Optimal. Leaf size=562 \[ \frac{a^{3/2} e \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 d^2}+\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x^2 (2 c d-b e)+8 c^2 d^2\right )}{16 c d^2 e}-\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2} d^2 e^2}+\frac{b e \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2} d^2}-\frac{e \left (8 a c+b^2+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2}+\frac{3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 \sqrt{c} d}+\frac{\left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d^2 e^2}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{2 d x^2}+\frac{3 \left (3 b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 d}-\frac{3 \sqrt{a} b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 d} \]
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Rubi [A] time = 0.923926, antiderivative size = 562, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {1251, 960, 732, 814, 843, 621, 206, 724, 734} \[ \frac{a^{3/2} e \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 d^2}+\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x^2 (2 c d-b e)+8 c^2 d^2\right )}{16 c d^2 e}-\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2} d^2 e^2}+\frac{b e \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2} d^2}-\frac{e \left (8 a c+b^2+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2}+\frac{3 \left (4 a c+b^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 \sqrt{c} d}+\frac{\left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d^2 e^2}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{2 d x^2}+\frac{3 \left (3 b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 d}-\frac{3 \sqrt{a} b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 960
Rule 732
Rule 814
Rule 843
Rule 621
Rule 206
Rule 724
Rule 734
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^3 \left (d+e x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^2 (d+e x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{\left (a+b x+c x^2\right )^{3/2}}{d x^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{d^2 x}+\frac{e^2 \left (a+b x+c x^2\right )^{3/2}}{d^2 (d+e x)}\right ) \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x^2} \, dx,x,x^2\right )}{2 d}-\frac{e \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{x} \, dx,x,x^2\right )}{2 d^2}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{2 d x^2}+\frac{3 \operatorname{Subst}\left (\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{x} \, dx,x,x^2\right )}{4 d}+\frac{e \operatorname{Subst}\left (\int \frac{(-2 a-b x) \sqrt{a+b x+c x^2}}{x} \, dx,x,x^2\right )}{4 d^2}-\frac{e \operatorname{Subst}\left (\int \frac{(b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{d+e x} \, dx,x,x^2\right )}{4 d^2}\\ &=\frac{3 \left (3 b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 d}-\frac{e \left (b^2+8 a c+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2 e}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{2 d x^2}-\frac{3 \operatorname{Subst}\left (\int \frac{-4 a b c-c \left (b^2+4 a c\right ) x}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{16 c d}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (4 c e (b d-2 a e)^2-d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )-\frac{1}{2} (2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{16 c d^2 e}-\frac{e \operatorname{Subst}\left (\int \frac{8 a^2 c-\frac{1}{2} b \left (b^2-12 a c\right ) x}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{16 c d^2}\\ &=\frac{3 \left (3 b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 d}-\frac{e \left (b^2+8 a c+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2 e}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{2 d x^2}+\frac{(3 a b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{4 d}+\frac{\left (3 \left (b^2+4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{16 d}-\frac{\left (a^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 d^2}+\frac{\left (b \left (b^2-12 a c\right ) e\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{32 c d^2}+\frac{\left (c d^2-b d e+a e^2\right )^2 \operatorname{Subst}\left (\int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 d^2 e^2}-\frac{\left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{32 c d^2 e^2}\\ &=\frac{3 \left (3 b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 d}-\frac{e \left (b^2+8 a c+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2 e}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{2 d x^2}-\frac{(3 a b) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{2 d}+\frac{\left (3 \left (b^2+4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{8 d}+\frac{\left (a^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x^2}{\sqrt{a+b x^2+c x^4}}\right )}{d^2}+\frac{\left (b \left (b^2-12 a c\right ) e\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{16 c d^2}-\frac{\left (c d^2-b d e+a e^2\right )^2 \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x^2}{\sqrt{a+b x^2+c x^4}}\right )}{d^2 e^2}-\frac{\left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{16 c d^2 e^2}\\ &=\frac{3 \left (3 b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{8 d}-\frac{e \left (b^2+8 a c+2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x^2\right ) \sqrt{a+b x^2+c x^4}}{16 c d^2 e}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{2 d x^2}-\frac{3 \sqrt{a} b \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 d}+\frac{a^{3/2} e \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 d^2}+\frac{3 \left (b^2+4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 \sqrt{c} d}+\frac{b \left (b^2-12 a c\right ) e \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2} d^2}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2} d^2 e^2}+\frac{\left (c d^2-b d e+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x^2}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x^2+c x^4}}\right )}{2 d^2 e^2}\\ \end{align*}
Mathematica [A] time = 0.513455, size = 240, normalized size = 0.43 \[ \frac{1}{4} \left (-\frac{2 \left (x^2 \left (e (a e-b d)+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right )+d e \sqrt{a+b x^2+c x^4} \left (a e-c d x^2\right )\right )}{d^2 e^2 x^2}+\frac{\sqrt{a} (2 a e-3 b d) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{d^2}-\frac{\sqrt{c} (2 c d-3 b e) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{e^2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 1207, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e x^{2} + d\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x^{3} \left (d + e x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e x^{2} + d\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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