Optimal. Leaf size=37 \[ \frac {c (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {642, 609} \[ \frac {c (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
Antiderivative was successfully verified.
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Rule 609
Rule 642
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx &=c \int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx\\ &=\frac {c (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 33, normalized size = 0.89 \[ \frac {c^2 x (d+e x) (2 d+e x)}{2 \sqrt {c (d+e x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 43, normalized size = 1.16 \[ \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (c e x^{2} + 2 \, c d x\right )}}{2 \, {\left (e x + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.08 \[ \frac {\left (e x +2 d \right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} x}{2 \left (e x +d \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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