Optimal. Leaf size=116 \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c (2 A c d-a B e)-4 b c (A e+B d)+3 b^2 B e\right )}{8 c^{5/2}}-\frac {\sqrt {a+b x+c x^2} (-4 c (A e+B d)+3 b B e-2 B c e x)}{4 c^2} \]
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Rubi [A] time = 0.09, antiderivative size = 115, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {779, 621, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 a B c e-4 b c (A e+B d)+8 A c^2 d+3 b^2 B e\right )}{8 c^{5/2}}-\frac {\sqrt {a+b x+c x^2} (-4 c (A e+B d)+3 b B e-2 B c e x)}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{\sqrt {a+b x+c x^2}} \, dx &=-\frac {(3 b B e-4 c (B d+A e)-2 B c e x) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {\left (8 A c^2 d+3 b^2 B e-4 a B c e-4 b c (B d+A e)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^2}\\ &=-\frac {(3 b B e-4 c (B d+A e)-2 B c e x) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {\left (8 A c^2 d+3 b^2 B e-4 a B c e-4 b c (B d+A 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 )}{4 c^2}\\ &=-\frac {(3 b B e-4 c (B d+A e)-2 B c e x) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {\left (8 A c^2 d+3 b^2 B e-4 a B c e-4 b c (B d+A e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 115, normalized size = 0.99 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \left (-2 a B c e-2 b c (A e+B d)+4 A c^2 d+\frac {3}{2} b^2 B e\right )+\sqrt {c} \sqrt {a+x (b+c x)} (4 A c e+B (-3 b e+4 c d+2 c e x))}{4 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.59, size = 120, normalized size = 1.03 \begin {gather*} \frac {\log \left (-2 c^{5/2} \sqrt {a+b x+c x^2}+b c^2+2 c^3 x\right ) \left (4 a B c e+4 A b c e-8 A c^2 d-3 b^2 B e+4 b B c d\right )}{8 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} (4 A c e-3 b B e+4 B c d+2 B c e x)}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 273, normalized size = 2.35 \begin {gather*} \left [\frac {{\left (4 \, {\left (B b c - 2 \, A c^{2}\right )} d - {\left (3 \, B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} e\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 (2 \, B c^{2} e x + 4 \, B c^{2} d - {\left (3 \, B b c - 4 \, A c^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{16 \, c^{3}}, \frac {{\left (4 \, {\left (B b c - 2 \, A c^{2}\right )} d - {\left (3 \, B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} e\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 (2 \, B c^{2} e x + 4 \, B c^{2} d - {\left (3 \, B b c - 4 \, A c^{2}\right )} e\right )} \sqrt {c x^{2} + b x + a}}{8 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 119, normalized size = 1.03 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x + a} {\left (\frac {2 \, B x e}{c} + \frac {4 \, B c d - 3 \, B b e + 4 \, A c e}{c^{2}}\right )} + \frac {{\left (4 \, B b c d - 8 \, A c^{2} d - 3 \, B b^{2} e + 4 \, B a c e + 4 \, A b c e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, 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 = 243, normalized size = 2.09 \begin {gather*} -\frac {A b 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 {A d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {B a 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 {3 B \,b^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {B b d \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 x}{2 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, A e}{c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B b e}{4 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B d}{c} \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 (A+B\,x\right )\,\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \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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