Optimal. Leaf size=685 \[ -\frac {(d e-c f) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (-\left (a^2 \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )\right )+a b \left (-2 c d \left (6 A f^2-7 B e f+6 C e^2\right )+d^2 e (B e-4 A f)+c^2 (-f) (4 C e-B f)\right )+b^2 \left (c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (B e-A f)+A d^2 e^2\right )\right )}{8 (b c-a d)^{5/2} (b e-a f)^{7/2}}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 a^3 C d f-a^2 b (-2 B d f+7 c C f+9 C d e)+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{12 b^2 (a+b x)^2 (b c-a d) (b e-a f)^2}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (8 a^4 C d^2 f^2-2 a^3 b d f (-2 B d f+7 c C f+13 C d e)-a^2 b^2 \left (4 d f (-2 A d f+B c f+4 B d e)-C \left (3 c^2 f^2+44 c d e f+33 d^2 e^2\right )\right )-a b^3 \left (2 c d \left (13 A f^2-14 B e f+30 C e^2\right )+d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)\right )-b^4 \left (-3 c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (3 B e-2 A f)+3 A d^2 e^2\right )\right )}{24 b^2 (a+b x) (b c-a d)^2 (b e-a f)^3}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)} \]
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Rubi [A] time = 1.78, antiderivative size = 685, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1613, 149, 151, 12, 93, 208} \[ -\frac {\sqrt {c+d x} \sqrt {e+f x} \left (-a^2 b^2 \left (4 d f (-2 A d f+B c f+4 B d e)-C \left (3 c^2 f^2+44 c d e f+33 d^2 e^2\right )\right )-2 a^3 b d f (-2 B d f+7 c C f+13 C d e)+8 a^4 C d^2 f^2-a b^3 \left (2 c d \left (13 A f^2-14 B e f+30 C e^2\right )+d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)\right )-b^4 \left (-3 c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (3 B e-2 A f)+3 A d^2 e^2\right )\right )}{24 b^2 (a+b x) (b c-a d)^2 (b e-a f)^3}-\frac {(d e-c f) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (a^2 \left (-\left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )\right )+a b \left (-2 c d \left (6 A f^2-7 B e f+6 C e^2\right )+d^2 e (B e-4 A f)+c^2 (-f) (4 C e-B f)\right )+b^2 \left (c^2 \left (5 A f^2-6 B e f+8 C e^2\right )-2 c d e (B e-A f)+A d^2 e^2\right )\right )}{8 (b c-a d)^{5/2} (b e-a f)^{7/2}}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (-a^2 b (-2 B d f+7 c C f+9 C d e)+4 a^3 C d f+a b^2 (-8 A d f+B c f+3 B d e+12 c C e)-b^3 (-5 A c f-3 A d e+6 B c e)\right )}{12 b^2 (a+b x)^2 (b c-a d) (b e-a f)^2}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 149
Rule 151
Rule 208
Rule 1613
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{(a+b x)^4 \sqrt {e+f x}} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}-\frac {\int \frac {\sqrt {c+d x} \left (-\frac {a^2 C (3 d e+c f)+b^2 (6 B c e-3 A d e-5 A c f)-a b (6 c C e+3 B d e+B c f-6 A d f)}{2 b}+\left (-3 b c C e+3 a C d e+3 a c C f+A b d f-a B d f-\frac {2 a^2 C d f}{b}\right ) x\right )}{(a+b x)^3 \sqrt {e+f x}} \, dx}{3 (b c-a d) (b e-a f)}\\ &=\frac {\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}-\frac {\int \frac {\frac {4 a^3 C d f (d e+c f)+a b^2 \left (d^2 e (3 B e-8 A f)+3 c^2 f (4 C e-B f)+4 c d \left (9 C e^2-4 B e f+4 A f^2\right )\right )+b^3 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a^2 b \left (2 B d f (d e+c f)-C \left (9 d^2 e^2+20 c d e f+3 c^2 f^2\right )\right )}{4 b}+\frac {d \left (4 a^3 C d f^2-a^2 b f (11 C d e+5 c C f-2 B d f)-b^3 \left (12 c C e^2-A d e f-c f (6 B e-5 A f)\right )+a b^2 (12 C e (d e+c f)-f (7 B d e+B c f-4 A d f))\right ) x}{2 b}}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx}{6 b (b c-a d) (b e-a f)^2}\\ &=\frac {\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac {\left (8 a^4 C d^2 f^2-2 a^3 b d f (13 C d e+7 c C f-2 B d f)-b^4 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)+2 c d \left (30 C e^2-14 B e f+13 A f^2\right )\right )-a^2 b^2 \left (4 d f (4 B d e+B c f-2 A d f)-C \left (33 d^2 e^2+44 c d e f+3 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b^2 (b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\int \frac {3 b (d e-c f) \left (b^2 \left (A d^2 e^2-2 c d e (B e-A f)+c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b \left (d^2 e (B e-4 A f)-c^2 f (4 C e-B f)-2 c d \left (6 