Optimal. Leaf size=431 \[ -\frac {g \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{16 c^{7/2} e^4}+\frac {g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{24 c^3 e^3}+\frac {g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{12 c^2 e^3}+\frac {(e f-d g)^4 \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^4 \sqrt {a e^2-b d e+c d^2}}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3} \]
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Rubi [A] time = 1.37, 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 = {1653, 843, 621, 206, 724} \[ \frac {g^2 \sqrt {a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{24 c^3 e^3}-\frac {g \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (4 d^2 e f g^2-d^3 g^3-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{16 c^{7/2} e^4}+\frac {g^3 (d+e x) \sqrt {a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{12 c^2 e^3}+\frac {(e f-d g)^4 \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^4 \sqrt {a e^2-b d e+c d^2}}+\frac {g^4 (d+e x)^2 \sqrt {a+b x+c x^2}}{3 c e^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 843
Rule 1653
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 ) \operatorname {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 ) \operatorname {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 = 0.89, size = 553, normalized size = 1.28 \[ \frac {\frac {6 e^2 g (e f-d g) \left (\left (-4 c g (a g+2 b f)+3 b^2 g^2+8 c^2 f^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+6 \sqrt {c} g \sqrt {a+x (b+c x)} (2 c f-b g)\right )}{c^{5/2}}+\frac {e^3 g \left (\frac {2 g \sqrt {a+x (b+c x)} \left (-2 c g (8 a g+27 b f+5 b g x)+15 b^2 g^2+4 c^2 f (16 f+5 g x)\right )}{c^2}+\frac {3 (2 c f-b g) \left (-4 c g (3 a g+2 b f)+5 b^2 g^2+8 c^2 f^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{5/2}}\right )}{c}+\frac {24 e g (2 c f-b g) (e f-d g)^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{3/2}}+\frac {48 (e f-d g)^4 \tanh ^{-1}\left (\frac {-2 a e+b (d-e x)+2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{\sqrt {e (a e-b d)+c d^2}}+\frac {24 e^2 g^2 (f+g x) \sqrt {a+x (b+c x)} (e f-d g)}{c}+\frac {48 e g^2 \sqrt {a+x (b+c x)} (e f-d g)^2}{c}+\frac {48 g (e f-d g)^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}+\frac {16 e^3 g^2 (f+g x)^2 \sqrt {a+x (b+c x)}}{c}}{48 e^4} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1597, normalized size = 3.71 \[ \text {result too large to display} \]
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] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^4}{\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {\left (f + g x\right )^{4}}{\left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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