Optimal. Leaf size=129 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4}}+\frac{4}{d \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.108493, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {693, 694, 329, 298, 203, 206} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{5/4}}+\frac{4}{d \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 693
Rule 694
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )} \, dx &=\frac{4}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}+\frac{\int \frac{\sqrt{b d+2 c d x}}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) d^2}\\ &=\frac{4}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{2 c \left (b^2-4 a c\right ) d^3}\\ &=\frac{4}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}+\frac{\operatorname{Subst}\left (\int \frac{x^2}{a-\frac{b^2}{4 c}+\frac{x^4}{4 c d^2}} \, dx,x,\sqrt{d (b+2 c x)}\right )}{c \left (b^2-4 a c\right ) d^3}\\ &=\frac{4}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c} d-x^2} \, dx,x,\sqrt{d (b+2 c x)}\right )}{\left (b^2-4 a c\right ) d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c} d+x^2} \, dx,x,\sqrt{d (b+2 c x)}\right )}{\left (b^2-4 a c\right ) d}\\ &=\frac{4}{\left (b^2-4 a c\right ) d \sqrt{b d+2 c d x}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )}{\left (b^2-4 a c\right )^{5/4} d^{3/2}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )}{\left (b^2-4 a c\right )^{5/4} d^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0447679, size = 54, normalized size = 0.42 \[ \frac{4 \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d \left (b^2-4 a c\right ) \sqrt{d (b+2 c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.195, size = 341, normalized size = 2.6 \begin{align*} -{\frac{\sqrt{2}}{2\,d \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({ \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-{\frac{\sqrt{2}}{d \left ( 4\,ac-{b}^{2} \right ) }\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+{\frac{\sqrt{2}}{d \left ( 4\,ac-{b}^{2} \right ) }\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-4\,{\frac{1}{d \left ( 4\,ac-{b}^{2} \right ) \sqrt{2\,cdx+bd}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.38624, size = 1912, normalized size = 14.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.15078, size = 671, normalized size = 5.2 \begin{align*} -\frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} - \frac{\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} + \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \log \left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d^{3} - 8 \, \sqrt{2} a b^{2} c d^{3} + 16 \, \sqrt{2} a^{2} c^{2} d^{3}} - \frac{{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \log \left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{\sqrt{2} b^{4} d^{3} - 8 \, \sqrt{2} a b^{2} c d^{3} + 16 \, \sqrt{2} a^{2} c^{2} d^{3}} + \frac{4}{{\left (b^{2} d - 4 \, a c d\right )} \sqrt{2 \, c d x + b d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]