Optimal. Leaf size=431 \[ \frac {g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt {a+b x+c x^2}}{24 c^3 e^3}+\frac {g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt {a+b x+c x^2}}{12 c^2 e^3}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}-\frac {g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{7/2} e^4}+\frac {(e f-d g)^4 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^4 \sqrt {c d^2-b d e+a e^2}} \]
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Rubi [A]
time = 0.86, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1667, 857, 635,
212, 738} \begin {gather*} -\frac {g \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{16 c^{7/2} e^4}+\frac {g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{24 c^3 e^3}+\frac {g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{12 c^2 e^3}+\frac {(e f-d g)^4 \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^4 \sqrt {a e^2-b d e+c d^2}}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 857
Rule 1667
Rubi steps
\begin {align*} \int \frac {(f+g x)^4}{(d+e x) \sqrt {a+b x+c x^2}} \, dx &=\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}+\frac {\int \frac {\frac {1}{2} e \left (6 c e^3 f^4-d^2 (b d+4 a e) g^4\right )-\frac {1}{2} e g \left (d e (7 b d+8 a e) g^3-c \left (24 e^3 f^3-2 d^3 g^3\right )\right ) x-\frac {1}{2} e^2 g^2 \left (e (11 b d+4 a e) g^2-c \left (36 e^2 f^2-10 d^2 g^2\right )\right ) x^2+\frac {1}{2} e^3 g^3 (24 c e f-14 c d g-5 b e g) x^3}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{3 c e^4}\\ &=\frac {g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt {a+b x+c x^2}}{12 c^2 e^3}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}+\frac {\int \frac {\frac {1}{4} e^4 \left (24 c^2 e^3 f^4+5 b d e (b d+2 a e) g^4-2 c d g^3 (b d (12 e f-5 d g)+6 a e (4 e f-d g))\right )+\frac {1}{2} e^4 g \left (5 b e^2 (2 b d+a e) g^3+2 c^2 \left (24 e^3 f^3-12 d^2 e f g^2+5 d^3 g^3\right )-c e g^2 (b d (48 e f-19 d g)+2 a e (12 e f+d g))\right ) x+\frac {1}{4} e^5 g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) x^2}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{6 c^2 e^7}\\ &=\frac {g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt {a+b x+c x^2}}{24 c^3 e^3}+\frac {g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt {a+b x+c x^2}}{12 c^2 e^3}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}+\frac {\int \frac {\frac {3}{8} e^6 \left (16 c^3 e^3 f^4-5 b^3 d e^2 g^4+6 b c d e g^3 (4 b e f-b d g+2 a e g)-8 c^2 d g^2 \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right )-\frac {3}{8} e^6 g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{6 c^3 e^9}\\ &=\frac {g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt {a+b x+c x^2}}{24 c^3 e^3}+\frac {g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt {a+b x+c x^2}}{12 c^2 e^3}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}+\frac {(e f-d g)^4 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^4}-\frac {\left (g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^3 e^4}\\ &=\frac {g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt {a+b x+c x^2}}{24 c^3 e^3}+\frac {g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt {a+b x+c x^2}}{12 c^2 e^3}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}-\frac {\left (2 (e f-d g)^4\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^4}-\frac {\left (g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\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^3 e^4}\\ &=\frac {g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt {a+b x+c x^2}}{24 c^3 e^3}+\frac {g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt {a+b x+c x^2}}{12 c^2 e^3}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3}-\frac {g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{7/2} e^4}+\frac {(e f-d g)^4 \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^4 \sqrt {c d^2-b d e+a e^2}}\\ \end {align*}
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Mathematica [A]
time = 2.17, size = 380, normalized size = 0.88 \begin {gather*} \frac {\frac {2 e g^2 \sqrt {a+x (b+c x)} \left (15 b^2 e^2 g^2-2 c e g (8 a e g+b (36 e f-9 d g+5 e g x))+4 c^2 \left (6 d^2 g^2-3 d e g (8 f+g x)+2 e^2 \left (18 f^2+6 f g x+g^2 x^2\right )\right )\right )}{c^3}+\frac {96 \sqrt {-c d^2+b d e-a e^2} (e f-d g)^4 \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{c d^2+e (-b d+a e)}+\frac {3 g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{c^{7/2}}}{48 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 755, normalized size = 1.75
method | result | size |
default | \(-\frac {g \left (-e^{3} g^{3} \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )+\left (d \,e^{2} g^{3}-4 e^{3} f \,g^{2}\right ) \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+\left (-d^{2} e \,g^{3}+4 d \,e^{2} f \,g^{2}-6 e^{3} f^{2} g \right ) \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {d^{3} g^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {4 d^{2} e f \,g^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+\frac {6 d \,e^{2} f^{2} g \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {4 e^{3} f^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}\right )}{e^{4}}-\frac {\left (d^{4} g^{4}-4 d^{3} e f \,g^{3}+6 d^{2} e^{2} f^{2} g^{2}-4 d \,e^{3} f^{3} g +e^{4} f^{4}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{5} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\) | \(755\) |
risch | \(\text {Expression too large to display}\) | \(1461\) |
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 \begin {gather*} \text {Exception raised: ValueError} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x\right )^{4}}{\left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 (f+g\,x\right )}^4}{\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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