Optimal. Leaf size=202 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a-b x) \sqrt {c+d x^2+e x}}{x (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (2 a d+b e) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{2 \sqrt {d} (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a e+2 b c) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{2 \sqrt {c} (a+b x)} \]
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Rubi [A] time = 0.19, antiderivative size = 202, 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, 812, 843, 621, 206, 724} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a-b x) \sqrt {c+d x^2+e x}}{x (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (2 a d+b e) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{2 \sqrt {d} (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a e+2 b c) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{2 \sqrt {c} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
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^2} \, 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^2} \, dx}{2 a b+2 b^2 x}\\ &=-\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {-2 b (2 b c+a e)-2 b (2 a d+b e) x}{x \sqrt {c+e x+d x^2}} \, dx}{2 \left (2 a b+2 b^2 x\right )}\\ &=-\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x (a+b x)}+\frac {\left (b (2 b c+a e) \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 (2 a d+b e) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{2 a b+2 b^2 x}\\ &=-\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x (a+b x)}-\frac {\left (2 b (2 b c+a e) \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 (2 b (2 a d+b e) \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 a b+2 b^2 x}\\ &=-\frac {(a-b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x (a+b x)}+\frac {(2 a d+b e) \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 )}{2 \sqrt {d} (a+b x)}-\frac {(2 b c+a e) \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 )}{2 \sqrt {c} (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 155, normalized size = 0.77 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (\sqrt {c} x (2 a d+b e) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )+\sqrt {d} \left (2 \sqrt {c} (b x-a) \sqrt {c+x (d x+e)}-x (a e+2 b c) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+x (d x+e)}}\right )\right )\right )}{2 \sqrt {c} \sqrt {d} x (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 136, normalized size = 0.67 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (\frac {(b x-a) \sqrt {c+d x^2+e x}}{x}+\frac {(-2 a d-b e) \log \left (-2 \sqrt {d} \sqrt {c+d x^2+e x}+2 d x+e\right )}{2 \sqrt {d}}+\frac {(-a e-2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2+e x}-\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 647, normalized size = 3.20 \begin {gather*} \left [\frac {{\left (2 \, a c d + b c e\right )} \sqrt {d} x \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 ) + {\left (2 \, b c d + a d e\right )} \sqrt {c} x \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 ) + 4 \, {\left (b c d x - a c d\right )} \sqrt {d x^{2} + e x + c}}{4 \, c d x}, -\frac {2 \, {\left (2 \, a c d + b c e\right )} \sqrt {-d} x \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 ) - {\left (2 \, b c d + a d e\right )} \sqrt {c} x \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 ) - 4 \, {\left (b c d x - a c d\right )} \sqrt {d x^{2} + e x + c}}{4 \, c d x}, \frac {2 \, {\left (2 \, b c d + a d e\right )} \sqrt {-c} x \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 (2 \, a c d + b c e\right )} \sqrt {d} x \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 (b c d x - a c d\right )} \sqrt {d x^{2} + e x + c}}{4 \, c d x}, \frac {{\left (2 \, b c d + a d e\right )} \sqrt {-c} x \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 (2 \, a c d + b c e\right )} \sqrt {-d} x \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 (b c d x - a c d\right )} \sqrt {d x^{2} + e x + c}}{2 \, c d x}\right ] \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 249, normalized size = 1.23 \begin {gather*} \frac {\left (2 a c \,d^{2} x \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-a \sqrt {c}\, d^{\frac {3}{2}} e x \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )-2 b \,c^{\frac {3}{2}} d^{\frac {3}{2}} x \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )+b c d e x \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+2 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {5}{2}} x^{2}+2 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {3}{2}} e x +2 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {3}{2}} x -2 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,d^{\frac {3}{2}}\right ) \mathrm {csgn}\left (b x +a \right )}{2 c \,d^{\frac {3}{2}} x} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {d x^{2} + e x + c} \sqrt {{\left (b x + a\right )}^{2}}}{x^{2}}\,{d x} \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 = -0.00 \begin {gather*} \int \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x^2} \,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 {\sqrt {c + d x^{2} + e x} \sqrt {\left (a + b x\right )^{2}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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