Optimal. Leaf size=148 \[ \frac {e \left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e (4 a e+3 b d)+3 b^2 e^2+2 c e x (2 c d-b e)+8 c^2 d^2\right )}{6 c^2}+\frac {2}{3} (d+e x)^2 \sqrt {a+b x+c x^2} \]
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Rubi [A] time = 0.20, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {832, 779, 621, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-2 c e (4 a e+3 b d)+3 b^2 e^2+2 c e x (2 c d-b e)+8 c^2 d^2\right )}{6 c^2}+\frac {e \left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 c^{5/2}}+\frac {2}{3} (d+e x)^2 \sqrt {a+b x+c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^2}{\sqrt {a+b x+c x^2}} \, dx &=\frac {2}{3} (d+e x)^2 \sqrt {a+b x+c x^2}+\frac {\int \frac {(d+e x) (2 c (b d-2 a e)+2 c (2 c d-b e) x)}{\sqrt {a+b x+c x^2}} \, dx}{3 c}\\ &=\frac {2}{3} (d+e x)^2 \sqrt {a+b x+c x^2}+\frac {\left (8 c^2 d^2+3 b^2 e^2-2 c e (3 b d+4 a e)+2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{6 c^2}+\frac {\left (\left (b^2-4 a c\right ) e (2 c d-b e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{4 c^2}\\ &=\frac {2}{3} (d+e x)^2 \sqrt {a+b x+c x^2}+\frac {\left (8 c^2 d^2+3 b^2 e^2-2 c e (3 b d+4 a e)+2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{6 c^2}+\frac {\left (\left (b^2-4 a c\right ) e (2 c d-b e)\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 )}{2 c^2}\\ &=\frac {2}{3} (d+e x)^2 \sqrt {a+b x+c x^2}+\frac {\left (8 c^2 d^2+3 b^2 e^2-2 c e (3 b d+4 a e)+2 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{6 c^2}+\frac {\left (b^2-4 a c\right ) e (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{4 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 185, normalized size = 1.25 \begin {gather*} \frac {-8 a^2 c e^2+a \left (3 b^2 e^2-2 b c e (3 d+5 e x)+4 c^2 \left (3 d^2+3 d e x-e^2 x^2\right )\right )+x (b+c x) \left (3 b^2 e^2-2 b c e (3 d+e x)+4 c^2 \left (3 d^2+3 d e x+e^2 x^2\right )\right )}{6 c^2 \sqrt {a+x (b+c x)}}-\frac {e \left (b^2-4 a c\right ) (b e-2 c d) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{4 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 154, normalized size = 1.04 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-8 a c e^2+3 b^2 e^2-6 b c d e-2 b c e^2 x+12 c^2 d^2+12 c^2 d e x+4 c^2 e^2 x^2\right )}{6 c^2}+\frac {\left (-4 a b c e^2+8 a c^2 d e+b^3 e^2-2 b^2 c d e\right ) \log \left (-2 c^{5/2} \sqrt {a+b x+c x^2}+b c^2+2 c^3 x\right )}{4 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 335, normalized size = 2.26 \begin {gather*} \left [\frac {3 \, {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (4 \, c^{3} e^{2} x^{2} + 12 \, c^{3} d^{2} - 6 \, b c^{2} d e + {\left (3 \, b^{2} c - 8 \, a c^{2}\right )} e^{2} + 2 \, {\left (6 \, c^{3} d e - b c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{24 \, c^{3}}, -\frac {3 \, {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (4 \, c^{3} e^{2} x^{2} + 12 \, c^{3} d^{2} - 6 \, b c^{2} d e + {\left (3 \, b^{2} c - 8 \, a c^{2}\right )} e^{2} + 2 \, {\left (6 \, c^{3} d e - b c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{12 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 146, normalized size = 0.99 \begin {gather*} \frac {1}{6} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (2 \, x e^{2} + \frac {6 \, c^{2} d e - b c e^{2}}{c^{2}}\right )} x + \frac {12 \, c^{2} d^{2} - 6 \, b c d e + 3 \, b^{2} e^{2} - 8 \, a c e^{2}}{c^{2}}\right )} - \frac {{\left (2 \, b^{2} c d e - 8 \, a c^{2} d e - b^{3} e^{2} + 4 \, a b c e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{4 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 280, normalized size = 1.89 \begin {gather*} \frac {2 \sqrt {c \,x^{2}+b x +a}\, e^{2} x^{2}}{3}+\frac {a b \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}-\frac {2 a d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {b^{3} e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {5}{2}}}+\frac {b^{2} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, b \,e^{2} x}{3 c}+2 \sqrt {c \,x^{2}+b x +a}\, d e x -\frac {4 \sqrt {c \,x^{2}+b x +a}\, a \,e^{2}}{3 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, b^{2} e^{2}}{2 c^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, b d e}{c}+2 \sqrt {c \,x^{2}+b x +a}\, d^{2} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (b+2\,c\,x\right )\,{\left (d+e\,x\right )}^2}{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b + 2 c x\right ) \left (d + e x\right )^{2}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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