Optimal. Leaf size=787 \[ \frac {\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} e^2 (e f-d g)^2}+\frac {e (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} g^3 (e f-d g)^2}-\frac {3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} g^3 (e f-d g)}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^2 (e f-d g)^2}+\frac {3 (2 c f-b g) \sqrt {c f^2-b f g+a g^2} \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{2 g^3 (e f-d g)}-\frac {e \left (c f^2-b f g+a g^2\right )^{3/2} \tanh ^{-1}\left (\frac {b f-2 a g+(2 c f-b g) x}{2 \sqrt {c f^2-b f g+a g^2} \sqrt {a+b x+c x^2}}\right )}{g^3 (e f-d g)^2} \]
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Rubi [A]
time = 0.88, antiderivative size = 787, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {974, 748, 828,
857, 635, 212, 738, 746} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right )}{8 c e (e f-d g)^2}-\frac {e \sqrt {a+b x+c x^2} \left (-2 c g (5 b f-4 a g)+b^2 g^2-2 c g x (2 c f-b g)+8 c^2 f^2\right )}{8 c g^2 (e f-d g)^2}-\frac {3 \left (-4 c g (2 b f-a g)+b^2 g^2+8 c^2 f^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} g^3 (e f-d g)}-\frac {(2 c d-b e) \left (-4 c e (2 b d-3 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 )}{16 c^{3/2} e^2 (e f-d g)^2}+\frac {e (2 c f-b g) \left (-4 c g (2 b f-3 a g)-b^2 g^2+8 c^2 f^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} g^3 (e f-d g)^2}+\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^2 (e f-d g)^2}-\frac {e \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)^2}+\frac {3 (2 c f-b g) \sqrt {a g^2-b f g+c f^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 )}{2 g^3 (e f-d g)}+\frac {3 \sqrt {a+b x+c x^2} (-3 b g+4 c f-2 c g x)}{4 g^2 (e f-d g)}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(f+g x) (e f-d g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 748
Rule 828
Rule 857
Rule 974
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^2} \, dx &=\int \left (\frac {e^2 \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (d+e x)}-\frac {g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)^2}-\frac {e g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)}\right ) \, dx\\ &=\frac {e^2 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{(e f-d g)^2}-\frac {(e g) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{f+g x} \, dx}{(e f-d g)^2}-\frac {g \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(f+g x)^2} \, dx}{e f-d g}\\ &=\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}-\frac {e \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 f-d g)^2}+\frac {e \int \frac {(b f-2 a g+(2 c f-b g) x) \sqrt {a+b x+c x^2}}{f+g x} \, dx}{2 (e f-d g)^2}-\frac {3 \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{f+g x} \, dx}{2 (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 {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}+\frac {\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 (e f-d g)^2}-\frac {e \int \frac {\frac {1}{2} \left (4 c g (b f-2 a g)^2-f (2 c f-b g) \left (4 b c f-b^2 g-4 a c g\right )\right )-\frac {1}{2} (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) x}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{8 c g^2 (e f-d g)^2}+\frac {3 \int \frac {c \left (3 b^2 f g+4 a c f g-4 b \left (c f^2+a g^2\right )\right )-c \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) x}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{8 c g^2 (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 {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}+\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^2 (e f-d g)^2}-\frac {\left ((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 )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c e^2 (e f-d g)^2}+\frac {\left (3 (2 c f-b g) \left (c f^2-b f g+a g^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{2 g^3 (e f-d g)}-\frac {\left (e \left (c f^2-b f g+a g^2\right )^2\right ) \int \frac {1}{(f+g x) \sqrt {a+b x+c x^2}} \, dx}{g^3 (e f-d g)^2}+\frac {\left (e (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c g^3 (e f-d g)^2}-\frac {\left (3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 g^3 (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 {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}-\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^2 (e f-d g)^2}-\frac {\left ((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 )\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 )}{8 c e^2 (e f-d g)^2}-\frac {\left (3 (2 c f-b g) \left (c f^2-b f g+a g^2\right )\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 (2 e \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)^2}+\frac {\left (e (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a 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 )}{8 c g^3 (e f-d g)^2}-\frac {\left (3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a 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 g^3 (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 {a+b x+c x^2}}{8 c e (e f-d g)^2}+\frac {3 (4 c f-3 b g-2 c g x) \sqrt {a+b x+c x^2}}{4 g^2 (e f-d g)}-\frac {e \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt {a+b x+c x^2}}{8 c g^2 (e f-d g)^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)}-\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 {a+b x+c x^2}}\right )}{16 c^{3/2} e^2 (e f-d g)^2}+\frac {e (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} g^3 (e f-d g)^2}-\frac {3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} 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^2 (e f-d g)^2}+\frac {3 (2 c f-b g) \sqrt {c f^2-b f g+a g^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 )}{2 g^3 (e f-d g)}-\frac {e \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)^2}\\ \end {align*}
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Mathematica [A]
time = 10.98, size = 357, normalized size = 0.45 \begin {gather*} \frac {-\sqrt {c} (e f-d g)^2 (4 c e f+2 c d g-3 b e g) (f+g x) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \left (c d^2+e (-b d+a e)\right )^{3/2} g^3 (f+g x) \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 )+e \left (2 g (-e f+d g) \sqrt {a+x (b+c x)} (e g (b f-a g)+c d g (f+g x)-c e f (2 f+g x))-e \sqrt {c f^2+g (-b f+a g)} (2 c f (2 e f-3 d g)+g (-b e f+3 b d g-2 a e g)) (f+g x) \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 )}{2 e^2 g^3 (e f-d g)^2 (f+g x)} \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. \(2371\) vs.
\(2(717)=1434\).
time = 0.17, size = 2372, normalized size = 3.01
method | result | size |
default | \(\text {Expression too large to display}\) | \(2372\) |
risch | \(\text {Expression too large to display}\) | \(5154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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(-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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Giac [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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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 )}^2\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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