Optimal. Leaf size=309 \[ -\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2} \left (-80 a d^2+32 b c d+42 b d e x-35 b e^2\right )}{240 d^3 \left (a+b x^2\right )}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt{c+d x^2+e x} \left (-16 a d^2+12 b c d-7 b e^2\right )}{128 d^4 \left (a+b x^2\right )}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (4 c d-e^2\right ) \left (-16 a d^2+12 b c d-7 b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{256 d^{9/2} \left (a+b x^2\right )}+\frac{b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{5 d \left (a+b x^2\right )} \]
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Rubi [A] time = 0.625224, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6744, 1653, 779, 612, 621, 206} \[ -\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2} \left (-80 a d^2+32 b c d+42 b d e x-35 b e^2\right )}{240 d^3 \left (a+b x^2\right )}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} (2 d x+e) \sqrt{c+d x^2+e x} \left (-16 a d^2+12 b c d-7 b e^2\right )}{128 d^4 \left (a+b x^2\right )}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4} \left (4 c d-e^2\right ) \left (-16 a d^2+12 b c d-7 b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{256 d^{9/2} \left (a+b x^2\right )}+\frac{b x^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{5 d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 6744
Rule 1653
Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int x \left (2 a b+2 b^2 x^2\right ) \sqrt{c+e x+d x^2} \, dx}{2 a b+2 b^2 x^2}\\ &=\frac{b x^2 \left (c+e x+d x^2\right )^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}+\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int x \left (-2 b (2 b c-5 a d)-7 b^2 e x\right ) \sqrt{c+e x+d x^2} \, dx}{5 d \left (2 a b+2 b^2 x^2\right )}\\ &=\frac{b x^2 \left (c+e x+d x^2\right )^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}-\frac{\left (32 b c d-80 a d^2-35 b e^2+42 b d e x\right ) \left (c+e x+d x^2\right )^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{240 d^3 \left (a+b x^2\right )}+\frac{\left (b e \left (12 b c d-16 a d^2-7 b e^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}\right ) \int \sqrt{c+e x+d x^2} \, dx}{16 d^3 \left (2 a b+2 b^2 x^2\right )}\\ &=\frac{e \left (12 b c d-16 a d^2-7 b e^2\right ) (e+2 d x) \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{128 d^4 \left (a+b x^2\right )}+\frac{b x^2 \left (c+e x+d x^2\right )^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}-\frac{\left (32 b c d-80 a d^2-35 b e^2+42 b d e x\right ) \left (c+e x+d x^2\right )^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{240 d^3 \left (a+b x^2\right )}+\frac{\left (b e \left (4 c d-e^2\right ) \left (12 b c d-16 a d^2-7 b e^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}\right ) \int \frac{1}{\sqrt{c+e x+d x^2}} \, dx}{128 d^4 \left (2 a b+2 b^2 x^2\right )}\\ &=\frac{e \left (12 b c d-16 a d^2-7 b e^2\right ) (e+2 d x) \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{128 d^4 \left (a+b x^2\right )}+\frac{b x^2 \left (c+e x+d x^2\right )^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}-\frac{\left (32 b c d-80 a d^2-35 b e^2+42 b d e x\right ) \left (c+e x+d x^2\right )^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{240 d^3 \left (a+b x^2\right )}+\frac{\left (b e \left (4 c d-e^2\right ) \left (12 b c d-16 a d^2-7 b e^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{4 d-x^2} \, dx,x,\frac{e+2 d x}{\sqrt{c+e x+d x^2}}\right )}{64 d^4 \left (2 a b+2 b^2 x^2\right )}\\ &=\frac{e \left (12 b c d-16 a d^2-7 b e^2\right ) (e+2 d x) \sqrt{c+e x+d x^2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{128 d^4 \left (a+b x^2\right )}+\frac{b x^2 \left (c+e x+d x^2\right )^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}-\frac{\left (32 b c d-80 a d^2-35 b e^2+42 b d e x\right ) \left (c+e x+d x^2\right )^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{240 d^3 \left (a+b x^2\right )}+\frac{e \left (4 c d-e^2\right ) \left (12 b c d-16 a d^2-7 b e^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac{e+2 d x}{2 \sqrt{d} \sqrt{c+e x+d x^2}}\right )}{256 d^{9/2} \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.345258, size = 214, normalized size = 0.69 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (2 \sqrt{d} \sqrt{c+x (d x+e)} \left (80 a d^2 \left (8 c d+8 d^2 x^2+2 d e x-3 e^2\right )+b \left (-256 c^2 d^2+4 c d \left (32 d^2 x^2-58 d e x+115 e^2\right )-56 d^2 e^2 x^2+48 d^3 e x^3+384 d^4 x^4+70 d e^3 x-105 e^4\right )\right )+15 e \left (e^2-4 c d\right ) \left (16 a d^2-12 b c d+7 b e^2\right ) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+x (d x+e)}}\right )\right )}{3840 d^{9/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 