Optimal. Leaf size=211 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 a d e+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{8 d^{3/2} (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2+e x} (4 a d+2 b d x+b e)}{4 d (a+b x)}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{a+b x} \]
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Rubi [A] time = 0.22, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1000, 814, 843, 621, 206, 724} \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (4 a d e+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{8 d^{3/2} (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2+e x} (4 a d+2 b d x+b e)}{4 d (a+b x)}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{a+b x} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 814
Rule 843
Rule 1000
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (2 a b+2 b^2 x\right ) \sqrt {c+e x+d x^2}}{x} \, dx}{2 a b+2 b^2 x}\\ &=\frac {(4 a d+b e+2 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 d (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {-8 a b c d-b \left (4 a d e+b \left (4 c d-e^2\right )\right ) x}{x \sqrt {c+e x+d x^2}} \, dx}{4 d \left (2 a b+2 b^2 x\right )}\\ &=\frac {(4 a d+b e+2 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 d (a+b x)}+\frac {\left (2 a b c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{x \sqrt {c+e x+d x^2}} \, dx}{2 a b+2 b^2 x}+\frac {\left (b \left (4 b c d+4 a d e-b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{4 d \left (2 a b+2 b^2 x\right )}\\ &=\frac {(4 a d+b e+2 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 d (a+b x)}-\frac {\left (4 a b c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {2 c+e x}{\sqrt {c+e x+d x^2}}\right )}{2 a b+2 b^2 x}+\frac {\left (b \left (4 b c d+4 a d e-b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 d-x^2} \, dx,x,\frac {e+2 d x}{\sqrt {c+e x+d x^2}}\right )}{2 d \left (2 a b+2 b^2 x\right )}\\ &=\frac {(4 a d+b e+2 b d x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 d (a+b x)}+\frac {\left (4 b c d+4 a d e-b e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{a+b x}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 149, normalized size = 0.71 \[ \frac {\sqrt {(a+b x)^2} \left (\left (4 a d e+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )+2 \sqrt {d} \left (\sqrt {c+x (d x+e)} (4 a d+b (2 d x+e))-4 a \sqrt {c} d \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+x (d x+e)}}\right )\right )\right )}{8 d^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.54, size = 651, normalized size = 3.09 \[ \left [\frac {8 \, a \sqrt {c} d^{2} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - {\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d e x - 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 4 \, {\left (2 \, b d^{2} x + 4 \, a d^{2} + b d e\right )} \sqrt {d x^{2} + e x + c}}{16 \, d^{2}}, \frac {4 \, a \sqrt {c} d^{2} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} - 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - {\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) + 2 \, {\left (2 \, b d^{2} x + 4 \, a d^{2} + b d e\right )} \sqrt {d x^{2} + e x + c}}{8 \, d^{2}}, \frac {16 \, a \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) - {\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d e x - 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 4 \, {\left (2 \, b d^{2} x + 4 \, a d^{2} + b d e\right )} \sqrt {d x^{2} + e x + c}}{16 \, d^{2}}, \frac {8 \, a \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) - {\left (4 \, b c d + 4 \, a d e - b e^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) + 2 \, {\left (2 \, b d^{2} x + 4 \, a d^{2} + b d e\right )} \sqrt {d x^{2} + e x + c}}{8 \, d^{2}}\right ] \]
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 [C] time = 0.01, size = 214, normalized size = 1.01 \[ -\frac {\left (8 a \sqrt {c}\, d^{\frac {5}{2}} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )-4 a \,d^{2} e \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-4 b c \,d^{2} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+b d \,e^{2} \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-4 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {5}{2}} x -8 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {5}{2}}-2 \sqrt {d \,x^{2}+e x +c}\, b \,d^{\frac {3}{2}} e \right ) \mathrm {csgn}\left (b x +a \right )}{8 d^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{2} + e x + c} \sqrt {{\left (b x + a\right )}^{2}}}{x}\,{d x} \]
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 \[ \int \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x} \,d x \]
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 \[ \int \frac {\sqrt {c + d x^{2} + e x} \sqrt {\left (a + b x\right )^{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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