Optimal. Leaf size=441 \[ \frac {\log \left (d+f x^2\right ) \left (A b f \left (-f \left (b^2 d-3 a^2 f\right )-6 a c d f+3 c^2 d^2\right )-B (c d-a f) \left (-f \left (3 b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )\right )}{2 f^4}-\frac {x^2 \left (A b f \left (-6 a c f+b^2 (-f)+3 c^2 d\right )-B \left (-3 c f \left (b^2 d-a^2 f\right )+3 a b^2 f^2-3 a c^2 d f+c^3 d^2\right )\right )}{2 f^3}-\frac {x \left (-A c \left (3 a^2 f^2-3 a c d f+c^2 d^2\right )+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2+b^3 B d f\right )}{f^3}+\frac {c x^4 \left (3 A b c f-B \left (-3 a c f-3 b^2 f+c^2 d\right )\right )}{4 f^2}+\frac {x^3 \left (-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)+3 A b^2 c f+b^3 B f\right )}{3 f^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (3 A b^2 d f (c d-a f)-A (c d-a f)^3-3 b B d (c d-a f)^2+b^3 B d^2 f\right )}{\sqrt {d} f^{7/2}}+\frac {c^2 x^5 (A c+3 b B)}{5 f}+\frac {B c^3 x^6}{6 f} \]
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Rubi [A] time = 0.62, antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1012, 635, 205, 260} \begin {gather*} -\frac {x^2 \left (A b f \left (-6 a c f+b^2 (-f)+3 c^2 d\right )-B \left (-3 c f \left (b^2 d-a^2 f\right )+3 a b^2 f^2-3 a c^2 d f+c^3 d^2\right )\right )}{2 f^3}+\frac {\log \left (d+f x^2\right ) \left (A b f \left (-f \left (b^2 d-3 a^2 f\right )-6 a c d f+3 c^2 d^2\right )-B (c d-a f) \left (-f \left (3 b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )\right )}{2 f^4}-\frac {x \left (-A c \left (3 a^2 f^2-3 a c d f+c^2 d^2\right )+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2+b^3 B d f\right )}{f^3}+\frac {c x^4 \left (3 A b c f-B \left (-3 a c f-3 b^2 f+c^2 d\right )\right )}{4 f^2}+\frac {x^3 \left (-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)+3 A b^2 c f+b^3 B f\right )}{3 f^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (3 A b^2 d f (c d-a f)-A (c d-a f)^3-3 b B d (c d-a f)^2+b^3 B d^2 f\right )}{\sqrt {d} f^{7/2}}+\frac {c^2 x^5 (A c+3 b B)}{5 f}+\frac {B c^3 x^6}{6 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 1012
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\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+f x^2} \, dx &=\int \left (-\frac {b^3 B d f+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2-A c \left (c^2 d^2-3 a c d f+3 a^2 f^2\right )}{f^3}-\frac {\left (A b f \left (3 c^2 d-b^2 f-6 a c f\right )-B \left (c^3 d^2-3 a c^2 d f+3 a b^2 f^2-3 c f \left (b^2 d-a^2 f\right )\right )\right ) x}{f^3}+\frac {\left (b^3 B f+3 A b^2 c f-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)\right ) x^2}{f^2}+\frac {c \left (3 A b c f-B \left (c^2 d-3 b^2 f-3 a c f\right )\right ) x^3}{f^2}+\frac {c^2 (3 b B+A c) x^4}{f}+\frac {B c^3 x^5}{f}-\frac {-b^3 B d^2 f-3 A b^2 d f (c d-a f)+3 b B d (c d-a f)^2+A (c d-a f)^3-\left (A b f \left (3 c^2 d^2-6 a c d f-f \left (b^2 d-3 a^2 f\right )\right )-B (c d-a f) \left (c^2 d^2-2 a c d f-f \left (3 b^2 d-a^2 f\right )\right )\right ) x}{f^3 \left (d+f x^2\right )}\right ) \, dx\\ &=-\frac {\left (b^3 B d f+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2-A c \left (c^2 d^2-3 a c d f+3 a^2 f^2\right )\right ) x}{f^3}-\frac {\left (A b f \left (3 c^2 d-b^2 f-6 a c f\right )-B \left (c^3 d^2-3 a c^2 d f+3 a b^2 f^2-3 c f \left (b^2 d-a^2 f\right )\right )\right ) x^2}{2 f^3}+\frac {\left (b^3 B f+3 A b^2 c f-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)\right ) x^3}{3 f^2}+\frac {c \left (3 A b c f-B \left (c^2 d-3 b^2 f-3 a c f\right )\right ) x^4}{4 f^2}+\frac {c^2 (3 b B+A c) x^5}{5 f}+\frac {B c^3 x^6}{6 f}-\frac {\int \frac {-b^3 B d^2 f-3 A b^2 d f (c d-a f)+3 b B d (c d-a f)^2+A (c d-a f)^3-\left (A b f \left (3 c^2 d^2-6 a c d f-f \left (b^2 d-3 a^2 f\right )\right )-B (c d-a f) \left (c^2 d^2-2 a c d f-f \left (3 b^2 d-a^2 f\right )\right )\right ) x}{d+f x^2} \, dx}{f^3}\\ &=-\frac {\left (b^3 B d f+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2-A c \left (c^2 d^2-3 a c d f+3 a^2 f^2\right )\right ) x}{f^3}-\frac {\left (A b f \left (3 c^2 d-b^2 f-6 a c f\right )-B \left (c^3 d^2-3 a c^2 d f+3 a b^2 f^2-3 c f \left (b^2 d-a^2 f\right )\right )\right ) x^2}{2 f^3}+\frac {\left (b^3 B f+3 A b^2 c f-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)\right ) x^3}{3 f^2}+\frac {c \left (3 A b c f-B \left (c^2 d-3 b^2 f-3 a c f\right )\right ) x^4}{4 f^2}+\frac {c^2 (3 b B+A c) x^5}{5 f}+\frac {B c^3 x^6}{6 f}+\frac {\left (b^3 B d^2 f+3 A b^2 d f (c d-a f)-3 b B d (c d-a f)^2-A (c d-a f)^3\right ) \int \frac {1}{d+f x^2} \, dx}{f^3}+\frac {\left (A b f \left (3 c^2 d^2-6 a c d f-f \left (b^2 d-3 a^2 f\right )\right )-B (c d-a f) \left (c^2 d^2-2 a c d f-f \left (3 b^2 d-a^2 f\right )\right )\right ) \int \frac {x}{d+f x^2} \, dx}{f^3}\\ &=-\frac {\left (b^3 B d f+3 A b^2 f (c d-a f)-3 b B (c d-a f)^2-A c \left (c^2 d^2-3 a c d f+3 a^2 f^2\right )\right ) x}{f^3}-\frac {\left (A b f \left (3 c^2 d-b^2 f-6 a c f\right )-B \left (c^3 d^2-3 a c^2 d f+3 a b^2 f^2-3 c f \left (b^2 d-a^2 f\right )\right )\right ) x^2}{2 f^3}+\frac {\left (b^3 B f+3 A b^2 c f-A c^2 (c d-3 a f)-3 b B c (c d-2 a f)\right ) x^3}{3 f^2}+\frac {c \left (3 A b c f-B \left (c^2 d-3 b^2 f-3 a c f\right )\right ) x^4}{4 f^2}+\frac {c^2 (3 b B+A c) x^5}{5 f}+\frac {B c^3 x^6}{6 f}+\frac {\left (b^3 B d^2 f+3 A b^2 d f (c d-a f)-3 b B d (c d-a f)^2-A (c d-a f)^3\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d} f^{7/2}}+\frac {\left (A b f \left (3 c^2 d^2-6 a c d f-f \left (b^2 d-3 a^2 f\right )\right )-B (c d-a f) \left (c^2 d^2-2 a c d f-f \left (3 b^2 d-a^2 f\right )\right )\right ) \log \left (d+f x^2\right )}{2 f^4}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 422, normalized size = 0.96 \begin {gather*} \frac {f x \left (3 b \left (4 B \left (15 a^2 f^2+10 a c f \left (f x^2-3 d\right )+c^2 \left (15 d^2-5 d f x^2+3 f^2 x^4\right )\right )+15 A c f x \left (4 a f-2 c d+c f x^2\right )\right )+c \left (4 A \left (45 a^2 f^2+15 a c f \left (f x^2-3 d\right )+c^2 \left (15 d^2-5 d f x^2+3 f^2 x^4\right )\right )+5 B x \left (18 a^2 f^2+9 a c f \left (f x^2-2 d\right )+c^2 \left (6 d^2-3 d f x^2+2 f^2 x^4\right )\right )\right )+15 b^2 f \left (4 A \left (3 a f-3 c d+c f x^2\right )+3 B x \left (2 a f-2 c d+c f x^2\right )\right )+10 b^3 f \left (3 A f x-6 B d+2 B f x^2\right )\right )-30 \log \left (d+f x^2\right ) \left (A b f \left (-3 a^2 f^2+6 a c d f+b^2 d f-3 c^2 d^2\right )+B (c d-a f) \left (a^2 f^2-2 a c d f-3 b^2 d f+c^2 d^2\right )\right )}{60 f^4}+\frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (3 A b^2 d f (c d-a f)-A (c d-a f)^3-3 b B d (c d-a f)^2+b^3 B d^2 f\right )}{\sqrt {d} f^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{d+f x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 1014, normalized size = 2.30 \begin {gather*} \left [\frac {10 \, B c^{3} d f^{3} x^{6} + 12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d f^{3} x^{5} - 15 \, {\left (B c^{3} d^{2} f^{2} - 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d f^{3}\right )} x^{4} - 20 \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} f^{2} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d f^{3}\right )} x^{3} + 30 \, {\left (B c^{3} d^{3} f - 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} f^{2} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d f^{3}\right )} x^{2} - 30 \, {\left (A a^{3} f^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} f - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d f^{2}\right )} \sqrt {-d f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-d f} x - d}{f x^{2} + d}\right ) + 60 \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} f - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} f^{2} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d f^{3}\right )} x - 30 \, {\left (B c^{3} d^{4} - 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} f + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} f^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d f^{3}\right )} \log \left (f x^{2} + d\right )}{60 \, d f^{4}}, \frac {10 \, B c^{3} d f^{3} x^{6} + 12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d f^{3} x^{5} - 15 \, {\left (B c^{3} d^{2} f^{2} - 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d f^{3}\right )} x^{4} - 20 \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} f^{2} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d f^{3}\right )} x^{3} + 30 \, {\left (B c^{3} d^{3} f - 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} f^{2} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d f^{3}\right )} x^{2} + 60 \, {\left (A a^{3} f^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} f - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d f^{2}\right )} \sqrt {d f} \arctan \left (\frac {\sqrt {d f} x}{d}\right ) + 60 \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} f - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} f^{2} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d f^{3}\right )} x - 30 \, {\left (B c^{3} d^{4} - 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} f + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} f^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d f^{3}\right )} \log \left (f x^{2} + d\right )}{60 \, d f^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 623, normalized size = 1.41 \begin {gather*} -\frac {{\left (3 \, B b c^{2} d^{3} + A c^{3} d^{3} - B b^{3} d^{2} f - 6 \, B a b c d^{2} f - 3 \, A b^{2} c d^{2} f - 3 \, A a c^{2} d^{2} f + 3 \, B a^{2} b d f^{2} + 3 \, A a b^{2} d f^{2} + 3 \, A a^{2} c d f^{2} - A a^{3} f^{3}\right )} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f} f^{3}} - \frac {{\left (B c^{3} d^{3} - 3 \, B b^{2} c d^{2} f - 3 \, B a