Optimal. Leaf size=257 \[ -\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) \left (35 a^2 d^2-64 a b c d+24 b^2 c^2\right )}{24 a b^3}-\frac{d x \sqrt{a+b x^2} \left (290 a^2 b c d^2-105 a^3 d^3-248 a b^2 c^2 d+48 b^3 c^3\right )}{48 a b^4}+\frac{d \left (120 a^2 b c d^2-35 a^3 d^3-144 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )^2 (6 b c-7 a d)}{6 a b^2}+\frac{x \left (c+d x^2\right )^3 (b c-a d)}{a b \sqrt{a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.259272, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {413, 528, 388, 217, 206} \[ -\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) \left (35 a^2 d^2-64 a b c d+24 b^2 c^2\right )}{24 a b^3}-\frac{d x \sqrt{a+b x^2} \left (290 a^2 b c d^2-105 a^3 d^3-248 a b^2 c^2 d+48 b^3 c^3\right )}{48 a b^4}+\frac{d \left (120 a^2 b c d^2-35 a^3 d^3-144 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )^2 (6 b c-7 a d)}{6 a b^2}+\frac{x \left (c+d x^2\right )^3 (b c-a d)}{a b \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 413
Rule 528
Rule 388
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\int \frac{\left (c+d x^2\right )^2 \left (a c d-d (6 b c-7 a d) x^2\right )}{\sqrt{a+b x^2}} \, dx}{a b}\\ &=-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\int \frac{\left (c+d x^2\right ) \left (a c d (12 b c-7 a d)-d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x^2\right )}{\sqrt{a+b x^2}} \, dx}{6 a b^2}\\ &=-\frac{d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt{a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\int \frac{a c d \left (72 b^2 c^2-92 a b c d+35 a^2 d^2\right )-d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x^2}{\sqrt{a+b x^2}} \, dx}{24 a b^3}\\ &=-\frac{d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt{a+b x^2}}{48 a b^4}-\frac{d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt{a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\left (d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right )\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{16 b^4}\\ &=-\frac{d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt{a+b x^2}}{48 a b^4}-\frac{d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt{a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\left (d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{16 b^4}\\ &=-\frac{d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt{a+b x^2}}{48 a b^4}-\frac{d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt{a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}\\ \end{align*}
Mathematica [A] time = 5.19608, size = 172, normalized size = 0.67 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (3 d^2 \left (19 a^2 d^2-56 a b c d+48 b^2 c^2\right )+2 b d^3 x^2 (24 b c-11 a d)+\frac{48 (b c-a d)^4}{a \left (a+b x^2\right )}+8 b^2 d^4 x^4\right )+3 d \left (120 a^2 b c d^2-35 a^3 d^3-144 a b^2 c^2 d+64 b^3 c^3\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{48 b^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.017, size = 340, normalized size = 1.3 \begin{align*}{\frac{{d}^{4}{x}^{7}}{6\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{7\,a{d}^{4}{x}^{5}}{24\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{d}^{4}{a}^{2}{x}^{3}}{48\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{a}^{3}{d}^{4}x}{16\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{35\,{a}^{3}{d}^{4}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{c{d}^{3}{x}^{5}}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,c{d}^{3}a{x}^{3}}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,c{d}^{3}{a}^{2}x}{2\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,c{d}^{3}{a}^{2}}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+3\,{\frac{{c}^{2}{d}^{2}{x}^{3}}{b\sqrt{b{x}^{2}+a}}}+9\,{\frac{{c}^{2}{d}^{2}ax}{{b}^{2}\sqrt{b{x}^{2}+a}}}-9\,{\frac{{c}^{2}{d}^{2}a\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{5/2}}}-4\,{\frac{{c}^{3}dx}{b\sqrt{b{x}^{2}+a}}}+4\,{\frac{{c}^{3}d\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{3/2}}}+{\frac{{c}^{4}x}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \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 [A] time = 2.32665, size = 1272, normalized size = 4.95 \begin{align*} \left [-\frac{3 \,{\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} +{\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (8 \, a b^{4} d^{4} x^{7} + 2 \,{\left (24 \, a b^{4} c d^{3} - 7 \, a^{2} b^{3} d^{4}\right )} x^{5} +{\left (144 \, a b^{4} c^{2} d^{2} - 120 \, a^{2} b^{3} c d^{3} + 35 \, a^{3} b^{2} d^{4}\right )} x^{3} + 3 \,{\left (16 \, b^{5} c^{4} - 64 \, a b^{4} c^{3} d + 144 \, a^{2} b^{3} c^{2} d^{2} - 120 \, a^{3} b^{2} c d^{3} + 35 \, a^{4} b d^{4}\right )} x\right )} \sqrt{b x^{2} + a}}{96 \,{\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}, -\frac{3 \,{\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} +{\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, a b^{4} d^{4} x^{7} + 2 \,{\left (24 \, a b^{4} c d^{3} - 7 \, a^{2} b^{3} d^{4}\right )} x^{5} +{\left (144 \, a b^{4} c^{2} d^{2} - 120 \, a^{2} b^{3} c d^{3} + 35 \, a^{3} b^{2} d^{4}\right )} x^{3} + 3 \,{\left (16 \, b^{5} c^{4} - 64 \, a b^{4} c^{3} d + 144 \, a^{2} b^{3} c^{2} d^{2} - 120 \, a^{3} b^{2} c d^{3} + 35 \, a^{4} b d^{4}\right )} x\right )} \sqrt{b x^{2} + a}}{48 \,{\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}\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{\left (c + d x^{2}\right )^{4}}{\left (a + b x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15544, size = 317, normalized size = 1.23 \begin{align*} \frac{{\left ({\left (2 \,{\left (\frac{4 \, d^{4} x^{2}}{b} + \frac{24 \, a b^{6} c d^{3} - 7 \, a^{2} b^{5} d^{4}}{a b^{7}}\right )} x^{2} + \frac{144 \, a b^{6} c^{2} d^{2} - 120 \, a^{2} b^{5} c d^{3} + 35 \, a^{3} b^{4} d^{4}}{a b^{7}}\right )} x^{2} + \frac{3 \,{\left (16 \, b^{7} c^{4} - 64 \, a b^{6} c^{3} d + 144 \, a^{2} b^{5} c^{2} d^{2} - 120 \, a^{3} b^{4} c d^{3} + 35 \, a^{4} b^{3} d^{4}\right )}}{a b^{7}}\right )} x}{48 \, \sqrt{b x^{2} + a}} - \frac{{\left (64 \, b^{3} c^{3} d - 144 \, a b^{2} c^{2} d^{2} + 120 \, a^{2} b c d^{3} - 35 \, a^{3} d^{4}\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]