Optimal. Leaf size=492 \[ \frac{5 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 e^6 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{96 e^3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{5 \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{64 e^5 (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{5 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^6}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4} \]
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Rubi [A] time = 0.734763, antiderivative size = 492, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {732, 810, 812, 843, 621, 206, 724} \[ \frac{5 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 e^6 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{96 e^3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{5 \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{64 e^5 (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{5 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^6}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4} \]
Antiderivative was successfully verified.
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Rule 732
Rule 810
Rule 812
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx &=-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}+\frac{5 \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx}{8 e}\\ &=-\frac{5 \left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac{5 \int \frac{\left (\frac{1}{2} \left (12 b^2 c d e+16 a c^2 d e+b^3 e^2-4 b c \left (4 c d^2+5 a e^2\right )\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^2} \, dx}{32 e^3 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{5 \left (64 c^3 d^3+b^3 e^3+4 b c e^2 (4 b d-5 a e)-16 c^2 d e (5 b d-4 a e)+2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{64 e^5 \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac{5 \left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}+\frac{5 \int \frac{\frac{1}{2} \left (-16 b^3 c d e^2-b^4 e^3-64 b c^2 d \left (c d^2+2 a e^2\right )+16 a c^2 e \left (4 c d^2+3 a e^2\right )+8 b^2 c e \left (10 c d^2+3 a e^2\right )\right )-32 c^2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{64 e^5 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{5 \left (64 c^3 d^3+b^3 e^3+4 b c e^2 (4 b d-5 a e)-16 c^2 d e (5 b d-4 a e)+2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{64 e^5 \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac{5 \left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac{\left (5 c^2 (2 c d-b e)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 e^6}+\frac{\left (5 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{128 e^6 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{5 \left (64 c^3 d^3+b^3 e^3+4 b c e^2 (4 b d-5 a e)-16 c^2 d e (5 b d-4 a e)+2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{64 e^5 \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac{5 \left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac{\left (5 c^2 (2 c d-b e)\right ) \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 )}{e^6}-\frac{\left (5 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right )\right ) \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}{\sqrt{a+b x+c x^2}}\right )}{64 e^6 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{5 \left (64 c^3 d^3+b^3 e^3+4 b c e^2 (4 b d-5 a e)-16 c^2 d e (5 b d-4 a e)+2 c e \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{64 e^5 \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac{5 \left (16 c^2 d^3-b e^2 (b d-4 a e)-4 c d e (3 b d-a e)+3 e \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{96 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4}-\frac{5 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^6}+\frac{5 \left (128 c^4 d^4-b^4 e^4-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)+48 c^2 e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{128 e^6 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 3.9276, size = 593, normalized size = 1.21 \[ \frac{\frac{2 e \sqrt{a+x (b+c x)} \left (2 c e^2 \left (-4 a^2 e^2 \left (11 d^2+20 d e x+27 e^2 x^2\right )-2 a b e \left (169 d^2 e x+45 d^3+191 d e^2 x^2+139 e^3 x^3\right )+b^2 d \left (435 d^2 e x+120 d^3+566 d e^2 x^2+323 e^3 x^3\right )\right )+e^3 \left (8 a^2 b e^2 (d-17 e x)-48 a^3 e^3+2 a b^2 e \left (5 d^2+18 d e x-59 e^2 x^2\right )+b^3 \left (55 d^2 e x+15 d^3+73 d e^2 x^2-15 e^3 x^3\right )\right )-8 c^2 e \left (b d \left (665 d^2 e^2 x^2+530 d^3 e x+150 d^4+327 d e^3 x^3+24 e^4 x^4\right )-a e \left (448 d^2 e^2 x^2+355 d^3 e x+100 d^4+235 d e^3 x^3+24 e^4 x^4\right )\right )+16 c^3 d^2 \left (260 d^2 e^2 x^2+210 d^3 e x+60 d^4+125 d e^3 x^3+12 e^4 x^4\right )\right )}{(d+e x)^4 \left (e (a e-b d)+c d^2\right )}-\frac{15 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}-960 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{384 e^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.252, size = 28635, normalized size = 58.2 \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(-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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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