Optimal. Leaf size=64 \[ \frac {2 \sqrt {c+d x} f^{a+b \sqrt {c+d x}}}{b d \log (f)}-\frac {2 f^{a+b \sqrt {c+d x}}}{b^2 d \log ^2(f)} \]
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Rubi [A] time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2207, 2176, 2194} \[ \frac {2 \sqrt {c+d x} f^{a+b \sqrt {c+d x}}}{b d \log (f)}-\frac {2 f^{a+b \sqrt {c+d x}}}{b^2 d \log ^2(f)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2207
Rubi steps
\begin {align*} \int f^{a+b \sqrt {c+d x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int f^{a+b x} x \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 f^{a+b \sqrt {c+d x}} \sqrt {c+d x}}{b d \log (f)}-\frac {2 \operatorname {Subst}\left (\int f^{a+b x} \, dx,x,\sqrt {c+d x}\right )}{b d \log (f)}\\ &=-\frac {2 f^{a+b \sqrt {c+d x}}}{b^2 d \log ^2(f)}+\frac {2 f^{a+b \sqrt {c+d x}} \sqrt {c+d x}}{b d \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.66 \[ \frac {2 f^{a+b \sqrt {c+d x}} \left (b \log (f) \sqrt {c+d x}-1\right )}{b^2 d \log ^2(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 42, normalized size = 0.66 \[ \frac {2 \, {\left (\sqrt {d x + c} b \log \relax (f) - 1\right )} e^{\left (\sqrt {d x + c} b \log \relax (f) + a \log \relax (f)\right )}}{b^{2} d \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 774, normalized size = 12.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int f^{a +\sqrt {d x +c}\, b}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 43, normalized size = 0.67 \[ \frac {2 \, {\left (\sqrt {d x + c} b f^{a} \log \relax (f) - f^{a}\right )} f^{\sqrt {d x + c} b}}{b^{2} d \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 38, normalized size = 0.59 \[ \frac {f^{a+b\,\sqrt {c+d\,x}}\,\left (2\,b\,\ln \relax (f)\,\sqrt {c+d\,x}-2\right )}{b^2\,d\,{\ln \relax (f)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 76, normalized size = 1.19 \[ \begin {cases} x & \text {for}\: b = 0 \wedge d = 0 \wedge f = 1 \\f^{a + b \sqrt {c}} x & \text {for}\: d = 0 \\f^{a} x & \text {for}\: b = 0 \\x & \text {for}\: f = 1 \\\frac {2 f^{a} f^{b \sqrt {c + d x}} \sqrt {c + d x}}{b d \log {\relax (f )}} - \frac {2 f^{a} f^{b \sqrt {c + d x}}}{b^{2} d \log {\relax (f )}^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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