {\left (c d x^{2} + 2 \, a d\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d x e - 3 \, c d^{2} + 4 \, a e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (x e^{3} + d e^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (c d^{2} x e + c d^{3} - a x e^{3} - a d e^{2}\right )} \sqrt {c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {c d^{2} - a e^{2}} \sqrt {x e + d} e^{\left (-\frac {1}{2}\right )}}{\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}\right ) e^{\left (-\frac {1}{2}\right )} - \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d x e - 3 \, c d^{2} + 4 \, a e^{2}\right )} \sqrt {x e + d}\right )}}{3 \, {\left (x e^{3} + d e^{2}\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 {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 401 vs.
\(2 (160) = 320\).
time = 0.96, size = 401, normalized size = 2.22 \begin {gather*} -\frac {2}{3} \, {\left ({\left (3 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c d^{2} e^{5} - 3 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a e^{7} - {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{4}\right )} e^{\left (-6\right )} - \frac {3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \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 )}{\sqrt {c d^{2} e - a e^{3}}}\right )} e^{\left (-2\right )} - \frac {2 \, {\left (3 \, c^{2} d^{4} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e - 6 \, a c d^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e^{3} - 4 \, \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} c d^{2} + 3 \, a^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e^{5} + 4 \, \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} a e^{2}\right )} e^{\left (-3\right )}}{3 \, \sqrt {c d^{2} e - a e^{3}}} \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 {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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