Optimal. Leaf size=826 \[ -\frac{\sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left (-2 d f \left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^3+b \left (C \left (d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d f e+c^2 f^2\right )\right )\right ) a^2+b^2 \left (-f^2 (4 C e-B f) c^3-d f \left (40 C e^2-23 B f e+18 A f^2\right ) c^2-d^2 e \left (4 C e^2-23 B f e+12 A f^2\right ) c+d^3 e^2 (B e-18 A f)\right ) a+b^3 \left (f \left (8 C e^2-6 B f e+5 A f^2\right ) c^3+d e \left (8 C e^2-4 B f e+3 A f^2\right ) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{8 (b c-a d)^{7/2} (b e-a f)^{7/2}}+\frac{\left (4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-34 c d f e+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right ) a^2-b^3 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-29 B f e+22 A f^2\right ) c+d^2 e (3 B e-44 A f)\right ) a-b^4 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac{\left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2} \]
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Rubi [A] time = 2.43334, antiderivative size = 826, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{\sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left (-2 d f \left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^3+b \left (C \left (d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d f e+c^2 f^2\right )\right )\right ) a^2+b^2 \left (-f^2 (4 C e-B f) c^3-d f \left (40 C e^2-23 B f e+18 A f^2\right ) c^2-d^2 e \left (4 C e^2-23 B f e+12 A f^2\right ) c+d^3 e^2 (B e-18 A f)\right ) a+b^3 \left (f \left (8 C e^2-6 B f e+5 A f^2\right ) c^3+d e \left (8 C e^2-4 B f e+3 A f^2\right ) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{8 (b c-a d)^{7/2} (b e-a f)^{7/2}}+\frac{\left (4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-34 c d f e+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right ) a^2-b^3 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-29 B f e+22 A f^2\right ) c+d^2 e (3 B e-44 A f)\right ) a-b^4 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac{\left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 1613
Rule 151
Rule 12
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{(a+b x)^4 \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}}{3 b (b c-a d) (b e-a f) (a+b x)^3}-\frac{\int \frac{-\frac{a^2 C (d e+c f)-a b (6 c C e+B d e+B c f-6 A d f)+b^2 (6 B c e-5 A (d e+c f))}{2 b}+\left (-3 b c C e+3 a C d e+3 a c C f+2 A b d f-2 a B d f-\frac{a^2 C d f}{b}\right ) x}{(a+b x)^3 \sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 (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}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac{\int \frac{\frac{2 a^3 C d f (d e+c f)+b^3 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^2 e (3 B e-34 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-23 B e f+17 A f^2\right )\right )+a^2 b \left (C \left (3 d^2 e^2-10 c d e f+3 c^2 f^2\right )+8 d f (3 A d f-B (d e+c f))\right )}{4 b}+\frac{d f \left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) x}{2 b}}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x}} \, dx}{6 (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}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac{\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 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-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac{\int \frac{3 \left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \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 )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right )}{8 (a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{6 (b c-a d)^3 (b e-a f)^3}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac{\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 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-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac{\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \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 )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 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)^3 (b e-a f)^3}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac{\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 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-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac{\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \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 )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 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)^3 (b e-a f)^3}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac{\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac{\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 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-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac{\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \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 )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\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)^{7/2} (b e-a f)^{7/2}}\\ \end{align*}
Mathematica [A] time = 6.08097, size = 800, normalized size = 0.97 \[ -\frac{-24 C (b c-a d)^2 (b e-a f)^2 (b d e+b c f-2 a d f) \tan ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{a d-b c} \sqrt{e+f x}}\right ) (a+b x)^3-6 (b B-2 a C) (b e-a f) \left (3 b (a d-b c)^{3/2} \sqrt{b e-a f} (b d e+b c f-2 a d f) \sqrt{c+d x} \sqrt{e+f x}-(b c-a d) \left (\left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right ) b^2-8 a d f (d e+c f) b+8 a^2 d^2 f^2\right ) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{a d-b c} \sqrt{e+f x}}\right )\right ) (a+b x)^2-24 b C (a d-b c)^{5/2} (b e-a f)^{5/2} \sqrt{c+d x} \sqrt{e+f x} (a+b x)^2-\left (A b^2+a (a C-b B)\right ) \left (10 b (a d-b c)^{3/2} (b e-a f)^{3/2} (b d e+b c f-2 a d f) \sqrt{c+d x} \sqrt{e+f x}-(a+b x) \left (-b \sqrt{a d-b c} \sqrt{b e-a f} \sqrt{c+d x} \sqrt{e+f x} \left (\left (15 d^2 e^2+14 c d f e+15 c^2 f^2\right ) b^2-44 a d f (d e+c f) b+44 a^2 d^2 f^2\right )-3 (b d e+b c f-2 a d f) \left (\left (5 d^2 e^2-2 c d f e+5 c^2 f^2\right ) b^2-8 a d f (d e+c f) b+8 a^2 d^2 f^2\right ) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{a d-b c} \sqrt{e+f x}}\right )\right )\right ) (a+b x)-12 b (b B-2 a C) (a d-b c)^{5/2} (b e-a f)^{5/2} \sqrt{c+d x} \sqrt{e+f x} (a+b x)-8 b \left (A b^2+a (a C-b B)\right ) (a d-b c)^{5/2} (b e-a f)^{5/2} \sqrt{c+d x} \sqrt{e+f x}}{24 b^2 (a d-b c)^{7/2} (b e-a f)^{7/2} (a+b x)^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.246, size = 18802, normalized size = 22.8 \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(-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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Giac [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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