Optimal. Leaf size=288 \[ -\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (4 a c d-a e^2+8 b c^2\right ) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{8 c^{3/2} \left (a+b x^2\right )}-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{2 c x^2 \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2+e x} (2 x (a d+2 b c)+a e)}{4 c x \left (a+b x^2\right )}+\frac {b e \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{2 \sqrt {d} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.78, antiderivative size = 288, 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, 812, 843, 621, 206, 724} \begin {gather*} -\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (4 a c d-a e^2+8 b c^2\right ) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{8 c^{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}}{2 c x^2 \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} (2 x (a d+2 b c)+a e)}{4 c x \left (a+b x^2\right )}+\frac {b e \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{2 \sqrt {d} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
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^3} \, 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^3} \, 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}}{2 c x^2 \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 (2 b c+a d) x) \sqrt {c+e x+d x^2}}{x^2} \, dx}{2 c \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {(a e+2 (2 b c+a d) x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 c x \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}}{2 c x^2 \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {b \left (8 b c^2+4 a c d-a e^2\right )+4 b^2 c e x}{x \sqrt {c+e x+d x^2}} \, dx}{4 c \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {(a e+2 (2 b c+a d) x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 c x \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}}{2 c x^2 \left (a+b x^2\right )}+\frac {\left (b^2 e \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int \frac {1}{\sqrt {c+e x+d x^2}} \, dx}{2 a b+2 b^2 x^2}+\frac {\left (b \left (8 b c^2+4 a c d-a e^2\right ) \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}{4 c \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {(a e+2 (2 b c+a d) x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 c x \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}}{2 c x^2 \left (a+b x^2\right )}+\frac {\left (2 b^2 e \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 a b+2 b^2 x^2}-\frac {\left (b \left (8 b c^2+4 a c d-a e^2\right ) \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 c \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {(a e+2 (2 b c+a d) x) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 c x \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}}{2 c x^2 \left (a+b x^2\right )}+\frac {b e \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 )}{2 \sqrt {d} \left (a+b x^2\right )}-\frac {\left (8 b c^2+4 a c d-a e^2\right ) \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 )}{8 c^{3/2} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 177, normalized size = 0.61 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (4 b c^{3/2} e x^2 \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+x (d x+e)}}\right )-\sqrt {d} \left (x^2 \left (4 a c d-a e^2+8 b c^2\right ) \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+x (d x+e)}}\right )+2 \sqrt {c} \sqrt {c+x (d x+e)} \left (2 a c+a e x-4 b c x^2\right )\right )\right )}{8 c^{3/2} \sqrt {d} x^2 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.84, size = 160, normalized size = 0.56 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (\frac {\left (-4 a c d+a e^2-8 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2+e x}-\sqrt {d} x}{\sqrt {c}}\right )}{4 c^{3/2}}+\frac {\sqrt {c+d x^2+e x} \left (-2 a c-a e x+4 b c x^2\right )}{4 c x^2}-\frac {b e \log \left (-2 \sqrt {d} \sqrt {c+d x^2+e x}+2 d x+e\right )}{2 \sqrt {d}}\right )}{a+b x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.42, size = 749, normalized size = 2.60 \begin {gather*} \left [\frac {4 \, b c^{2} \sqrt {d} e x^{2} \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 (8 \, b c^{2} d + 4 \, a c d^{2} - a d e^{2}\right )} \sqrt {c} x^{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 ) + 4 \, {\left (4 \, b c^{2} d x^{2} - a c d e x - 2 \, a c^{2} d\right )} \sqrt {d x^{2} + e x + c}}{16 \, c^{2} d x^{2}}, -\frac {8 \, b c^{2} \sqrt {-d} e x^{2} \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 (8 \, b c^{2} d + 4 \, a c d^{2} - a d e^{2}\right )} \sqrt {c} x^{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 ) - 4 \, {\left (4 \, b c^{2} d x^{2} - a c d e x - 2 \, a c^{2} d\right )} \sqrt {d x^{2} + e x + c}}{16 \, c^{2} d x^{2}}, \frac {2 \, b c^{2} \sqrt {d} e x^{2} \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 (8 \, b c^{2} d + 4 \, a c d^{2} - a d e^{2}\right )} \sqrt {-c} x^{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 ) + 2 \, {\left (4 \, b c^{2} d x^{2} - a c d e x - 2 \, a c^{2} d\right )} \sqrt {d x^{2} + e x + c}}{8 \, c^{2} d x^{2}}, -\frac {4 \, b c^{2} \sqrt {-d} e x^{2} \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 (8 \, b c^{2} d + 4 \, a c d^{2} - a d e^{2}\right )} \sqrt {-c} x^{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 ) - 2 \, {\left (4 \, b c^{2} d x^{2} - a c d e x - 2 \, a c^{2} d\right )} \sqrt {d x^{2} + e x + c}}{8 \, c^{2} d x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 370, normalized size = 1.28 \begin {gather*} -\frac {b e \log \left ({\left | -2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} d - \sqrt {d} e \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{2 \, \sqrt {d}} + \sqrt {d x^{2} + x e + c} b \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {{\left (8 \, b c^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, a c d \mathrm {sgn}\left (b x^{2} + a\right ) - a e^{2} \mathrm {sgn}\left (b x^{2} + a\right )\right )} \arctan \left (-\frac {\sqrt {d} x - \sqrt {d x^{2} + x e + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c} c} + \frac {4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{3} a c d \mathrm {sgn}\left (b x^{2} + a\right ) + 8 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{2} a c \sqrt {d} e \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} a c^{2} d \mathrm {sgn}\left (b x^{2} + a\right ) + {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{3} a e^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + {\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )} a c e^{2} \mathrm {sgn}\left (b x^{2} + a\right )}{4 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + x e + c}\right )}^{2} - c\right )}^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 329, normalized size = 1.14 \begin {gather*} \frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-4 a \,c^{\frac {3}{2}} d^{\frac {5}{2}} x^{2} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )+a \sqrt {c}\, d^{\frac {3}{2}} e^{2} x^{2} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )-8 b \,c^{\frac {5}{2}} d^{\frac {3}{2}} x^{2} \ln \left (\frac {e x +2 c +2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {c}}{x}\right )+4 b \,c^{2} d e \,x^{2} \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}} e \,x^{3}+4 \sqrt {d \,x^{2}+e x +c}\, a c \,d^{\frac {5}{2}} x^{2}-2 \sqrt {d \,x^{2}+e x +c}\, a \,d^{\frac {3}{2}} e^{2} x^{2}+8 \sqrt {d \,x^{2}+e x +c}\, b \,c^{2} d^{\frac {3}{2}} x^{2}+2 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a \,d^{\frac {3}{2}} e x -4 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} a c \,d^{\frac {3}{2}}\right )}{8 \left (b \,x^{2}+a \right ) c^{2} d^{\frac {3}{2}} x^{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 = -0.00 \begin {gather*} \int \frac {\sqrt {{\left (b\,x^2+a\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x^3} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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