Optimal. Leaf size=406 \[ -\frac{\left (-\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{c^3}-\frac{\left (d+e x^2\right )^{3/2} (b e+c d)}{3 c^2 e^2}+\frac{\left (d+e x^2\right )^{5/2}}{5 c e^2} \]
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Rubi [A] time = 8.59299, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1251, 897, 1287, 1166, 208} \[ -\frac{\left (-\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (\frac{-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^3 c d+b^4 (-e)}{\sqrt{b^2-4 a c}}+2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{c^3}-\frac{\left (d+e x^2\right )^{3/2} (b e+c d)}{3 c^2 e^2}+\frac{\left (d+e x^2\right )^{5/2}}{5 c e^2} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 897
Rule 1287
Rule 1166
Rule 208
Rubi steps
\begin{align*} \int \frac{x^7 \sqrt{d+e x^2}}{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3 \sqrt{d+e x}}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-\frac{d}{e}+\frac{x^2}{e}\right )^3}{\frac{c d^2-b d e+a e^2}{e^2}-\frac{(2 c d-b e) x^2}{e^2}+\frac{c x^4}{e^2}} \, dx,x,\sqrt{d+e x^2}\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\left (b^2-a c\right ) e}{c^3}-\frac{(c d+b e) x^2}{c^2 e}+\frac{x^4}{c e}-\frac{\left (b^2-a c\right ) \left (c d^2-b d e+a e^2\right )-\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x^2}{c^3 e \left (\frac{c d^2-b d e+a e^2}{e^2}-\frac{(2 c d-b e) x^2}{e^2}+\frac{c x^4}{e^2}\right )}\right ) \, dx,x,\sqrt{d+e x^2}\right )}{e}\\ &=\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{c^3}-\frac{(c d+b e) \left (d+e x^2\right )^{3/2}}{3 c^2 e^2}+\frac{\left (d+e x^2\right )^{5/2}}{5 c e^2}-\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-a c\right ) \left (c d^2-b d e+a e^2\right )+\left (-b^2 c d+a c^2 d+b^3 e-2 a b c e\right ) x^2}{\frac{c d^2-b d e+a e^2}{e^2}-\frac{(2 c d-b e) x^2}{e^2}+\frac{c x^4}{e^2}} \, dx,x,\sqrt{d+e x^2}\right )}{c^3 e^2}\\ &=\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{c^3}-\frac{(c d+b e) \left (d+e x^2\right )^{3/2}}{3 c^2 e^2}+\frac{\left (d+e x^2\right )^{5/2}}{5 c e^2}+\frac{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e-\frac{b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{\sqrt{b^2-4 a c}}{2 e}-\frac{2 c d-b e}{2 e^2}+\frac{c x^2}{e^2}} \, dx,x,\sqrt{d+e x^2}\right )}{2 c^3 e^2}+\frac{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e+\frac{b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b^2-4 a c}}{2 e}-\frac{2 c d-b e}{2 e^2}+\frac{c x^2}{e^2}} \, dx,x,\sqrt{d+e x^2}\right )}{2 c^3 e^2}\\ &=\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{c^3}-\frac{(c d+b e) \left (d+e x^2\right )^{3/2}}{3 c^2 e^2}+\frac{\left (d+e x^2\right )^{5/2}}{5 c e^2}-\frac{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e-\frac{b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}-\frac{\left (b^2 c d-a c^2 d-b^3 e+2 a b c e+\frac{b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right )}{\sqrt{2} c^{7/2} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\\ \end{align*}
