Optimal. Leaf size=119 \[ -\frac {64 d^2 \sqrt {c d^2-c e^2 x^2}}{15 c e \sqrt {d+e x}}-\frac {16 d \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{15 c e}-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e} \]
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Rubi [A] time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {657, 649} \[ -\frac {64 d^2 \sqrt {c d^2-c e^2 x^2}}{15 c e \sqrt {d+e x}}-\frac {16 d \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{15 c e}-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e} \]
Antiderivative was successfully verified.
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Rule 649
Rule 657
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx &=-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e}+\frac {1}{5} (8 d) \int \frac {(d+e x)^{3/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx\\ &=-\frac {16 d \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{15 c e}-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e}+\frac {1}{15} \left (32 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-c e^2 x^2}} \, dx\\ &=-\frac {64 d^2 \sqrt {c d^2-c e^2 x^2}}{15 c e \sqrt {d+e x}}-\frac {16 d \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}{15 c e}-\frac {2 (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}{5 c e}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 59, normalized size = 0.50 \[ -\frac {2 (d-e x) \sqrt {d+e x} \left (43 d^2+14 d e x+3 e^2 x^2\right )}{15 e \sqrt {c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 58, normalized size = 0.49 \[ -\frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (3 \, e^{2} x^{2} + 14 \, d e x + 43 \, d^{2}\right )} \sqrt {e x + d}}{15 \, {\left (c e^{2} x + c d e\right )}} \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{\sqrt {-c e^{2} x^{2} + c d^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 55, normalized size = 0.46 \[ -\frac {2 \left (-e x +d \right ) \left (3 e^{2} x^{2}+14 d x e +43 d^{2}\right ) \sqrt {e x +d}}{15 \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 58, normalized size = 0.49 \[ \frac {2 \, {\left (3 \, \sqrt {c} e^{3} x^{3} + 11 \, \sqrt {c} d e^{2} x^{2} + 29 \, \sqrt {c} d^{2} e x - 43 \, \sqrt {c} d^{3}\right )}}{15 \, \sqrt {-e x + d} c e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 79, normalized size = 0.66 \[ -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}}{5\,c}+\frac {86\,d^2\,\sqrt {d+e\,x}}{15\,c\,e^2}+\frac {28\,d\,x\,\sqrt {d+e\,x}}{15\,c\,e}\right )}{x+\frac {d}{e}} \]
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 \[ \int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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