Optimal. Leaf size=160 \[ -\frac{256 d^3 \sqrt{c d^2-c e^2 x^2}}{35 c e \sqrt{d+e x}}-\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e} \]
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Rubi [A] time = 0.0800169, antiderivative size = 160, normalized size of antiderivative = 1., 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 = {657, 649} \[ -\frac{256 d^3 \sqrt{c d^2-c e^2 x^2}}{35 c e \sqrt{d+e x}}-\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e} \]
Antiderivative was successfully verified.
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Rule 657
Rule 649
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx &=-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}+\frac{1}{7} (12 d) \int \frac{(d+e x)^{5/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx\\ &=-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}+\frac{1}{35} \left (96 d^2\right ) \int \frac{(d+e x)^{3/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx\\ &=-\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}+\frac{1}{35} \left (128 d^3\right ) \int \frac{\sqrt{d+e x}}{\sqrt{c d^2-c e^2 x^2}} \, dx\\ &=-\frac{256 d^3 \sqrt{c d^2-c e^2 x^2}}{35 c e \sqrt{d+e x}}-\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}\\ \end{align*}
Mathematica [A] time = 0.0753387, size = 70, normalized size = 0.44 \[ -\frac{2 (d-e x) \sqrt{d+e x} \left (71 d^2 e x+177 d^3+27 d e^2 x^2+5 e^3 x^3\right )}{35 e \sqrt{c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 66, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 5\,{e}^{3}{x}^{3}+27\,d{e}^{2}{x}^{2}+71\,{d}^{2}xe+177\,{d}^{3} \right ) }{35\,e}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06837, size = 77, normalized size = 0.48 \begin{align*} \frac{2 \,{\left (5 \, e^{4} x^{4} + 22 \, d e^{3} x^{3} + 44 \, d^{2} e^{2} x^{2} + 106 \, d^{3} e x - 177 \, d^{4}\right )}}{35 \, \sqrt{-e x + d} \sqrt{c} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11261, size = 154, normalized size = 0.96 \begin{align*} -\frac{2 \,{\left (5 \, e^{3} x^{3} + 27 \, d e^{2} x^{2} + 71 \, d^{2} e x + 177 \, d^{3}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{35 \,{\left (c e^{2} x + c d e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{\frac{7}{2}}}{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{-c e^{2} x^{2} + c d^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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