Optimal. Leaf size=160 \[ -\frac {64 d^2 \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{35 c e}-\frac {24 d (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{35 c e}-\frac {2 (d+e x)^{5/2} \sqrt {c d^2-c e^2 x^2}}{7 c e}-\frac {256 d^3 \sqrt {c d^2-c e^2 x^2}}{35 c e \sqrt {d+e x}} \]
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Rubi [A] time = 0.08, 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 = {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 649
Rule 657
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*}
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Mathematica [A] time = 0.07, size = 70, normalized size = 0.44 \[ -\frac {2 (d-e x) \sqrt {d+e x} \left (177 d^3+71 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right )}{35 e \sqrt {c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 69, normalized size = 0.43 \[ -\frac {2 \, {\left (5 \, e^{3} x^{3} + 27 \, d e^{2} x^{2} + 71 \, d^{2} e x + 177 \, d^{3}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{35 \, {\left (c e^{2} x + c d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {-c e^{2} x^{2} + c d^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 66, normalized size = 0.41 \[ -\frac {2 \left (-e x +d \right ) \left (5 e^{3} x^{3}+27 e^{2} x^{2} d +71 d^{2} x e +177 d^{3}\right ) \sqrt {e x +d}}{35 \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 57, normalized size = 0.36 \[ \frac {2 \, {\left (5 \, e^{4} x^{4} + 22 \, d e^{3} x^{3} + 44 \, d^{2} e^{2} x^{2} + 106 \, d^{3} e x - 177 \, d^{4}\right )}}{35 \, \sqrt {-e x + d} \sqrt {c} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 98, normalized size = 0.61 \[ -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {354\,d^3\,\sqrt {d+e\,x}}{35\,c\,e^2}+\frac {54\,d\,x^2\,\sqrt {d+e\,x}}{35\,c}+\frac {2\,e\,x^3\,\sqrt {d+e\,x}}{7\,c}+\frac {142\,d^2\,x\,\sqrt {d+e\,x}}{35\,c\,e}\right )}{x+\frac {d}{e}} \]
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 \[ \int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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