Optimal. Leaf size=35 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt {b} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, 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 = {247, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt {b} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-a}+b (c+d x)^2} \, dx &=\frac {\operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 35, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt [4]{-a}}\right )}{\sqrt [4]{-a} \sqrt {b} d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 279, normalized size = 7.97 \[ \left [\frac {\sqrt {\frac {\sqrt {-a}}{a b}} \log \left (\frac {b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {-a} + 2 \, {\left (a b d x + a b c + {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \sqrt {-a}\right )} \sqrt {\frac {\sqrt {-a}}{a b}} - a}{b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} + a}\right )}{2 \, d}, \frac {\sqrt {-\frac {\sqrt {-a}}{a b}} \arctan \left ({\left (b d x + b c\right )} \sqrt {-\frac {\sqrt {-a}}{a b}}\right )}{d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.45, size = 30, normalized size = 0.86 \[ \frac {\arctan \left (\frac {b d x + b c}{\left (-a\right )^{\frac {1}{4}} \sqrt {b}}\right )}{\left (-a\right )^{\frac {1}{4}} \sqrt {b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 42, normalized size = 1.20 \[ \frac {\arctan \left (\frac {2 b \,d^{2} x +2 b d c}{2 \sqrt {\sqrt {-a}\, b}\, d}\right )}{\sqrt {\sqrt {-a}\, b}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.40, size = 66, normalized size = 1.89 \[ \frac {\log \left (\frac {b d^{2} x + b c d - \sqrt {-\sqrt {-a} b} d}{b d^{2} x + b c d + \sqrt {-\sqrt {-a} b} d}\right )}{2 \, \sqrt {-\sqrt {-a} b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.10, size = 31, normalized size = 0.89 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,c+\sqrt {b}\,d\,x}{{\left (-a\right )}^{1/4}}\right )}{{\left (-a\right )}^{1/4}\,\sqrt {b}\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.22, size = 92, normalized size = 2.63 \[ \frac {- \frac {\sqrt {- \frac {1}{b \sqrt {- a}}} \log {\left (x + \frac {c - \sqrt {- a} \sqrt {- \frac {1}{b \sqrt {- a}}}}{d} \right )}}{2} + \frac {\sqrt {- \frac {1}{b \sqrt {- a}}} \log {\left (x + \frac {c + \sqrt {- a} \sqrt {- \frac {1}{b \sqrt {- a}}}}{d} \right )}}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________