Integrand size = 29, antiderivative size = 119 \[ \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {64 d^2 \sqrt {d+e x}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663} \[ \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}+\frac {64 d^2 \sqrt {d+e x}}{3 c e \sqrt {c d^2-c e^2 x^2}} \]
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Rule 663
Rule 671
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{3} (8 d) \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx \\ & = -\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{3} \left (32 d^2\right ) \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx \\ & = \frac {64 d^2 \sqrt {d+e x}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.46 \[ \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (-23 d^2+10 d e x+e^2 x^2\right )}{3 c e \sqrt {c \left (d^2-e^2 x^2\right )}} \]
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Time = 2.72 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(\frac {2 \left (-e x +d \right ) \left (-x^{2} e^{2}-10 d e x +23 d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e \left (-c \,x^{2} e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}\) | \(55\) |
default | \(\frac {2 \sqrt {c \left (-x^{2} e^{2}+d^{2}\right )}\, \left (-x^{2} e^{2}-10 d e x +23 d^{2}\right )}{3 \sqrt {e x +d}\, c^{2} \left (-e x +d \right ) e}\) | \(59\) |
risch | \(\frac {2 \left (e x +11 d \right ) \left (-e x +d \right ) \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{3 e \sqrt {-c \left (e x -d \right )}\, \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, c}+\frac {8 d^{2} \sqrt {-\frac {c \left (x^{2} e^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{e \sqrt {c \left (-e x +d \right )}\, \sqrt {-c \left (x^{2} e^{2}-d^{2}\right )}\, c}\) | \(156\) |
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Time = 0.41 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.55 \[ \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (e^{2} x^{2} + 10 \, d e x - 23 \, d^{2}\right )} \sqrt {e x + d}}{3 \, {\left (c^{2} e^{3} x^{2} - c^{2} d^{2} e\right )}} \]
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\[ \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.36 \[ \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\sqrt {c} e^{2} x^{2} + 10 \, \sqrt {c} d e x - 23 \, \sqrt {c} d^{2}\right )}}{3 \, \sqrt {-e x + d} c^{2} e} \]
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Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=-\frac {32 \, \sqrt {2} d^{2}}{3 \, \sqrt {c d} c e} + \frac {8 \, d^{2}}{\sqrt {-{\left (e x + d\right )} c + 2 \, c d} c e} + \frac {2 \, {\left (12 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d} c^{7} d e^{2} - {\left (-{\left (e x + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{6} e^{2}\right )}}{3 \, c^{9} e^{3}} \]
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Time = 9.87 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.72 \[ \int \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}}{3\,c^2\,e}-\frac {46\,d^2\,\sqrt {d+e\,x}}{3\,c^2\,e^3}+\frac {20\,d\,x\,\sqrt {d+e\,x}}{3\,c^2\,e^2}\right )}{x^2-\frac {d^2}{e^2}} \]
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