Optimal. Leaf size=717 \[ -\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (24 a b c f-180 a c^2 d+9 b^2 c d-4 b^3 f\right )+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{x \sqrt{a+b x^2+c x^4} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right )}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{x \sqrt{a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d+9 b^2 c d-4 b^3 f\right )}{315 c^2}-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{256 c^3}+\frac{3 \left (b^2-4 a c\right )^2 (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac{\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} (2 c e-b g)}{32 c^2}+\frac{x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c} \]
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Rubi [A] time = 0.595828, antiderivative size = 717, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1673, 1176, 1197, 1103, 1195, 1247, 640, 612, 621, 206} \[ -\frac{x \sqrt{a+b x^2+c x^4} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right )}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (24 a b c f-180 a c^2 d+9 b^2 c d-4 b^3 f\right )+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d+18 b^3 c d-8 b^4 f\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{x \sqrt{a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d+9 b^2 c d-4 b^3 f\right )}{315 c^2}-\frac{3 \left (b^2-4 a c\right ) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4} (2 c e-b g)}{256 c^3}+\frac{3 \left (b^2-4 a c\right )^2 (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac{\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} (2 c e-b g)}{32 c^2}+\frac{x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c} \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1176
Rule 1197
Rule 1103
Rule 1195
Rule 1247
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx &=\int \left (d+f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx+\int x \left (e+g x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx\\ &=\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{1}{2} \operatorname{Subst}\left (\int (e+g x) \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )+\frac{\int \left (a (18 c d-b f)+\left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4} \, dx}{21 c}\\ &=\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{\int \frac{-a \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )+\left (-18 b^3 c d+144 a b c^2 d+8 b^4 f-57 a b^2 c f+84 a^2 c^2 f\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{315 c^2}+\frac{(2 c e-b g) \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{4 c}\\ &=\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{(2 c e-b g) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{\left (\sqrt{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{315 c^{5/2}}-\frac{\left (\sqrt{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )\right )\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{315 c^{5/2}}-\frac{\left (3 \left (b^2-4 a c\right ) (2 c e-b g)\right ) \operatorname{Subst}\left (\int \sqrt{a+b x+c x^2} \, dx,x,x^2\right )}{64 c^2}\\ &=-\frac{\left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) x \sqrt{a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \left (b^2-4 a c\right ) (2 c e-b g) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^3}+\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{(2 c e-b g) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\left (3 \left (b^2-4 a c\right )^2 (2 c e-b g)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{512 c^3}\\ &=-\frac{\left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) x \sqrt{a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \left (b^2-4 a c\right ) (2 c e-b g) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^3}+\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{(2 c e-b g) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\left (3 \left (b^2-4 a c\right )^2 (2 c e-b g)\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 )}{256 c^3}\\ &=-\frac{\left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) x \sqrt{a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{3 \left (b^2-4 a c\right ) (2 c e-b g) \left (b+2 c x^2\right ) \sqrt{a+b x^2+c x^4}}{256 c^3}+\frac{x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{(2 c e-b g) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac{x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac{g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac{3 \left (b^2-4 a c\right )^2 (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f+\sqrt{a} \sqrt{c} \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [B] time = 0.095, size = 3038, normalized size = 4.2 \begin{align*} \text{output 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{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (g x^{3} + f x^{2} + e x + d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c g x^{7} + c f x^{6} +{\left (c e + b g\right )} x^{5} +{\left (c d + b f\right )} x^{4} +{\left (b e + a g\right )} x^{3} + a e x +{\left (b d + a f\right )} x^{2} + a d\right )} \sqrt{c x^{4} + b x^{2} + a}, x\right ) \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 \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}} \left (d + e x + f x^{2} + g x^{3}\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{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (g x^{3} + f x^{2} + e x + d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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