Optimal. Leaf size=270 \[ \frac {g \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e g (a e g-b d g+3 b e f)+3 b^2 e^2 g^2+8 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )}{8 c^{5/2} e^3}+\frac {3 g^2 \sqrt {a+b x+c x^2} (-b e g-2 c d g+4 c e f)}{4 c^2 e^2}+\frac {(e f-d g)^3 \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^3 \sqrt {a e^2-b d e+c d^2}}+\frac {g^3 (d+e x) \sqrt {a+b x+c x^2}}{2 c e^2} \]
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Rubi [A] time = 0.71, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {g \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e g (a e g-b d g+3 b e f)+3 b^2 e^2 g^2+8 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )}{8 c^{5/2} e^3}+\frac {3 g^2 \sqrt {a+b x+c x^2} (-b e g-2 c d g+4 c e f)}{4 c^2 e^2}+\frac {(e f-d g)^3 \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^3 \sqrt {a e^2-b d e+c d^2}}+\frac {g^3 (d+e x) \sqrt {a+b x+c x^2}}{2 c e^2} \]
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)^3}{(d+e x) \sqrt {a+b x+c x^2}} \, dx &=\frac {g^3 (d+e x) \sqrt {a+b x+c x^2}}{2 c e^2}+\frac {\int \frac {\frac {1}{2} e \left (4 c e^2 f^3-d (b d+2 a e) g^3\right )-e g \left (e (2 b d+a e) g^2-c \left (6 e^2 f^2-d^2 g^2\right )\right ) x+\frac {3}{2} e^2 g^2 (4 c e f-2 c d g-b e g) x^2}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2 c e^3}\\ &=\frac {3 g^2 (4 c e f-2 c d g-b e g) \sqrt {a+b x+c x^2}}{4 c^2 e^2}+\frac {g^3 (d+e x) \sqrt {a+b x+c x^2}}{2 c e^2}+\frac {\int \frac {\frac {1}{4} e^3 \left (8 c^2 e^2 f^3+3 b^2 d e g^3-4 c d g^2 (3 b e f-b d g+a e g)\right )+\frac {1}{4} e^3 g \left (3 b^2 e^2 g^2-4 c e g (3 b e f-b d g+a e g)+8 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{2 c^2 e^5}\\ &=\frac {3 g^2 (4 c e f-2 c d g-b e g) \sqrt {a+b x+c x^2}}{4 c^2 e^2}+\frac {g^3 (d+e x) \sqrt {a+b x+c x^2}}{2 c e^2}+\frac {(e f-d g)^3 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^3}+\frac {\left (g \left (3 b^2 e^2 g^2-4 c e g (3 b e f-b d g+a e g)+8 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^2 e^3}\\ &=\frac {3 g^2 (4 c e f-2 c d g-b e g) \sqrt {a+b x+c x^2}}{4 c^2 e^2}+\frac {g^3 (d+e x) \sqrt {a+b x+c x^2}}{2 c e^2}-\frac {\left (2 (e f-d g)^3\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^3}+\frac {\left (g \left (3 b^2 e^2 g^2-4 c e g (3 b e f-b d g+a e g)+8 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\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 )}{4 c^2 e^3}\\ &=\frac {3 g^2 (4 c e f-2 c d g-b e g) \sqrt {a+b x+c x^2}}{4 c^2 e^2}+\frac {g^3 (d+e x) \sqrt {a+b x+c x^2}}{2 c e^2}+\frac {g \left (3 b^2 e^2 g^2-4 c e g (3 b e f-b d g+a e g)+8 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2} e^3}+\frac {(e f-d g)^3 \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^3 \sqrt {c d^2-b d e+a e^2}}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 358, normalized size = 1.33 \[ \frac {\frac {e^2 g \left (-4 c g (a g+2 b f)+3 b^2 g^2+8 c^2 f^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{5/2}}+\frac {4 e g (2 c f-b g) (e f-d g) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{3/2}}+\frac {6 e^2 g^2 \sqrt {a+x (b+c x)} (2 c f-b g)}{c^2}+\frac {8 (e f-d g)^3 \tanh ^{-1}\left (\frac {-2 a e+b (d-e x)+2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{\sqrt {e (a e-b d)+c d^2}}+\frac {8 e g^2 \sqrt {a+x (b+c x)} (e f-d g)}{c}+\frac {8 g (e f-d g)^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}+\frac {4 e^2 g^2 (f+g x) \sqrt {a+x (b+c x)}}{c}}{8 e^3} \]
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 = 0.02, size = 1007, normalized size = 3.73 \[ \frac {d^{3} g^{3} \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -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 )}{\sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e^{4}}-\frac {3 d^{2} f \,g^{2} \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -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 )}{\sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e^{3}}+\frac {3 d \,f^{2} g \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -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 )}{\sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e^{2}}-\frac {f^{3} \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -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 )}{\sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e}-\frac {a \,g^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}} e}+\frac {3 b^{2} g^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}} e}+\frac {b d \,g^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}} e^{2}}-\frac {3 b f \,g^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}} e}+\frac {d^{2} g^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}\, e^{3}}-\frac {3 d f \,g^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}\, e^{2}}+\frac {3 f^{2} g \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}\, e}+\frac {\sqrt {c \,x^{2}+b x +a}\, g^{3} x}{2 c e}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b \,g^{3}}{4 c^{2} e}-\frac {\sqrt {c \,x^{2}+b x +a}\, d \,g^{3}}{c \,e^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, f \,g^{2}}{c e} \]
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 \[ \int \frac {{\left (f+g\,x\right )}^3}{\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {\left (f + g x\right )^{3}}{\left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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