Optimal. Leaf size=96 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^2 (a+b x)} \]
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Rubi [A] time = 0.0374104, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{3 e^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) \sqrt{d+e x} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b (b d-a e) \sqrt{d+e x}}{e}+\frac{b^2 (d+e x)^{3/2}}{e}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{2 (b d-a e) (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x)}+\frac{2 b (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0226549, size = 48, normalized size = 0.5 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} (5 a e-2 b d+3 b e x)}{15 e^2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 43, normalized size = 0.5 \begin{align*}{\frac{6\,bxe+10\,ae-4\,bd}{15\,{e}^{2} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10817, size = 62, normalized size = 0.65 \begin{align*} \frac{2 \,{\left (3 \, b e^{2} x^{2} - 2 \, b d^{2} + 5 \, a d e +{\left (b d e + 5 \, a e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43663, size = 108, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (3 \, b e^{2} x^{2} - 2 \, b d^{2} + 5 \, a d e +{\left (b d e + 5 \, a e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.5209, size = 49, normalized size = 0.51 \begin{align*} \frac{2 a \left (d + e x\right )^{\frac{3}{2}}}{3 e} - \frac{2 b d \left (d + e x\right )^{\frac{3}{2}}}{3 e^{2}} + \frac{2 b \left (d + e x\right )^{\frac{5}{2}}}{5 e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12019, size = 73, normalized size = 0.76 \begin{align*} \frac{2}{15} \,{\left ({\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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