c^{2} d^{2} f - 3 \, A b c^{2} d^{2} f + 3 \, B a b^{2} d f^{2} + A b^{3} d f^{2} + 3 \, B a^{2} c d f^{2} + 6 \, A a b c d f^{2} - B a^{3} f^{3} - 3 \, A a^{2} b f^{3}\right )} \log \left (f x^{2} + d\right )}{2 \, f^{4}} + \frac {10 \, B c^{3} f^{5} x^{6} + 36 \, B b c^{2} f^{5} x^{5} + 12 \, A c^{3} f^{5} x^{5} - 15 \, B c^{3} d f^{4} x^{4} + 45 \, B b^{2} c f^{5} x^{4} + 45 \, B a c^{2} f^{5} x^{4} + 45 \, A b c^{2} f^{5} x^{4} - 60 \, B b c^{2} d f^{4} x^{3} - 20 \, A c^{3} d f^{4} x^{3} + 20 \, B b^{3} f^{5} x^{3} + 120 \, B a b c f^{5} x^{3} + 60 \, A b^{2} c f^{5} x^{3} + 60 \, A a c^{2} f^{5} x^{3} + 30 \, B c^{3} d^{2} f^{3} x^{2} - 90 \, B b^{2} c d f^{4} x^{2} - 90 \, B a c^{2} d f^{4} x^{2} - 90 \, A b c^{2} d f^{4} x^{2} + 90 \, B a b^{2} f^{5} x^{2} + 30 \, A b^{3} f^{5} x^{2} + 90 \, B a^{2} c f^{5} x^{2} + 180 \, A a b c f^{5} x^{2} + 180 \, B b c^{2} d^{2} f^{3} x + 60 \, A c^{3} d^{2} f^{3} x - 60 \, B b^{3} d f^{4} x - 360 \, B a b c d f^{4} x - 180 \, A b^{2} c d f^{4} x - 180 \, A a c^{2} d f^{4} x + 180 \, B a^{2} b f^{5} x + 180 \, A a b^{2} f^{5} x + 180 \, A a^{2} c f^{5} x}{60 \, f^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 822, normalized size = 1.86 \begin {gather*} \frac {B \,c^{3} x^{6}}{6 f}+\frac {A \,c^{3} x^{5}}{5 f}+\frac {3 B b \,c^{2} x^{5}}{5 f}+\frac {3 A b \,c^{2} x^{4}}{4 f}+\frac {3 B a \,c^{2} x^{4}}{4 f}+\frac {3 B \,b^{2} c \,x^{4}}{4 f}-\frac {B \,c^{3} d \,x^{4}}{4 f^{2}}+\frac {A a \,c^{2} x^{3}}{f}+\frac {A \,b^{2} c \,x^{3}}{f}-\frac {A \,c^{3} d \,x^{3}}{3 f^{2}}+\frac {2 B a b c \,x^{3}}{f}+\frac {B \,b^{3} x^{3}}{3 f}-\frac {B b \,c^{2} d \,x^{3}}{f^{2}}+\frac {A \,a^{3} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}}-\frac {3 A \,a^{2} c d \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f}-\frac {3 A a \,b^{2} d \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f}+\frac {3 A a b c \,x^{2}}{f}+\frac {3 A a \,c^{2} d^{2} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f^{2}}+\frac {A \,b^{3} x^{2}}{2 f}+\frac {3 A \,b^{2} c \,d^{2} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f^{2}}-\frac {3 A b \,c^{2} d \,x^{2}}{2 f^{2}}-\frac {A \,c^{3} d^{3} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f^{3}}-\frac {3 B \,a^{2} b d \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f}+\frac {3 B \,a^{2} c \,x^{2}}{2 f}+\frac {3 B a \,b^{2} x^{2}}{2 f}+\frac {6 B a b c \,d^{2} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f^{2}}-\frac {3 B a \,c^{2} d \,x^{2}}{2 f^{2}}+\frac {B \,b^{3} d^{2} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f^{2}}-\frac {3 B \,b^{2} c d \,x^{2}}{2 f^{2}}-\frac {3 B b \,c^{2} d^{3} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f}\, f^{3}}+\frac {B \,c^{3} d^{2} x^{2}}{2 f^{3}}+\frac {3 A \,a^{2} b \ln \left (f \,x^{2}+d \right )}{2 f}+\frac {3 A \,a^{2} c x}{f}+\frac {3 A a \,b^{2} x}{f}-\frac {3 A a b c d \ln \left (f \,x^{2}+d \right )}{f^{2}}-\frac {3 A a \,c^{2} d x}{f^{2}}-\frac {A \,b^{3} d \ln \left (f \,x^{2}+d \right )}{2 f^{2}}-\frac {3 A \,b^{2} c d x}{f^{2}}+\frac {3 A b \,c^{2} d^{2} \ln \left (f \,x^{2}+d \right )}{2 f^{3}}+\frac {A \,c^{3} d^{2} x}{f^{3}}+\frac {B \,a^{3} \ln \left (f \,x^{2}+d \right )}{2 f}+\frac {3 B \,a^{2} b x}{f}-\frac {3 