Mathematica [B] time = 10.8281, size = 943, normalized size = 2.32 \[ \frac{\frac{c \left (105 \left (-b^3+\sqrt{b^2-4 a c} b^2+3 a c b-a c \sqrt{b^2-4 a c}\right ) \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}}\right ) e^3+\sqrt{2} \sqrt{\frac{c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}} \left (105 b^3 e^3-35 b^2 \left (3 \sqrt{b^2-4 a c} e+c \left (e x^2+d\right )\right ) e^2+7 b c \left (\left (e x^2+d\right ) \left (-5 c d+5 \sqrt{b^2-4 a c} e+3 c \left (e x^2+d\right )\right )-45 a e^2\right ) e+c \left (35 a \left (3 \sqrt{b^2-4 a c} e+2 c \left (e x^2+d\right )\right ) e^2+c \left (e x^2+d\right ) \left (7 \sqrt{b^2-4 a c} e \left (5 d-3 \left (e x^2+d\right )\right )+c \left (-70 d^2+84 \left (e x^2+d\right ) d-30 \left (e x^2+d\right )^2\right )\right )\right )\right )\right ) \left (e x^2+d\right )^{9/2}}{210 \sqrt{2} \sqrt{b^2-4 a c} e^4 \left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right )^3 \left (-\frac{2 c d-b e}{e^2}-\frac{\sqrt{b^2-4 a c}}{e}\right ) \left (\frac{c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )^{9/2}}+\frac{2 c d^3 \left (\frac{3 \left (b^3+\sqrt{b^2-4 a c} b^2-3 a c b-a c \sqrt{b^2-4 a c}\right ) e^3 \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right ) \left (e x^2+d\right )^3}{4 \sqrt{2} d^3 \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right )^3 \left (\frac{c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^{9/2}}+\frac{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (-105 b^3 e^3+35 b^2 \left (c \left (e x^2+d\right )-3 \sqrt{b^2-4 a c} e\right ) e^2-7 b c \left (\left (e x^2+d\right ) \left (-5 c d-5 \sqrt{b^2-4 a c} e+3 c \left (e x^2+d\right )\right )-45 a e^2\right ) e+c \left (35 a \left (3 \sqrt{b^2-4 a c} e-2 c \left (e x^2+d\right )\right ) e^2+c \left (e x^2+d\right ) \left (7 \sqrt{b^2-4 a c} e \left (5 d-3 \left (e x^2+d\right )\right )+c \left (70 d^2-84 \left (e x^2+d\right ) d+30 \left (e x^2+d\right )^2\right )\right )\right )\right )}{140 c^4 d^3 \left (e x^2+d\right )}\right ) \left (e x^2+d\right )^{3/2}}{3 \sqrt{b^2-4 a c} e^4 \left (\frac{\sqrt{b^2-4 a c}}{e}-\frac{2 c d-b e}{e^2}\right )}}{e} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.049, size = 496, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{5\,ce} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{2\,d}{15\,c{e}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{b}{3\,{c}^{2}e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{xa}{2\,{c}^{2}}\sqrt{e}}-{\frac{x{b}^{2}}{2\,{c}^{3}}\sqrt{e}}-{\frac{a}{2\,{c}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{{b}^{2}}{2\,{c}^{3}}\sqrt{e{x}^{2}+d}}-{\frac{ad}{2\,{c}^{2}} \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{-1}}+{\frac{{b}^{2}d}{2\,{c}^{3}} \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{-1}}-{\frac{1}{4\,{c}^{3}}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{6}+ \left ( 16\,a{e}^{2}-8\,deb+6\,c{d}^{2} \right ){{\it \_Z}}^{4}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){{\it \_Z}}^{2}+c{d}^{4} \right ) }{\frac{ \left ( -2\,abce+a{c}^{2}d+{b}^{3}e-{b}^{2}cd \right ){{\it \_R}}^{6}+ \left ( -4\,{a}^{2}c{e}^{2}+4\,a{b}^{2}{e}^{2}+2\,abcde-3\,a{c}^{2}{d}^{2}-3\,{b}^{3}de+3\,{b}^{2}c{d}^{2} \right ){{\it \_R}}^{4}+d \left ( 4\,{a}^{2}c{e}^{2}-4\,a{b}^{2}{e}^{2}-2\,abcde+3\,a{c}^{2}{d}^{2}+3\,{b}^{3}de-3\,{b}^{2}c{d}^{2} \right ){{\it \_R}}^{2}+2\,abc{d}^{3}e-a{c}^{2}{d}^{4}-{b}^{3}{d}^{3}e+{b}^{2}c{d}^{4}}{{{\it \_R}}^{7}c+3\,{{\it \_R}}^{5}be-3\,{{\it \_R}}^{5}cd+8\,{{\it \_R}}^{3}a{e}^{2}-4\,{{\it \_R}}^{3}bde+3\,{{\it \_R}}^{3}c{d}^{2}+{\it \_R}\,b{d}^{2}e-{\it \_R}\,c{d}^{3}}\ln \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x-{\it \_R} \right ) }} \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{\sqrt{e x^{2} + d} x^{7}}{c x^{4} + b x^{2} + a}\,{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{x^{7} \sqrt{d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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