Optimal. Leaf size=354 \[ -\frac{2 \left (-c x \left (2 c^2 \left (-16 a^2 j-6 a b i+b^2 h\right )+b^2 c (28 a j+b i)-c^3 (8 b g-8 a h)-4 b^4 j+16 c^4 f\right )-4 b c^2 \left (8 a^2 j+a c h+2 c^2 f\right )+24 a^2 c^3 i+2 b^2 c^2 (2 c g-3 a i)-b^3 c (c h-10 a j)+b^4 c i+b^5 (-j)\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{2 \left (-x \left (c^2 \left (2 a^2 j+3 a b i+b^2 h\right )-b^2 c (4 a j+b i)-c^3 (2 a h+b g)+b^4 j+2 c^4 f\right )-b c \left (-3 a^2 j+a c h+c^2 f\right )+a b^2 c i-a b^3 j+2 a c^2 (c g-a i)\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{j \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]
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Rubi [A] time = 0.377244, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {1660, 12, 621, 206} \[ -\frac{2 \left (-c x \left (2 c^2 \left (-16 a^2 j-6 a b i+b^2 h\right )+b^2 c (28 a j+b i)-c^3 (8 b g-8 a h)-4 b^4 j+16 c^4 f\right )-4 b c^2 \left (8 a^2 j+a c h+2 c^2 f\right )+24 a^2 c^3 i+2 b^2 c^2 (2 c g-3 a i)-b^3 c (c h-10 a j)+b^4 c i+b^5 (-j)\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{2 \left (-x \left (c^2 \left (2 a^2 j+3 a b i+b^2 h\right )-b^2 c (4 a j+b i)-c^3 (2 a h+b g)+b^4 j+2 c^4 f\right )-b c \left (-3 a^2 j+a c h+c^2 f\right )+a b^2 c i-a b^3 j+2 a c^2 (c g-a i)\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{j \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{f+g x+h x^2+365 x^3+j x^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 \left (c^3 \left (b f+\frac{a^2 (730 c-3 b j)}{c^2}-\frac{a \left (365 b^2 c+2 c^3 g-b c^2 h-b^3 j\right )}{c^3}\right )-\left (365 b^3 c-b c^2 (1095 a-c g)-b^4 j-b^2 c (c h-4 a j)-2 c^2 \left (c^2 f-a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{-\frac{365 b^3 c+4 b c^3 g-b^4 j-b^2 c (c h-a j)-4 c^2 \left (2 c^2 f+a c h-a^2 j\right )}{2 c^3}-\frac{3 \left (b^2-4 a c\right ) (365 c-b j) x}{2 c^2}+\frac{3}{2} \left (4 a-\frac{b^2}{c}\right ) j x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 \left (c^3 \left (b f+\frac{a^2 (730 c-3 b j)}{c^2}-\frac{a \left (365 b^2 c+2 c^3 g-b c^2 h-b^3 j\right )}{c^3}\right )-\left (365 b^3 c-b c^2 (1095 a-c g)-b^4 j-b^2 c (c h-4 a j)-2 c^2 \left (c^2 f-a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (365 b^4 c+8760 a^2 c^3-b^2 \left (2190 a c^2-4 c^3 g\right )-b^5 j-b^3 c (c h-10 a j)-4 b c^2 \left (2 c^2 f+a c h+8 a^2 j\right )-c \left (365 b^3 c-4 b c^2 (1095 a+2 c g)-4 b^4 j+2 b^2 c (c h+14 a j)+8 c^2 \left (2 c^2 f+a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{4 \int \frac{3 \left (b^2-4 a c\right )^2 j}{4 c^2 \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}\\ &=-\frac{2 \left (c^3 \left (b f+\frac{a^2 (730 c-3 b j)}{c^2}-\frac{a \left (365 b^2 c+2 c^3 g-b c^2 h-b^3 j\right )}{c^3}\right )-\left (365 b^3 c-b c^2 (1095 a-c g)-b^4 j-b^2 c (c h-4 a j)-2 c^2 \left (c^2 f-a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (365 b^4 c+8760 a^2 c^3-b^2 \left (2190 a c^2-4 c^3 g\right )-b^5 j-b^3 c (c h-10 a j)-4 b c^2 \left (2 c^2 f+a c h+8 a^2 j\right )-c \left (365 b^3 c-4 b c^2 (1095 a+2 c g)-4 b^4 j+2 b^2 c (c h+14 a j)+8 c^2 \left (2 c^2 f+a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{j \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{c^2}\\ &=-\frac{2 \left (c^3 \left (b f+\frac{a^2 (730 c-3 b j)}{c^2}-\frac{a \left (365 b^2 c+2 c^3 g-b c^2 h-b^3 j\right )}{c^3}\right )-\left (365 b^3 c-b c^2 (1095 a-c g)-b^4 j-b^2 c (c h-4 a j)-2 c^2 \left (c^2 f-a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (365 b^4 c+8760 a^2 c^3-b^2 \left (2190 a c^2-4 c^3 g\right )-b^5 j-b^3 c (c h-10 a j)-4 b c^2 \left (2 c^2 f+a c h+8 a^2 j\right )-c \left (365 b^3 c-4 b c^2 (1095 a+2 c g)-4 b^4 j+2 b^2 c (c h+14 a j)+8 c^2 \left (2 c^2 f+a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{(2 j) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{c^2}\\ &=-\frac{2 \left (c^3 \left (b f+\frac{a^2 (730 c-3 b j)}{c^2}-\frac{a \left (365 b^2 c+2 c^3 g-b c^2 h-b^3 j\right )}{c^3}\right )-\left (365 b^3 c-b c^2 (1095 a-c g)-b^4 j-b^2 c (c h-4 a j)-2 c^2 \left (c^2 f-a c h+a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (365 b^4 c+8760 a^2 c^3-b^2 \left (2190 a c^2-4 c^3 g\right )-b^5 j-b^3 c (c h-10 a j)-4 b c^2 \left (2 c^2 f+a c h+8 a^2 j\right )-c \left (365 b^3 c-4 b c^2 (1095 a+2 c g)-4 b^4 j+2 b^2 c (c h+14 a j)+8 c^2 \left (2 c^2 f+a c h-4 a^2 j\right )\right ) x\right )}{3 c^3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{j \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.23512, size = 316, normalized size = 0.89 \[ \frac{-\frac{2 \left (b c \left (-3 a^2 j+a c (h+3 i x)+c^2 (f-g x)\right )+2 c^2 \left (a^2 (i+j x)-a c (g+h x)+c^2 f x\right )+b^2 c (c h x-a (i+4 j x))+b^3 (a j-c i x)+b^4 j x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}}+\frac{2 \left (4 b c^2 \left (8 a^2 j+a c (h-3 i x)+2 c^2 (f-g x)\right )+8 c^3 \left (a^2 (-(3 i+4 j x))+a c h x+2 c^2 f x\right )+2 b^2 c^2 (3 a i+14 a j x-2 c g+c h x)+b^3 c (c (h+i x)-10 a j)-b^4 c (i+4 j x)+b^5 j\right )}{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)}}+3 \sqrt{c} j \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{3 c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 1406, normalized size = 4. \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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(-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 = 1.16615, size = 628, normalized size = 1.77 \begin{align*} \frac{2 \,{\left ({\left ({\left (\frac{{\left (16 \, c^{5} f - 8 \, b c^{4} g + 2 \, b^{2} c^{3} h + 8 \, a c^{4} h + b^{3} c^{2} i - 12 \, a b c^{3} i - 4 \, b^{4} c j + 28 \, a b^{2} c^{2} j - 32 \, a^{2} c^{3} j\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (8 \, b c^{4} f - 4 \, b^{2} c^{3} g + b^{3} c^{2} h + 4 \, a b c^{3} h - 2 \, a b^{2} c^{2} i - 8 \, a^{2} c^{3} i - b^{5} j + 6 \, a b^{3} c j\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (2 \, b^{2} c^{3} f + 8 \, a c^{4} f - b^{3} c^{2} g - 4 \, a b c^{3} g + 4 \, a b^{2} c^{2} h - 8 \, a^{2} b c^{2} i - 2 \, a b^{4} j + 14 \, a^{2} b^{2} c j - 8 \, a^{3} c^{2} j\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} c^{2} f - 12 \, a b c^{3} f + 2 \, a b^{2} c^{2} g + 8 \, a^{2} c^{3} g - 8 \, a^{2} b c^{2} h + 16 \, a^{3} c^{2} i + 3 \, a^{2} b^{3} j - 20 \, a^{3} b c j}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} - \frac{j \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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