Optimal. Leaf size=258 \[ \frac{x \sqrt{c+d x^2} \left (32 a^2 d^2-52 a b c d+19 b^2 c^2\right )}{16 b^4}+\frac{\left (120 a^2 b c d^2-64 a^3 d^3-60 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^5 \sqrt{d}}+\frac{d x^3 \sqrt{c+d x^2} (7 b c-8 a d)}{8 b^3}-\frac{\sqrt{a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^5}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.448375, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {467, 581, 582, 523, 217, 206, 377, 205} \[ \frac{x \sqrt{c+d x^2} \left (32 a^2 d^2-52 a b c d+19 b^2 c^2\right )}{16 b^4}+\frac{\left (120 a^2 b c d^2-64 a^3 d^3-60 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^5 \sqrt{d}}+\frac{d x^3 \sqrt{c+d x^2} (7 b c-8 a d)}{8 b^3}-\frac{\sqrt{a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^5}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 467
Rule 581
Rule 582
Rule 523
Rule 217
Rule 206
Rule 377
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{\int \frac{x^2 \left (c+d x^2\right )^{3/2} \left (3 c+8 d x^2\right )}{a+b x^2} \, dx}{2 b}\\ &=\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{\int \frac{x^2 \sqrt{c+d x^2} \left (6 c (3 b c-4 a d)+6 d (7 b c-8 a d) x^2\right )}{a+b x^2} \, dx}{12 b^2}\\ &=\frac{d (7 b c-8 a d) x^3 \sqrt{c+d x^2}}{8 b^3}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{\int \frac{x^2 \left (6 c \left (12 b^2 c^2-37 a b c d+24 a^2 d^2\right )+6 d \left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{48 b^3}\\ &=\frac{\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt{c+d x^2}}{16 b^4}+\frac{d (7 b c-8 a d) x^3 \sqrt{c+d x^2}}{8 b^3}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac{\int \frac{6 a c d \left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right )-6 d \left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) x^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{96 b^4 d}\\ &=\frac{\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt{c+d x^2}}{16 b^4}+\frac{d (7 b c-8 a d) x^3 \sqrt{c+d x^2}}{8 b^3}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac{\left (a (3 b c-8 a d) (b c-a d)^2\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{2 b^5}+\frac{\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \int \frac{1}{\sqrt{c+d x^2}} \, dx}{16 b^5}\\ &=\frac{\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt{c+d x^2}}{16 b^4}+\frac{d (7 b c-8 a d) x^3 \sqrt{c+d x^2}}{8 b^3}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac{\left (a (3 b c-8 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{2 b^5}+\frac{\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{16 b^5}\\ &=\frac{\left (19 b^2 c^2-52 a b c d+32 a^2 d^2\right ) x \sqrt{c+d x^2}}{16 b^4}+\frac{d (7 b c-8 a d) x^3 \sqrt{c+d x^2}}{8 b^3}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}-\frac{\sqrt{a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^5}+\frac{\left (5 b^3 c^3-60 a b^2 c^2 d+120 a^2 b c d^2-64 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^5 \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.279052, size = 219, normalized size = 0.85 \[ \frac{b x \sqrt{c+d x^2} \left (72 a^2 d^2+2 b d x^2 (13 b c-12 a d)+\frac{24 a (b c-a d)^2}{a+b x^2}-108 a b c d+33 b^2 c^2+8 b^2 d^2 x^4\right )+\frac{3 \left (120 a^2 b c d^2-64 a^3 d^3-60 a b^2 c^2 d+5 b^3 c^3\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}+24 \sqrt{a} (8 a d-3 b c) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{48 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.022, size = 7611, normalized size = 29.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{4}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 17.5238, size = 3663, normalized size = 14.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.23077, size = 703, normalized size = 2.72 \begin{align*} \frac{1}{48} \,{\left (2 \,{\left (\frac{4 \, d^{2} x^{2}}{b^{2}} + \frac{13 \, b^{12} c d^{5} - 12 \, a b^{11} d^{6}}{b^{14} d^{4}}\right )} x^{2} + \frac{3 \,{\left (11 \, b^{12} c^{2} d^{4} - 36 \, a b^{11} c d^{5} + 24 \, a^{2} b^{10} d^{6}\right )}}{b^{14} d^{4}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (3 \, a b^{3} c^{3} \sqrt{d} - 14 \, a^{2} b^{2} c^{2} d^{\frac{3}{2}} + 19 \, a^{3} b c d^{\frac{5}{2}} - 8 \, a^{4} d^{\frac{7}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt{a b c d - a^{2} d^{2}} b^{5}} - \frac{{\left (5 \, b^{3} c^{3} \sqrt{d} - 60 \, a b^{2} c^{2} d^{\frac{3}{2}} + 120 \, a^{2} b c d^{\frac{5}{2}} - 64 \, a^{3} d^{\frac{7}{2}}\right )} \log \left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{32 \, b^{5} d} - \frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b^{3} c^{3} \sqrt{d} - 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} b^{2} c^{2} d^{\frac{3}{2}} + 5 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{3} b c d^{\frac{5}{2}} - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{4} d^{\frac{7}{2}} - a b^{3} c^{4} \sqrt{d} + 2 \, a^{2} b^{2} c^{3} d^{\frac{3}{2}} - a^{3} b c^{2} d^{\frac{5}{2}}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]