C e^2-7 B e f+6 A f^2\right )\right )-a^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )}{8 (a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{6 b (b c-a d)^2 (b e-a f)^3}\\ &=\frac {\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac {\left (8 a^4 C d^2 f^2-2 a^3 b d f (13 C d e+7 c C f-2 B d f)-b^4 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)+2 c d \left (30 C e^2-14 B e f+13 A f^2\right )\right )-a^2 b^2 \left (4 d f (4 B d e+B c f-2 A d f)-C \left (33 d^2 e^2+44 c d e f+3 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b^2 (b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left ((d e-c f) \left (b^2 \left (A d^2 e^2-2 c d e (B e-A f)+c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b \left (d^2 e (B e-4 A f)-c^2 f (4 C e-B f)-2 c d \left (6 C e^2-7 B e f+6 A f^2\right )\right )-a^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{16 (b c-a d)^2 (b e-a f)^3}\\ &=\frac {\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac {\left (8 a^4 C d^2 f^2-2 a^3 b d f (13 C d e+7 c C f-2 B d f)-b^4 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)+2 c d \left (30 C e^2-14 B e f+13 A f^2\right )\right )-a^2 b^2 \left (4 d f (4 B d e+B c f-2 A d f)-C \left (33 d^2 e^2+44 c d e f+3 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b^2 (b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left ((d e-c f) \left (b^2 \left (A d^2 e^2-2 c d e (B e-A f)+c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b \left (d^2 e (B e-4 A f)-c^2 f (4 C e-B f)-2 c d \left (6 C e^2-7 B e f+6 A f^2\right )\right )-a^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-b c+a d-(-b e+a f) x^2} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{8 (b c-a d)^2 (b e-a f)^3}\\ &=\frac {\left (4 a^3 C d f-b^3 (6 B c e-3 A d e-5 A c f)+a b^2 (12 c C e+3 B d e+B c f-8 A d f)-a^2 b (9 C d e+7 c C f-2 B d f)\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b^2 (b c-a d) (b e-a f)^2 (a+b x)^2}-\frac {\left (8 a^4 C d^2 f^2-2 a^3 b d f (13 C d e+7 c C f-2 B d f)-b^4 \left (3 A d^2 e^2-2 c d e (3 B e-2 A f)-3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-10 A f)+3 c^2 f (4 C e-B f)+2 c d \left (30 C e^2-14 B e f+13 A f^2\right )\right )-a^2 b^2 \left (4 d f (4 B d e+B c f-2 A d f)-C \left (33 d^2 e^2+44 c d e f+3 c^2 f^2\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b^2 (b c-a d)^2 (b e-a f)^3 (a+b x)}-\frac {\left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}-\frac {(d e-c f) \left (b^2 \left (A d^2 e^2-2 c d e (B e-A f)+c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b \left (d^2 e (B e-4 A f)-c^2 f (4 C e-B f)-2 c d \left (6 C e^2-7 B e f+6 A f^2\right )\right )-a^2 \left (2 d f (3 B d e+B c f-4 A d f)-C \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{8 (b c-a d)^{5/2} (b e-a f)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 6.34, size = 729, normalized size = 1.06 \[ -\frac {\left (a^2 C-a b B+A b^2\right ) \left (-\frac {3 \left (8 a^2 d^2 f^2-4 a b d f (3 c f+d e)+b^2 \left (5 c^2 f^2+2 c d e f+d^2 e^2\right )\right ) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{(a+b x) (a f-b e)}-\frac {(d e-c f) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{\sqrt {a d-b c} (a f-b e)^{3/2}}\right )}{8 (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (\frac {1}{2} b (-6 a d f+5 b c f+3 b d e)-a b d f\right )}{2 (a+b x)^2 (b c-a d) (b e-a f)}\right )}{3 b^2 (b c-a d) (b e-a f)}-\frac {(c+d x)^{3/2} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^3 (b c-a d) (b e-a f)}+\frac {(b B-2 a C) (-4 a d f+3 b c f+b d e) \left (\frac {\sqrt {c+d x} \sqrt {e+f x}}{(a+b x) (b e-a f)}+\frac {(d e-c f) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{\sqrt {a d-b c} (a f-b e)^{3/2}}\right )}{4 b^2 (b c-a d) (b e-a f)}-\frac {C \sqrt {c+d x} \sqrt {e+f x}}{b^2 (a+b x) (b e-a f)}-\frac {C (d e-c f) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{b^2 \sqrt {a d-b c} (a f-b e)^{3/2}}-\frac {(c+d x)^{3/2} \sqrt {e+f x} (b B-2 a C)}{2 b (a+b x)^2 (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 15990, normalized size = 23.34 \[ \text {output too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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