442, normalized size = 1.4 \begin{align*}{\frac{1}{3840\,b{x}^{2}+3840\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 768\,{d}^{9/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}{x}^{2}b-672\,{d}^{7/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}xbe+1280\,{d}^{9/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}a-512\,{d}^{7/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}bc+560\,{d}^{5/2} \left ({x}^{2}d+ex+c \right ) ^{3/2}b{e}^{2}-960\,{d}^{9/2}\sqrt{{x}^{2}d+ex+c}xae+720\,{d}^{7/2}\sqrt{{x}^{2}d+ex+c}xbce-420\,{d}^{5/2}\sqrt{{x}^{2}d+ex+c}xb{e}^{3}-480\,{d}^{7/2}\sqrt{{x}^{2}d+ex+c}a{e}^{2}+360\,{d}^{5/2}\sqrt{{x}^{2}d+ex+c}bc{e}^{2}-210\,{d}^{3/2}\sqrt{{x}^{2}d+ex+c}b{e}^{4}-960\,\ln \left ( 1/2\,{\frac{2\,\sqrt{{x}^{2}d+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) ac{d}^{4}e+240\,\ln \left ( 1/2\,{\frac{2\,\sqrt{{x}^{2}d+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) a{d}^{3}{e}^{3}+720\,\ln \left ( 1/2\,{\frac{2\,\sqrt{{x}^{2}d+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) b{c}^{2}{d}^{3}e-600\,\ln \left ( 1/2\,{\frac{2\,\sqrt{{x}^{2}d+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) bc{d}^{2}{e}^{3}+105\,\ln \left ( 1/2\,{\frac{2\,\sqrt{{x}^{2}d+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ) bd{e}^{5} \right ){d}^{-{\frac{11}{2}}}} \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 \sqrt{d x^{2} + e x + c} \sqrt{{\left (b x^{2} + a\right )}^{2}} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05082, size = 1118, normalized size = 3.62 \begin{align*} \left [\frac{15 \,{\left (7 \, b e^{5} - 8 \,{\left (5 \, b c d - 2 \, a d^{2}\right )} e^{3} + 16 \,{\left (3 \, b c^{2} d^{2} - 4 \, a c d^{3}\right )} e\right )} \sqrt{d} \log \left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, \sqrt{d x^{2} + e x + c}{\left (2 \, d x + e\right )} \sqrt{d} + 4 \, c d + e^{2}\right ) + 4 \,{\left (384 \, b d^{5} x^{4} + 48 \, b d^{4} e x^{3} - 256 \, b c^{2} d^{3} + 640 \, a c d^{4} - 105 \, b d e^{4} + 20 \,{\left (23 \, b c d^{2} - 12 \, a d^{3}\right )} e^{2} + 8 \,{\left (16 \, b c d^{4} + 80 \, a d^{5} - 7 \, b d^{3} e^{2}\right )} x^{2} + 2 \,{\left (35 \, b d^{2} e^{3} - 4 \,{\left (29 \, b c d^{3} - 20 \, a d^{4}\right )} e\right )} x\right )} \sqrt{d x^{2} + e x + c}}{7680 \, d^{5}}, -\frac{15 \,{\left (7 \, b e^{5} - 8 \,{\left (5 \, b c d - 2 \, a d^{2}\right )} e^{3} + 16 \,{\left (3 \, b c^{2} d^{2} - 4 \, a c d^{3}\right )} e\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{d x^{2} + e x + c}{\left (2 \, d x + e\right )} \sqrt{-d}}{2 \,{\left (d^{2} x^{2} + d e x + c d\right )}}\right ) - 2 \,{\left (384 \, b d^{5} x^{4} + 48 \, b d^{4} e x^{3} - 256 \, b c^{2} d^{3} + 640 \, a c d^{4} - 105 \, b d e^{4} + 20 \,{\left (23 \, b c d^{2} - 12 \, a d^{3}\right )} e^{2} + 8 \,{\left (16 \, b c d^{4} + 80 \, a d^{5} - 7 \, b d^{3} e^{2}\right )} x^{2} + 2 \,{\left (35 \, b d^{2} e^{3} - 4 \,{\left (29 \, b c d^{3} - 20 \, a d^{4}\right )} e\right )} x\right )} \sqrt{d x^{2} + e x + c}}{3840 \, d^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.1943, size = 486, normalized size = 1.57 \begin{align*} \frac{1}{1920} \, \sqrt{d x^{2} + x e + c}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, b x \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{b e \mathrm{sgn}\left (b x^{2} + a\right )}{d}\right )} x + \frac{16 \, b c d^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 80 \, a d^{4} \mathrm{sgn}\left (b x^{2} + a\right ) - 7 \, b d^{2} e^{2} \mathrm{sgn}\left (b x^{2} + a\right )}{d^{4}}\right )} x - \frac{116 \, b c d^{2} e \mathrm{sgn}\left (b x^{2} + a\right ) - 80 \, a d^{3} e \mathrm{sgn}\left (b x^{2} + a\right ) - 35 \, b d e^{3} \mathrm{sgn}\left (b x^{2} + a\right )}{d^{4}}\right )} x - \frac{256 \, b c^{2} d^{2} \mathrm{sgn}\left (b x^{2} + a\right ) - 640 \, a c d^{3} \mathrm{sgn}\left (b x^{2} + a\right ) - 460 \, b c d e^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 240 \, a d^{2} e^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 105 \, b e^{4} \mathrm{sgn}\left (b x^{2} + a\right )}{d^{4}}\right )} - \frac{{\left (48 \, b c^{2} d^{2} e \mathrm{sgn}\left (b x^{2} + a\right ) - 64 \, a c d^{3} e \mathrm{sgn}\left (b x^{2} + a\right ) - 40 \, b c d e^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 16 \, a d^{2} e^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 7 \, b e^{5} \mathrm{sgn}\left (b x^{2} + a\right )\right )} \log \left ({\left | -2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + x e + c}\right )} \sqrt{d} - e \right |}\right )}{256 \, d^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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