Optimal. Leaf size=294 \[ \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (8 a d^2+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{8 d^{3/2} \left (a+b x^2\right )}-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{c x \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2+e x} (e (4 a d+b c)+2 d x (2 a d+b c))}{4 c d \left (a+b x^2\right )}-\frac {a e \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{2 \sqrt {c} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.77, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {6744, 1650, 814, 843, 621, 206, 724} \[ \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (8 a d^2+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{8 d^{3/2} \left (a+b x^2\right )}-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{c x \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2+e x} (e (4 a d+b c)+2 d x (2 a d+b c))}{4 c d \left (a+b x^2\right )}-\frac {a e \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{2 \sqrt {c} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 814
Rule 843
Rule 1650
Rule 6744
Rubi steps
\begin {align*} \int \frac {\sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (2 a b+2 b^2 x^2\right ) \sqrt {c+e x+d x^2}}{x^2} \, dx}{2 a b+2 b^2 x^2}\\ &=-\frac {a \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}-\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {(-a b e-2 b (b c+2 a d) x) \sqrt {c+e x+d x^2}}{x} \, dx}{c \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {((b c+4 a d) e+2 d (b c+2 a d) x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 c d \left (a+b x^2\right )}-\frac {a \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {4 a b c d e+b c \left (4 b c d+8 a d^2-b e^2\right ) x}{x \sqrt {c+e x+d x^2}} \, dx}{4 c d \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {((b c+4 a d) e+2 d (b c+2 a d) x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 c d \left (a+b x^2\right )}-\frac {a \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}+\frac {\left (a b e \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int \frac {1}{x \sqrt {c+e x+d x^2}} \, dx}{2 a b+2 b^2 x^2}+\frac {\left (b \left (4 b c d+8 a d^2-b e^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{4 d \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {((b c+4 a d) e+2 d (b c+2 a d) x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 c d \left (a+b x^2\right )}-\frac {a \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}-\frac {\left (2 a b e \sqrt {a^2+2 a b x^2+b^2 x^4}\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^2}+\frac {\left (b \left (4 b c d+8 a d^2-b e^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\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^2\right )}\\ &=\frac {((b c+4 a d) e+2 d (b c+2 a d) x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 c d \left (a+b x^2\right )}-\frac {a \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}+\frac {\left (4 b c d+8 a d^2-b e^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{8 d^{3/2} \left (a+b x^2\right )}-\frac {a e \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{2 \sqrt {c} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 171, normalized size = 0.58 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \left (\sqrt {c} x \left (8 a d^2+4 b c d-b e^2\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )+2 \sqrt {d} \left (\sqrt {c} \sqrt {c+x (d x+e)} (b x (2 d x+e)-4 a d)-2 a d e x \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+x (d x+e)}}\right )\right )\right )}{8 \sqrt {c} d^{3/2} x \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.44, size = 731, normalized size = 2.49 \[ \left [\frac {4 \, a \sqrt {c} d^{2} e 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 ) - {\left (4 \, b c^{2} d + 8 \, a c d^{2} - b c e^{2}\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 (2 \, b c d^{2} x^{2} + b c d e x - 4 \, a c d^{2}\right )} \sqrt {d x^{2} + e x + c}}{16 \, c d^{2} x}, \frac {2 \, a \sqrt {c} d^{2} e 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 ) - {\left (4 \, b c^{2} d + 8 \, a c d^{2} - b c e^{2}\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 (2 \, b c d^{2} x^{2} + b c d e x - 4 \, a c d^{2}\right )} \sqrt {d x^{2} + e x + c}}{8 \, c d^{2} x}, \frac {8 \, a \sqrt {-c} d^{2} e 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 (4 \, b c^{2} d + 8 \, a c d^{2} - b c e^{2}\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 (2 \, b c d^{2} x^{2} + b c d e x - 4 \, a c d^{2}\right )} \sqrt {d x^{2} + e x + c}}{16 \, c d^{2} x}, \frac {4 \, a \sqrt {-c} d^{2} e 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 (4 \, b c^{2} d + 8 \, a c d^{2} - b c e^{2}\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 (2 \, b c d^{2} x^{2} + b c d e x - 4 \, a c d^{2}\right )} \sqrt {d x^{2} + e x + c}}{8 \, c d^{2} x}\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 [A] time = 0.01, size = 288, normalized size = 0.98 \[ -\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-8 a c \,d^{3} x \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+4 a \sqrt {c}\, d^{\frac {5}{2}} e x \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )-4 b \,c^{2} d^{2} x \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )+b c d \,e^{2} x \ln \left (\frac {2 d x +e +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}}{2 \sqrt {d}}\right )-8 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {7}{2}} x^{2}-4 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {5}{2}} x^{2}-8 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {5}{2}} e x -2 \sqrt {d \,x^{2}+e x +c}\, b c \,d^{\frac {3}{2}} e x +8 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,d^{\frac {5}{2}}\right )}{8 \left (b \,x^{2}+a \right ) c \,d^{\frac {5}{2}} x} \]
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 \[ \int \frac {\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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