Optimal. Leaf size=424 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (a^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+a b \left (-2 c d \left (4 A f^2-7 B e f+4 C e^2\right )+d^2 e (B e-8 A f)+c^2 (-f) (8 C e-B f)\right )+b^2 \left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )-2 c d e (2 B e-A f)+3 A d^2 e^2\right )\right )}{4 (b c-a d)^{5/2} (b e-a f)^{5/2}}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (2 B d f-5 C (c f+d e))+a b^2 (-6 A d f+B c f+B d e+8 c C e)-b^3 (4 B c e-3 A (c f+d e))\right )}{4 b (a+b x) (b c-a d)^2 (b e-a f)^2}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)} \]
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Rubi [A] time = 0.97, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {1613, 151, 12, 93, 208} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (a^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+a b \left (-2 c d \left (4 A f^2-7 B e f+4 C e^2\right )+d^2 e (B e-8 A f)+c^2 (-f) (8 C e-B f)\right )+b^2 \left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )-2 c d e (2 B e-A f)+3 A d^2 e^2\right )\right )}{4 (b c-a d)^{5/2} (b e-a f)^{5/2}}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (a^2 b (2 B d f-5 C (c f+d e))+2 a^3 C d f+a b^2 (-6 A d f+B c f+B d e+8 c C e)-b^3 (4 B c e-3 A (c f+d e))\right )}{4 b (a+b x) (b c-a d)^2 (b e-a f)^2}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 151
Rule 208
Rule 1613
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}-\frac {\int \frac {-\frac {a^2 C (d e+c f)-a b (4 c C e+B d e+B c f-4 A d f)+b^2 (4 B c e-3 A (d e+c f))}{2 b}+\left (-2 b c C e+2 a C d e+2 a c C f+A b d f-a B d f-\frac {a^2 C d f}{b}\right ) x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx}{2 (b c-a d) (b e-a f)}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {\left (2 a^3 C d f+a b^2 (8 c C e+B d e+B c f-6 A d f)-b^3 (4 B c e-3 A (d e+c f))+a^2 b (2 B d f-5 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b (b c-a d)^2 (b e-a f)^2 (a+b x)}+\frac {\int \frac {b^2 \left (3 A d^2 e^2-2 c d e (2 B e-A f)+c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )+a b \left (d^2 e (B e-8 A f)-c^2 f (8 C e-B f)-2 c d \left (4 C e^2-7 B e f+4 A f^2\right )\right )+a^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )}{4 (a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{2 (b c-a d)^2 (b e-a f)^2}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {\left (2 a^3 C d f+a b^2 (8 c C e+B d e+B c f-6 A d f)-b^3 (4 B c e-3 A (d e+c f))+a^2 b (2 B d f-5 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b (b c-a d)^2 (b e-a f)^2 (a+b x)}+\frac {\left (b^2 \left (3 A d^2 e^2-2 c d e (2 B e-A f)+c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )+a b \left (d^2 e (B e-8 A f)-c^2 f (8 C e-B f)-2 c d \left (4 C e^2-7 B e f+4 A f^2\right )\right )+a^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{8 (b c-a d)^2 (b e-a f)^2}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {\left (2 a^3 C d f+a b^2 (8 c C e+B d e+B c f-6 A d f)-b^3 (4 B c e-3 A (d e+c f))+a^2 b (2 B d f-5 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b (b c-a d)^2 (b e-a f)^2 (a+b x)}+\frac {\left (b^2 \left (3 A d^2 e^2-2 c d e (2 B e-A f)+c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )+a b \left (d^2 e (B e-8 A f)-c^2 f (8 C e-B f)-2 c d \left (4 C e^2-7 B e f+4 A f^2\right )\right )+a^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\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 )}{4 (b c-a d)^2 (b e-a f)^2}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {\left (2 a^3 C d f+a b^2 (8 c C e+B d e+B c f-6 A d f)-b^3 (4 B c e-3 A (d e+c f))+a^2 b (2 B d f-5 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b (b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {\left (b^2 \left (3 A d^2 e^2-2 c d e (2 B e-A f)+c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )+a b \left (d^2 e (B e-8 A f)-c^2 f (8 C e-B f)-2 c d \left (4 C e^2-7 B e f+4 A f^2\right )\right )+a^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\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 )}{4 (b c-a d)^{5/2} (b e-a f)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 2.09, size = 512, normalized size = 1.21 \[ \frac {\frac {\left (a (a C-b B)+A b^2\right ) \left (\frac {\left (8 a^2 d^2 f^2-8 a b d f (c f+d e)+b^2 \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{(a d-b c)^{3/2} (a f-b e)^{3/2}}+\frac {3 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{(a+b x) (b c-a d) (b e-a f)}\right )}{(b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \left (a (a C-b B)+A b^2\right )}{(a+b x)^2 (b c-a d) (b e-a f)}-\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f)}-\frac {4 (b B-2 a C) (-2 a d f+b c f+b d e) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{(a d-b c)^{3/2} (a f-b e)^{3/2}}+\frac {8 C \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{\sqrt {a d-b c} \sqrt {a f-b e}}}{4 b^2} \]
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 = 0.13, size = 7119, normalized size = 16.79 \[ \text {output too large to display} \]
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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