Optimal. Leaf size=35 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt {b} d} \]
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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {253, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt {b} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-a}+b (c+d x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-a}+b x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt {b} d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt {b} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 42, normalized size = 1.20
method | result | size |
default | \(\frac {\arctan \left (\frac {2 d^{2} b x +2 b c d}{2 d \sqrt {\sqrt {-a}\, b}}\right )}{d \sqrt {\sqrt {-a}\, b}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (27) = 54\).
time = 0.48, size = 66, normalized size = 1.89 \begin {gather*} \frac {\log \left (\frac {b d^{2} x + b c d - \sqrt {-\sqrt {-a} b} d}{b d^{2} x + b c d + \sqrt {-\sqrt {-a} b} d}\right )}{2 \, \sqrt {-\sqrt {-a} b} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 279, normalized size = 7.97 \begin {gather*} \left [\frac {\sqrt {\frac {\sqrt {-a}}{a b}} \log \left (\frac {b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {-a} + 2 \, {\left (a b d x + a b c + {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \sqrt {-a}\right )} \sqrt {\frac {\sqrt {-a}}{a b}} - a}{b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} + a}\right )}{2 \, d}, \frac {\sqrt {-\frac {\sqrt {-a}}{a b}} \arctan \left ({\left (b d x + b c\right )} \sqrt {-\frac {\sqrt {-a}}{a b}}\right )}{d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (31) = 62\).
time = 0.08, size = 92, normalized size = 2.63 \begin {gather*} \frac {- \frac {\sqrt {- \frac {1}{b \sqrt {- a}}} \log {\left (x + \frac {c - \sqrt {- a} \sqrt {- \frac {1}{b \sqrt {- a}}}}{d} \right )}}{2} + \frac {\sqrt {- \frac {1}{b \sqrt {- a}}} \log {\left (x + \frac {c + \sqrt {- a} \sqrt {- \frac {1}{b \sqrt {- a}}}}{d} \right )}}{2}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.83, size = 30, normalized size = 0.86 \begin {gather*} \frac {\arctan \left (\frac {b d x + b c}{\left (-a\right )^{\frac {1}{4}} \sqrt {b}}\right )}{\left (-a\right )^{\frac {1}{4}} \sqrt {b} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 31, normalized size = 0.89 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,c+\sqrt {b}\,d\,x}{{\left (-a\right )}^{1/4}}\right )}{{\left (-a\right )}^{1/4}\,\sqrt {b}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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