Optimal. Leaf size=160 \[ \frac {256 d^3 \sqrt {d+e x}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {64 d^2 (d+e x)^{3/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663}
\begin {gather*} -\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {64 d^2 (d+e x)^{3/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}+\frac {256 d^3 \sqrt {d+e x}}{5 c e \sqrt {c d^2-c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 671
Rubi steps
\begin {align*} \int \frac {(d+e x)^{9/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{5} (12 d) \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{5} \left (32 d^2\right ) \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {64 d^2 (d+e x)^{3/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{5} \left (128 d^3\right ) \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {256 d^3 \sqrt {d+e x}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {64 d^2 (d+e x)^{3/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {8 d (d+e x)^{5/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {c d^2-c e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 66, normalized size = 0.41 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (-91 d^3+43 d^2 e x+7 d e^2 x^2+e^3 x^3\right )}{5 c e \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 70, normalized size = 0.44
method | result | size |
gosper | \(\frac {2 \left (-e x +d \right ) \left (-e^{3} x^{3}-7 d \,e^{2} x^{2}-43 d^{2} e x +91 d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{5 e \left (-x^{2} c \,e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}\) | \(66\) |
default | \(\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (-e^{3} x^{3}-7 d \,e^{2} x^{2}-43 d^{2} e x +91 d^{3}\right )}{5 \sqrt {e x +d}\, c^{2} \left (-e x +d \right ) e}\) | \(70\) |
risch | \(\frac {2 \left (e^{2} x^{2}+8 d x e +51 d^{2}\right ) \left (-e x +d \right ) \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{5 e \sqrt {-c \left (e x -d \right )}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, c}+\frac {16 d^{3} \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{e \sqrt {c \left (-e x +d \right )}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, c}\) | \(167\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 44, normalized size = 0.28 \begin {gather*} -\frac {2 \, {\left (x^{3} e^{3} + 7 \, d x^{2} e^{2} + 43 \, d^{2} x e - 91 \, d^{3}\right )} e^{\left (-1\right )}}{5 \, \sqrt {-x e + d} c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.56, size = 76, normalized size = 0.48 \begin {gather*} \frac {2 \, {\left (x^{3} e^{3} + 7 \, d x^{2} e^{2} + 43 \, d^{2} x e - 91 \, d^{3}\right )} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d}}{5 \, {\left (c^{2} x^{2} e^{3} - c^{2} d^{2} e\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 (d + e x\right )^{\frac {9}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 140, normalized size = 0.88 \begin {gather*} -\frac {128 \, \sqrt {2} d^{3} e^{\left (-1\right )}}{5 \, \sqrt {c d} c} + \frac {16 \, d^{3} e^{\left (-1\right )}}{\sqrt {-{\left (x e + d\right )} c + 2 \, c d} c} + \frac {2 \, {\left (60 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{18} d^{2} e^{4} - 10 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{17} d e^{4} + {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{16} e^{4}\right )} e^{\left (-5\right )}}{5 \, c^{20}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.75, size = 104, normalized size = 0.65 \begin {gather*} \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x^3\,\sqrt {d+e\,x}}{5\,c^2}-\frac {182\,d^3\,\sqrt {d+e\,x}}{5\,c^2\,e^3}+\frac {14\,d\,x^2\,\sqrt {d+e\,x}}{5\,c^2\,e}+\frac {86\,d^2\,x\,\sqrt {d+e\,x}}{5\,c^2\,e^2}\right )}{x^2-\frac {d^2}{e^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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