Optimal. Leaf size=161 \[ -\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)}-\frac {b c^2 \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1015, 794, 201,
223, 212} \begin {gather*} -\frac {b c^2 \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 794
Rule 1015
Rubi steps
\begin {align*} \int x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x \left (2 a b+2 b^2 x\right ) \sqrt {c+d x^2} \, dx}{2 a b+2 b^2 x}\\ &=\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)}-\frac {\left (b^2 c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \sqrt {c+d x^2} \, dx}{2 d \left (2 a b+2 b^2 x\right )}\\ &=-\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)}-\frac {\left (b^2 c^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{4 d \left (2 a b+2 b^2 x\right )}\\ &=-\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)}-\frac {\left (b^2 c^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{4 d \left (2 a b+2 b^2 x\right )}\\ &=-\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)}-\frac {b c^2 \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 95, normalized size = 0.59 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (\sqrt {d} \sqrt {c+d x^2} \left (8 a \left (c+d x^2\right )+3 b x \left (c+2 d x^2\right )\right )+3 b c^2 \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )\right )}{24 d^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.08, size = 83, normalized size = 0.52
method | result | size |
default | \(\frac {\mathrm {csgn}\left (b x +a \right ) \left (6 \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {d}\, b x +8 a \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {d}-3 \sqrt {d \,x^{2}+c}\, \sqrt {d}\, b c x -3 \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right ) b \,c^{2}\right )}{24 d^{\frac {3}{2}}}\) | \(83\) |
risch | \(\frac {\left (6 b d \,x^{3}+8 a d \,x^{2}+3 b c x +8 a c \right ) \sqrt {d \,x^{2}+c}\, \sqrt {\left (b x +a \right )^{2}}}{24 d \left (b x +a \right )}-\frac {c^{2} b \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right ) \sqrt {\left (b x +a \right )^{2}}}{8 d^{\frac {3}{2}} \left (b x +a \right )}\) | \(97\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 157, normalized size = 0.98 \begin {gather*} \left [\frac {3 \, b c^{2} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (6 \, b d^{2} x^{3} + 8 \, a d^{2} x^{2} + 3 \, b c d x + 8 \, a c d\right )} \sqrt {d x^{2} + c}}{48 \, d^{2}}, \frac {3 \, b c^{2} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (6 \, b d^{2} x^{3} + 8 \, a d^{2} x^{2} + 3 \, b c d x + 8 \, a c d\right )} \sqrt {d x^{2} + c}}{24 \, d^{2}}\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 x \sqrt {c + d x^{2}} \sqrt {\left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.30, size = 98, normalized size = 0.61 \begin {gather*} \frac {b c^{2} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{8 \, d^{\frac {3}{2}}} + \frac {1}{24} \, \sqrt {d x^{2} + c} {\left ({\left (2 \, {\left (3 \, b x \mathrm {sgn}\left (b x + a\right ) + 4 \, a \mathrm {sgn}\left (b x + a\right )\right )} x + \frac {3 \, b c \mathrm {sgn}\left (b x + a\right )}{d}\right )} x + \frac {8 \, a c \mathrm {sgn}\left (b x + a\right )}{d}\right )} \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 = -0.01 \begin {gather*} \int x\,\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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