Optimal. Leaf size=473 \[ \frac{5 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^7 \sqrt{a e^2-b d e+c d^2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{12 e^4 (d+e x)^2}+\frac{5 \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{8 e^6 (d+e x)}-\frac{5 \sqrt{c} (2 c d-b e) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^7}+\frac{\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3} \]
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Rubi [A] time = 0.746819, antiderivative size = 473, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {812, 843, 621, 206, 724} \[ \frac{5 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^7 \sqrt{a e^2-b d e+c d^2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{12 e^4 (d+e x)^2}+\frac{5 \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{8 e^6 (d+e x)}-\frac{5 \sqrt{c} (2 c d-b e) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^7}+\frac{\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 812
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\frac{(4 c d-b e+2 c e x) \left (a+b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac{5 \int \frac{\left (3 \left (4 b c d-b^2 e-4 a c e\right )+12 c (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx}{18 e^2}\\ &=-\frac{5 \left (16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(4 c d-b e+2 c e x) \left (a+b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac{5 \int \frac{\left (-6 \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+3 a e^2\right )\right )+12 c \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^2} \, dx}{48 e^4}\\ &=\frac{5 \left (64 c^3 d^3-b^3 e^3-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)+2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{8 e^6 (d+e x)}-\frac{5 \left (16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(4 c d-b e+2 c e x) \left (a+b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac{5 \int \frac{6 \left (24 b^3 c d e^2-b^4 e^3-16 a c^2 e \left (4 c d^2+a e^2\right )+32 b c^2 d \left (2 c d^2+3 a e^2\right )-8 b^2 c e \left (10 c d^2+3 a e^2\right )\right )+48 c (2 c d-b e) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{96 e^6}\\ &=\frac{5 \left (64 c^3 d^3-b^3 e^3-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)+2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{8 e^6 (d+e x)}-\frac{5 \left (16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(4 c d-b e+2 c e x) \left (a+b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac{\left (5 c (2 c d-b e) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 e^7}+\frac{\left (5 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 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}{16 e^7}\\ &=\frac{5 \left (64 c^3 d^3-b^3 e^3-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)+2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{8 e^6 (d+e x)}-\frac{5 \left (16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(4 c d-b e+2 c e x) \left (a+b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac{\left (5 c (2 c d-b e) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right )\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^7}-\frac{\left (5 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 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 )}{8 e^7}\\ &=\frac{5 \left (64 c^3 d^3-b^3 e^3-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)+2 c e \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{8 e^6 (d+e x)}-\frac{5 \left (16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(4 c d-b e+2 c e x) \left (a+b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac{5 \sqrt{c} (2 c d-b e) \left (8 c^2 d^2+b^2 e^2-4 c e (2 b d-a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^7}+\frac{5 \left (128 c^4 d^4+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+16 c^2 e^2 \left (10 b^2 d^2-8 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 )}{16 e^7 \sqrt{c d^2-b d e+a e^2}}\\ \end{align*}
Mathematica [A] time = 2.73703, size = 823, normalized size = 1.74 \[ \frac{-120 \sqrt{c} (2 c d-b e) \left (8 c^3 d^4-4 c^2 e (4 b d-3 a e) d^2+c e^2 (3 b d-2 a e)^2+b^2 e^3 (a e-b d)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) (d+e x)^3-15 \sqrt{c d^2+e (a e-b d)} \left (128 c^4 d^4-128 c^3 e (2 b d-a e) d^2+b^4 e^4-8 b^2 c e^3 (4 b d-3 a e)+16 c^2 e^2 \left (10 b^2 d^2-8 a b e d+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{-b d-2 c x d+2 a e+b e x}{2 \sqrt{c d^2+e (a e-b d)} \sqrt{a+x (b+c x)}}\right ) (d+e x)^3+2 e \sqrt{a+x (b+c x)} \left (16 d^2 \left (60 d^5+150 e x d^4+110 e^2 x^2 d^3+15 e^3 x^3 d^2-3 e^4 x^4 d+e^5 x^5\right ) c^4+8 e \left (a e \left (160 d^5+405 e x d^4+303 e^2 x^2 d^3+44 e^3 x^3 d^2-6 e^4 x^4 d+2 e^5 x^5\right )-b d \left (270 d^5+680 e x d^4+505 e^2 x^2 d^3+72 e^3 x^3 d^2-14 e^4 x^4 d+2 e^5 x^5\right )\right ) c^3+2 e^2 \left (d \left (780 d^4+1985 e x d^3+1501 e^2 x^2 d^2+228 e^3 x^3 d-32 e^4 x^4\right ) b^2-2 a e \left (395 d^4+1014 e x d^3+777 e^2 x^2 d^2+112 e^3 x^3 d-16 e^4 x^4\right ) b+4 a^2 e^2 \left (39 d^3+102 e x d^2+83 e^2 x^2 d+14 e^3 x^3\right )\right ) c^2-e^3 \left (5 d \left (75 d^3+194 e x d^2+151 e^2 x^2 d+24 e^3 x^3\right ) b^3-2 a e \left (205 d^3+540 e x d^2+443 e^2 x^2 d+60 e^3 x^3\right ) b^2+4 a^2 e^2 \left (15 d^2+38 e x d+41 e^2 x^2\right ) b+8 a^3 e^3 (d+3 e x)\right ) c+b e^4 (b d-a e) \left (\left (15 d^2+40 e x d+33 e^2 x^2\right ) b^2+2 a e (5 d+13 e x) b+8 a^2 e^2\right )\right )}{48 e^7 \left (c d^2+e (a e-b d)\right ) (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 28593, normalized size = 60.5 \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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