Optimal. Leaf size=43 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a-b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} \sqrt{d}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0277071, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {63, 217, 203} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a-b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a-b x} \sqrt{c+d x}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{a d}{b}-\frac{d x^2}{b}}} \, dx,x,\sqrt{a-b x}\right )}{b}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a-b x}}{\sqrt{c+d x}}\right )}{b}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a-b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} \sqrt{d}}\\ \end{align*}
Mathematica [B] time = 0.0787922, size = 103, normalized size = 2.4 \[ \frac{2 \sqrt{-b} \sqrt{-a d-b c} \sqrt{\frac{b (c+d x)}{a d+b c}} \sin ^{-1}\left (\frac{\sqrt{-b} \sqrt{d} \sqrt{a-b x}}{\sqrt{b} \sqrt{-a d-b c}}\right )}{b^{3/2} \sqrt{d} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.007, size = 84, normalized size = 2. \begin{align*}{\sqrt{ \left ( -bx+a \right ) \left ( dx+c \right ) }\arctan \left ({\sqrt{bd} \left ( x-{\frac{ad-bc}{2\,bd}} \right ){\frac{1}{\sqrt{-d{x}^{2}b+ \left ( ad-bc \right ) x+ac}}}} \right ){\frac{1}{\sqrt{-bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{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 = 2.12206, size = 417, normalized size = 9.7 \begin{align*} \left [-\frac{\sqrt{-b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 4 \,{\left (2 \, b d x + b c - a d\right )} \sqrt{-b d} \sqrt{-b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d - a b d^{2}\right )} x\right )}{2 \, b d}, -\frac{\sqrt{b d} \arctan \left (\frac{{\left (2 \, b d x + b c - a d\right )} \sqrt{b d} \sqrt{-b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \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}{\sqrt{a - b x} \sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0829, size = 73, normalized size = 1.7 \begin{align*} \frac{2 \, b \log \left ({\left | -\sqrt{-b d} \sqrt{-b x + a} + \sqrt{b^{2} c +{\left (b x - a\right )} b d + a b d} \right |}\right )}{\sqrt{-b d}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]