B \,a^{2} c d \ln \left (f \,x^{2}+d \right )}{2 f^{2}}-\frac {3 B a \,b^{2} d \ln \left (f \,x^{2}+d \right )}{2 f^{2}}-\frac {6 B a b c d x}{f^{2}}+\frac {3 B a \,c^{2} d^{2} \ln \left (f \,x^{2}+d \right )}{2 f^{3}}-\frac {B \,b^{3} d x}{f^{2}}+\frac {3 B \,b^{2} c \,d^{2} \ln \left (f \,x^{2}+d \right )}{2 f^{3}}+\frac {3 B b \,c^{2} d^{2} x}{f^{3}}-\frac {B \,c^{3} d^{3} \ln \left (f \,x^{2}+d \right )}{2 f^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 471, normalized size = 1.07 \begin {gather*} \frac {{\left (A a^{3} f^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} f - 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d f^{2}\right )} \arctan \left (\frac {f x}{\sqrt {d f}}\right )}{\sqrt {d f} f^{3}} + \frac {10 \, B c^{3} f^{2} x^{6} + 12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} f^{2} x^{5} - 15 \, {\left (B c^{3} d f - 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} f^{2}\right )} x^{4} - 20 \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d f - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} f^{2}\right )} x^{3} + 30 \, {\left (B c^{3} d^{2} - 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d f + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} f^{2}\right )} x^{2} + 60 \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d f + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} f^{2}\right )} x}{60 \, f^{3}} - \frac {{\left (B c^{3} d^{3} - 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} f + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d f^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} f^{3}\right )} \log \left (f x^{2} + d\right )}{2 \, f^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.79, size = 552, normalized size = 1.25 \begin {gather*} x^2\,\left (\frac {3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3}{2\,f}-\frac {d\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{f}-\frac {B\,c^3\,d}{f^2}\right )}{2\,f}\right )+x\,\left (\frac {3\,B\,a^2\,b+3\,A\,c\,a^2+3\,A\,a\,b^2}{f}-\frac {d\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{f}-\frac {d\,\left (A\,c^3+3\,B\,b\,c^2\right )}{f^2}\right )}{f}\right )+x^3\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{3\,f}-\frac {d\,\left (A\,c^3+3\,B\,b\,c^2\right )}{3\,f^2}\right )+x^4\,\left (\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{4\,f}-\frac {B\,c^3\,d}{4\,f^2}\right )+\frac {x^5\,\left (A\,c^3+3\,B\,b\,c^2\right )}{5\,f}+\frac {B\,c^3\,x^6}{6\,f}+\frac {\ln \left (f\,x^2+d\right )\,\left (4\,B\,a^3\,d\,f^7+12\,A\,a^2\,b\,d\,f^7-12\,B\,a^2\,c\,d^2\,f^6-12\,B\,a\,b^2\,d^2\,f^6-24\,A\,a\,b\,c\,d^2\,f^6+12\,B\,a\,c^2\,d^3\,f^5-4\,A\,b^3\,d^2\,f^6+12\,B\,b^2\,c\,d^3\,f^5+12\,A\,b\,c^2\,d^3\,f^5-4\,B\,c^3\,d^4\,f^4\right )}{8\,d\,f^8}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {d}}\right )\,\left (A\,a^3\,f^3-3\,B\,a^2\,b\,d\,f^2-3\,A\,a^2\,c\,d\,f^2-3\,A\,a\,b^2\,d\,f^2+6\,B\,a\,b\,c\,d^2\,f+3\,A\,a\,c^2\,d^2\,f+B\,b^3\,d^2\,f+3\,A\,b^2\,c\,d^2\,f-3\,B\,b\,c^2\,d^3-A\,c^3\,d^3\right )}{\sqrt {d}\,f^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 23.62, size = 1962, normalized size = 4.45
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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