Optimal. Leaf size=119 \[ -\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e (d+e x)^{3/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e} \]
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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} \[ -\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e (d+e x)^{3/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e} \]
Antiderivative was successfully verified.
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Rule 649
Rule 657
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2} \, dx &=-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e}+\frac {1}{7} (8 d) \int \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2} \, dx\\ &=-\frac {16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e}+\frac {1}{35} \left (32 d^2\right ) \int \frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}} \, dx\\ &=-\frac {64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e (d+e x)^{3/2}}-\frac {16 d \left (c d^2-c e^2 x^2\right )^{3/2}}{35 c e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{7 c e}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 64, normalized size = 0.54 \[ \frac {2 \left (-71 d^3+17 d^2 e x+39 d e^2 x^2+15 e^3 x^3\right ) \sqrt {c \left (d^2-e^2 x^2\right )}}{105 e \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 67, normalized size = 0.56 \[ \frac {2 \, {\left (15 \, e^{3} x^{3} + 39 \, d e^{2} x^{2} + 17 \, d^{2} e x - 71 \, d^{3}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{105 \, {\left (e^{2} x + d e\right )}} \]
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 \[ \int \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (e x + d\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ -\frac {2 \left (-e x +d \right ) \left (15 e^{2} x^{2}+54 d e x +71 d^{2}\right ) \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{105 \sqrt {e x +d}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 68, normalized size = 0.57 \[ \frac {2 \, {\left (15 \, \sqrt {c} e^{3} x^{3} + 39 \, \sqrt {c} d e^{2} x^{2} + 17 \, \sqrt {c} d^{2} e x - 71 \, \sqrt {c} d^{3}\right )} {\left (e x + d\right )} \sqrt {-e x + d}}{105 \, {\left (e^{2} x + d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 85, normalized size = 0.71 \[ \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {26\,d\,x^2\,\sqrt {d+e\,x}}{35}-\frac {142\,d^3\,\sqrt {d+e\,x}}{105\,e^2}+\frac {2\,e\,x^3\,\sqrt {d+e\,x}}{7}+\frac {34\,d^2\,x\,\sqrt {d+e\,x}}{105\,e}\right )}{x+\frac {d}{e}} \]
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 \[ \int \sqrt {- c \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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