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 = 0.04, antiderivative size = 96, normalized size of antiderivative = 1.00, 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 43
Rule 646
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*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.50 \[ \frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} (5 a e-2 b d+3 b e x)}{15 e^2 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 46, normalized size = 0.48 \[ \frac {2 \, {\left (3 \, b e^{2} x^{2} - 2 \, b d^{2} + 5 \, a d e + {\left (b d e + 5 \, a e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 129, normalized size = 1.34 \[ \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b d e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 15 \, \sqrt {x e + d} a d \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 43, normalized size = 0.45 \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3 b e x +5 a e -2 b d \right ) \sqrt {\left (b x +a \right )^{2}}}{15 \left (b x +a \right ) e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 46, normalized size = 0.48 \[ \frac {2 \, {\left (3 \, b e^{2} x^{2} - 2 \, b d^{2} + 5 \, a d e + {\left (b d e + 5 \, a e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.98, size = 49, normalized size = 0.51 \[ \frac {2 a \left (d + e x\right )^{\frac {3}{2}}}{3 e} - \frac {2 b d \left (d + e x\right )^{\frac {3}{2}}}{3 e^{2}} + \frac {2 b \left (d + e x\right )^{\frac {5}{2}}}{5 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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