Optimal. Leaf size=886 \[ \frac{\tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )^{5/2}}{e^5 (e f-d g)}+\frac{\left (c x^2+b x+a\right )^{3/2} \left (c d^2-b e d+a e^2\right )}{3 e^2 (e f-d g)}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )}{16 c^{3/2} e^5 (e f-d g)}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{c x^2+b x+a} \left (c d^2-b e d+a e^2\right )}{8 c e^4 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{128 c^{3/2} e g^5 (e f-d g)}-\frac{\left (c f^2-b g f+a g^2\right )^{5/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{g^5 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{64 c e g^4 (e f-d g)} \]
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Rubi [A] time = 1.80028, antiderivative size = 886, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {895, 734, 814, 843, 621, 206, 724} \[ \frac{\tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )^{5/2}}{e^5 (e f-d g)}+\frac{\left (c x^2+b x+a\right )^{3/2} \left (c d^2-b e d+a e^2\right )}{3 e^2 (e f-d g)}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )}{16 c^{3/2} e^5 (e f-d g)}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{c x^2+b x+a} \left (c d^2-b e d+a e^2\right )}{8 c e^4 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{128 c^{3/2} e g^5 (e f-d g)}-\frac{\left (c f^2-b g f+a g^2\right )^{5/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{g^5 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{64 c e g^4 (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 895
Rule 734
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) (f+g x)} \, dx &=-\frac{\int \frac{(c d f-b e f+a e g-c (e f-d g) x) \left (a+b x+c x^2\right )^{3/2}}{f+g x} \, dx}{e (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{e (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}-\frac{\left (c d^2-b d e+a e^2\right ) \int \frac{(b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{d+e x} \, dx}{2 e^2 (e f-d g)}+\frac{\int \frac{\left (\frac{1}{2} c \left (f \left (8 b c f-3 b^2 g-4 a c g\right ) (e f-d g)+8 g (b f-2 a g) (c d f-b e f+a e g)\right )+\frac{1}{2} c \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{f+g x} \, dx}{8 c e g^2 (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{8 c e^4 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e f^2 g (9 b f-8 a g)-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c e g^4 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \int \frac{\frac{1}{2} \left (4 c e (b d-2 a e)^2-d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )-\frac{1}{2} (2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{8 c e^4 (e f-d g)}-\frac{\int \frac{\frac{1}{4} c \left (4 c g (b f-2 a g) \left (f \left (8 b c f-3 b^2 g-4 a c g\right ) (e f-d g)+8 g (b f-2 a g) (c d f-b e f+a e g)\right )-f \left (4 b c f-b^2 g-4 a c g\right ) \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right )\right )-\frac{1}{4} c \left (128 c^4 e f^5-320 c^3 e f^3 g (b f-a g)-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e f^2 g+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) x}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{32 c^2 e g^4 (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{8 c e^4 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e f^2 g (9 b f-8 a g)-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c e g^4 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right )^3 \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e^5 (e f-d g)}-\frac{\left ((2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16 c e^5 (e f-d g)}-\frac{\left (c f^2-b f g+a g^2\right )^3 \int \frac{1}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{g^5 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e f^3 g (b f-a g)-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e f^2 g+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{128 c e g^5 (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{8 c e^4 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e f^2 g (9 b f-8 a g)-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c e g^4 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}-\frac{\left (2 \left (c d^2-b d e+a e^2\right )^3\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^5 (e f-d g)}-\frac{\left ((2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 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 )}{8 c e^5 (e f-d g)}+\frac{\left (2 \left (c f^2-b f g+a g^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac{-b f+2 a g-(2 c f-b g) x}{\sqrt{a+b x+c x^2}}\right )}{g^5 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e f^3 g (b f-a g)-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e f^2 g+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\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 g^5 (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{8 c e^4 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e f^2 g (9 b f-8 a g)-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c e g^4 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} e^5 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e f^3 g (b f-a g)-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e f^2 g+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\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 g^5 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right )^{5/2} \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 )}{e^5 (e f-d g)}-\frac{\left (c f^2-b f g+a g^2\right )^{5/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b f g+a g^2} \sqrt{a+b x+c x^2}}\right )}{g^5 (e f-d g)}\\ \end{align*}
Mathematica [A] time = 2.69505, size = 647, normalized size = 0.73 \[ \frac{3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (240 c^2 e^2 g^2 (e f-d g) \left (a^2 e^2 g^2-2 a b e g (d g+e f)+b^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )-40 b^2 c e^3 g^3 (e f-d g) (-3 a e g+b d g+b e f)+320 c^3 e g \left (-a d^3 e g^4+a e^4 f^3 g+b d^4 g^4-b e^4 f^4\right )+5 b^4 e^4 g^4 (d g-e f)+128 c^4 \left (e^5 f^5-d^5 g^5\right )\right )+2 \sqrt{c} \left (-e g \sqrt{a+x (b+c x)} (d g-e f) \left (8 c^2 e g \left (a e g (-56 d g-56 e f+27 e g x)+b \left (54 d^2 g^2+2 d e g (27 f-13 g x)+e^2 \left (54 f^2-26 f g x+17 g^2 x^2\right )\right )\right )+2 b c e^2 g^2 (278 a e g+b (-132 d g-132 e f+59 e g x))+15 b^3 e^3 g^3-16 c^3 \left (-6 d^2 e g^2 (g x-2 f)+12 d^3 g^3+2 d e^2 g \left (6 f^2-3 f g x+2 g^2 x^2\right )+e^3 \left (-6 f^2 g x+12 f^3+4 f g^2 x^2-3 g^3 x^3\right )\right )\right )-192 c g^5 \left (e (a e-b d)+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )+192 c e^5 \left (g (a g-b f)+c f^2\right )^{5/2} \tanh ^{-1}\left (\frac{2 a g-b f+b g x-2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right )\right )}{384 c^{3/2} e^5 g^5 (e f-d g)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.372, size = 9052, normalized size = 10.2 \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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