Optimal. Leaf size=491 \[ \frac {\left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}{e^2 (e f-d g)}-\frac {\left (4 c e f^2-g (5 b e f-b d g-4 a e g)-2 c g (e f-d g) x\right ) \sqrt {a+b x+c x^2}}{4 e g^2 (e f-d g)}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} e^3 (e f-d g)}+\frac {\left (8 c^2 e f^3+b g^2 (3 b e f+b d g-4 a e g)-4 c g \left (3 b e f^2-a g (3 e f-d g)\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} e g^3 (e f-d g)}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^3 (e f-d g)}-\frac {\left (c f^2-b f g+a g^2\right )^{3/2} \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{g^3 (e f-d g)} \]
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Rubi [A]
time = 0.52, antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {909, 748, 857,
635, 212, 738, 828} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c g \left (3 b e f^2-a g (3 e f-d g)\right )+b g^2 (-4 a e g+b d g+3 b e f)+8 c^2 e f^3\right )}{8 \sqrt {c} e g^3 (e f-d g)}+\frac {\sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{e^2 (e f-d g)}+\frac {\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 )}{e^3 (e f-d g)}-\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} e^3 (e f-d g)}-\frac {\sqrt {a+b x+c x^2} \left (-g (-4 a e g-b d g+5 b e f)-2 c g x (e f-d g)+4 c e f^2\right )}{4 e g^2 (e f-d g)}-\frac {\left (a g^2-b f g+c f^2\right )^{3/2} \tanh ^{-1}\left (\frac {-2 a g+x (2 c f-b g)+b f}{2 \sqrt {a+b x+c x^2} \sqrt {a g^2-b f g+c f^2}}\right )}{g^3 (e f-d g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 748
Rule 828
Rule 857
Rule 909
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(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) \sqrt {a+b x+c x^2}}{f+g x} \, dx}{e (e f-d g)}+\frac {\left (c d^2-b d e+a e^2\right ) \int \frac {\sqrt {a+b x+c x^2}}{d+e x} \, dx}{e (e f-d g)}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}{e^2 (e f-d g)}-\frac {\left (4 c e f^2-g (5 b e f-b d g-4 a e g)-2 c g (e f-d g) x\right ) \sqrt {a+b x+c x^2}}{4 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}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2 e^2 (e f-d g)}+\frac {\int \frac {\frac {1}{2} c \left (f \left (4 b c f-b^2 g-4 a c g\right ) (e f-d g)+4 g (b f-2 a g) (c d f-b e f+a e g)\right )+\frac {1}{2} c \left (8 c^2 e f^3+b g^2 (3 b e f+b d g-4 a e g)-4 c g \left (3 b e f^2-a g (3 e f-d g)\right )\right ) x}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{4 c e g^2 (e f-d g)}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}{e^2 (e f-d g)}-\frac {\left (4 c e f^2-g (5 b e f-b d g-4 a e g)-2 c g (e f-d g) x\right ) \sqrt {a+b x+c x^2}}{4 e g^2 (e f-d g)}-\frac {\left ((2 c d-b e) \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 e^3 (e f-d g)}+\frac {\left (c d^2-b d e+a e^2\right )^2 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^3 (e f-d g)}-\frac {\left (c f^2-b f g+a g^2\right )^2 \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{g^3 (e f-d g)}+\frac {\left (8 c^2 e f^3+b g^2 (3 b e f+b d g-4 a e g)-4 c g \left (3 b e f^2-a g (3 e f-d g)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 e g^3 (e f-d g)}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}{e^2 (e f-d g)}-\frac {\left (4 c e f^2-g (5 b e f-b d g-4 a e g)-2 c g (e f-d g) x\right ) \sqrt {a+b x+c x^2}}{4 e g^2 (e f-d g)}-\frac {\left ((2 c d-b e) \left (c d^2-b d e+a e^2\right )\right ) \text {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^3 (e f-d g)}-\frac {\left (2 \left (c d^2-b d e+a e^2\right )^2\right ) \text {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^3 (e f-d g)}+\frac {\left (2 \left (c f^2-b f g+a g^2\right )^2\right ) \text {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^3 (e f-d g)}+\frac {\left (8 c^2 e f^3+b g^2 (3 b e f+b d g-4 a e g)-4 c g \left (3 b e f^2-a g (3 e f-d g)\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{4 e g^3 (e f-d g)}\\ &=\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}{e^2 (e f-d g)}-\frac {\left (4 c e f^2-g (5 b e f-b d g-4 a e g)-2 c g (e f-d g) x\right ) \sqrt {a+b x+c x^2}}{4 e g^2 (e f-d g)}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} e^3 (e f-d g)}+\frac {\left (8 c^2 e f^3+b g^2 (3 b e f+b d g-4 a e g)-4 c g \left (3 b e f^2-a g (3 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 )}{8 \sqrt {c} e g^3 (e f-d g)}+\frac {\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 )}{e^3 (e f-d g)}-\frac {\left (c f^2-b f g+a g^2\right )^{3/2} \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^3 (e f-d g)}\\ \end {align*}
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Mathematica [A]
time = 10.75, size = 323, normalized size = 0.66 \begin {gather*} \frac {\frac {\left (3 b^2 e^2 g^2-12 c e g (b e f+b d g-a e g)+8 c^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}+\frac {2 \left (e g (e f-d g) \sqrt {a+x (b+c x)} (5 b e g+c (-4 e f-4 d g+2 e g x))-4 \left (c d^2+e (-b d+a e)\right )^{3/2} g^3 \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )+4 e^3 \left (c f^2+g (-b f+a g)\right )^{3/2} \tanh ^{-1}\left (\frac {-b f+2 a g-2 c f x+b g x}{2 \sqrt {c f^2+g (-b f+a g)} \sqrt {a+x (b+c x)}}\right )\right )}{e f-d g}}{8 e^3 g^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1264\) vs.
\(2(449)=898\).
time = 0.19, size = 1265, normalized size = 2.58
method | result | size |
default | \(\text {Expression too large to display}\) | \(1265\) |
risch | \(\text {Expression too large to display}\) | \(2348\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right ) \left (f + g x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{\left (f+g\,x\right )\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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