Optimal. Leaf size=286 \[ \frac {\left (8 a d^2-b e^2-2 b d e x\right ) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d^2 \left (a+b x^2\right )}+\frac {b \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {e \left (8 a d^2-b \left (4 c d-e^2\right )\right ) \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{16 d^{5/2} \left (a+b x^2\right )}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{a+b x^2} \]
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Rubi [A]
time = 0.50, antiderivative size = 286, 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 = {6876, 1667,
828, 857, 635, 212, 738} \begin {gather*} \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2+e x} \left (8 a d^2-2 b d e x-b e^2\right )}{8 d^2 \left (a+b x^2\right )}+\frac {e \sqrt {a^2+2 a b x^2+b^2 x^4} \left (8 a d^2-b \left (4 c d-e^2\right )\right ) \tanh ^{-1}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{16 d^{5/2} \left (a+b x^2\right )}+\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2+e x\right )^{3/2}}{3 d \left (a+b x^2\right )}-\frac {a \sqrt {c} \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 )}{a+b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 828
Rule 857
Rule 1667
Rule 6876
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} \, 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} \, dx}{2 a b+2 b^2 x^2}\\ &=\frac {b \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (6 a b d-3 b^2 e x\right ) \sqrt {c+e x+d x^2}}{x} \, dx}{3 d \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {\left (8 a d^2-b e^2-2 b d e x\right ) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d^2 \left (a+b x^2\right )}+\frac {b \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}-\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {-24 a b c d^2-\frac {3}{2} b e \left (8 a d^2-b \left (4 c d-e^2\right )\right ) x}{x \sqrt {c+e x+d x^2}} \, dx}{12 d^2 \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {\left (8 a d^2-b e^2-2 b d e x\right ) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d^2 \left (a+b x^2\right )}+\frac {b \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {\left (2 a b c \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 e \left (8 a d^2-b \left (4 c d-e^2\right )\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}{8 d^2 \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {\left (8 a d^2-b e^2-2 b d e x\right ) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d^2 \left (a+b x^2\right )}+\frac {b \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}-\frac {\left (4 a b c \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \text {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 e \left (8 a d^2-b \left (4 c d-e^2\right )\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \text {Subst}\left (\int \frac {1}{4 d-x^2} \, dx,x,\frac {e+2 d x}{\sqrt {c+e x+d x^2}}\right )}{4 d^2 \left (2 a b+2 b^2 x^2\right )}\\ &=\frac {\left (8 a d^2-b e^2-2 b d e x\right ) \sqrt {c+e x+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 d^2 \left (a+b x^2\right )}+\frac {b \left (c+e x+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {e \left (8 a d^2-b \left (4 c d-e^2\right )\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 )}{16 d^{5/2} \left (a+b x^2\right )}-\frac {a \sqrt {c} \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 )}{a+b x^2}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 174, normalized size = 0.61 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (2 \sqrt {d} \sqrt {c+x (e+d x)} \left (24 a d^2+b \left (8 c d-3 e^2+2 d e x+8 d^2 x^2\right )\right )+96 a \sqrt {c} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} x-\sqrt {c+x (e+d x)}}{\sqrt {c}}\right )-3 e \left (8 a d^2+b \left (-4 c d+e^2\right )\right ) \log \left (e+2 d x-2 \sqrt {d} \sqrt {c+x (e+d x)}\right )\right )}{48 d^{5/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 251, normalized size = 0.88
method | result | size |
default | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (16 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} d^{\frac {5}{2}} b -12 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {5}{2}} b e x -48 d^{\frac {7}{2}} \sqrt {c}\, \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a +48 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {7}{2}} a -6 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {3}{2}} b \,e^{2}+24 d^{3} \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) a e -12 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b c \,d^{2} e +3 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b d \,e^{3}\right )}{48 \left (b \,x^{2}+a \right ) d^{\frac {7}{2}}}\) | \(251\) |
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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Fricas [A]
time = 0.84, size = 771, normalized size = 2.70 \begin {gather*} \left [\frac {48 \, a \sqrt {c} d^{3} \log \left (\frac {4 \, c d x^{2} + x^{2} e^{2} + 8 \, c x e - 4 \, \sqrt {d x^{2} + x e + c} {\left (x e + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) + 3 \, {\left (b e^{3} - 4 \, {\left (b c d - 2 \, a d^{2}\right )} e\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d x e + 4 \, \sqrt {d x^{2} + x e + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 4 \, {\left (8 \, b d^{3} x^{2} + 2 \, b d^{2} x e + 8 \, b c d^{2} + 24 \, a d^{3} - 3 \, b d e^{2}\right )} \sqrt {d x^{2} + x e + c}}{96 \, d^{3}}, \frac {24 \, a \sqrt {c} d^{3} \log \left (\frac {4 \, c d x^{2} + x^{2} e^{2} + 8 \, c x e - 4 \, \sqrt {d x^{2} + x e + c} {\left (x e + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - 3 \, {\left (b e^{3} - 4 \, {\left (b c d - 2 \, a d^{2}\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + x e + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d x e + c d\right )}}\right ) + 2 \, {\left (8 \, b d^{3} x^{2} + 2 \, b d^{2} x e + 8 \, b c d^{2} + 24 \, a d^{3} - 3 \, b d e^{2}\right )} \sqrt {d x^{2} + x e + c}}{48 \, d^{3}}, \frac {96 \, a \sqrt {-c} d^{3} \arctan \left (\frac {\sqrt {d x^{2} + x e + c} {\left (x e + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c x e + c^{2}\right )}}\right ) + 3 \, {\left (b e^{3} - 4 \, {\left (b c d - 2 \, a d^{2}\right )} e\right )} \sqrt {d} \log \left (8 \, d^{2} x^{2} + 8 \, d x e + 4 \, \sqrt {d x^{2} + x e + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + 4 \, {\left (8 \, b d^{3} x^{2} + 2 \, b d^{2} x e + 8 \, b c d^{2} + 24 \, a d^{3} - 3 \, b d e^{2}\right )} \sqrt {d x^{2} + x e + c}}{96 \, d^{3}}, \frac {48 \, a \sqrt {-c} d^{3} \arctan \left (\frac {\sqrt {d x^{2} + x e + c} {\left (x e + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c x e + c^{2}\right )}}\right ) - 3 \, {\left (b e^{3} - 4 \, {\left (b c d - 2 \, a d^{2}\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{2} + x e + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d x e + c d\right )}}\right ) + 2 \, {\left (8 \, b d^{3} x^{2} + 2 \, b d^{2} x e + 8 \, b c d^{2} + 24 \, a d^{3} - 3 \, b d e^{2}\right )} \sqrt {d x^{2} + x e + c}}{48 \, d^{3}}\right ] \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^{2}\right )^{2}}}{x}\, dx \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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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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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