Optimal. Leaf size=388 \[ \frac{5 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{3/2} e^6}-\frac{5 \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-4 a e)+4 b c e^2 (12 b d-11 a e)-b^3 e^3+64 c^3 d^3\right )}{64 c e^5}-\frac{5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^6}-\frac{5 \left (a+b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)} \]
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Rubi [A] time = 0.663218, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {732, 814, 843, 621, 206, 724} \[ \frac{5 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{3/2} e^6}-\frac{5 \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-4 a e)+4 b c e^2 (12 b d-11 a e)-b^3 e^3+64 c^3 d^3\right )}{64 c e^5}-\frac{5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^6}-\frac{5 \left (a+b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 732
Rule 814
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)^2} \, dx &=-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac{5 \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e}\\ &=-\frac{5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}-\frac{5 \int \frac{\left (c \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+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} \, dx}{16 c e^3}\\ &=-\frac{5 \left (64 c^3 d^3-b^3 e^3+4 b c e^2 (12 b d-11 a e)-16 c^2 d e (7 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 c e^5}-\frac{5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac{5 \int \frac{\frac{1}{2} c \left (d \left (4 b c d-b^2 e-4 a c e\right ) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right )+4 c e (b d-2 a e) \left (7 b^2 d e+4 a c d e-8 b \left (c d^2+a e^2\right )\right )\right )+\frac{1}{2} c \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 ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{64 c^2 e^5}\\ &=-\frac{5 \left (64 c^3 d^3-b^3 e^3+4 b c e^2 (12 b d-11 a e)-16 c^2 d e (7 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 c e^5}-\frac{5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}-\frac{\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2\right ) \int \frac{1}{(d+e x) \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}{\sqrt{a+b x+c x^2}} \, dx}{128 c e^6}\\ &=-\frac{5 \left (64 c^3 d^3-b^3 e^3+4 b c e^2 (12 b d-11 a e)-16 c^2 d e (7 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 c e^5}-\frac{5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac{\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2\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 )}{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-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{64 c e^6}\\ &=-\frac{5 \left (64 c^3 d^3-b^3 e^3+4 b c e^2 (12 b d-11 a e)-16 c^2 d e (7 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 c e^5}-\frac{5 (8 c d-7 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)}+\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+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{3/2} e^6}-\frac{5 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^{3/2} \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 )}{2 e^6}\\ \end{align*}
Mathematica [A] time = 1.17093, size = 370, normalized size = 0.95 \[ \frac{5 \left (\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )+2 \sqrt{c} \left (e \sqrt{a+x (b+c x)} \left (8 c^2 e (a e (3 e x-8 d)+2 b d (7 d-2 e x))+2 b c e^2 (22 a e-24 b d+b e x)+b^3 e^3+32 c^3 d^2 (e x-2 d)\right )+32 c (2 c d-b e) \left (e (a e-b d)+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )\right )\right )}{128 c^{3/2} e^6}+\frac{5 (a+x (b+c x))^{3/2} (7 b e-8 c d+6 c e x)}{24 e^3}-\frac{(a+x (b+c x))^{5/2}}{e (d+e x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.233, size = 6711, normalized size = 17.3 \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(-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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