Optimal. Leaf size=148 \[ \frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {a x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{2 (a+b x)}+\frac {a c \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d} (a+b x)} \]
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Rubi [A] time = 0.06, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {970, 641, 195, 217, 206} \begin {gather*} \frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {a x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{2 (a+b x)}+\frac {a c \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 641
Rule 970
Rubi steps
\begin {align*} \int \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 \left (2 a b+2 b^2 x\right ) \sqrt {c+d x^2} \, dx}{2 a b+2 b^2 x}\\ &=\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {\left (2 a b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \sqrt {c+d x^2} \, dx}{2 a b+2 b^2 x}\\ &=\frac {a x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{2 (a+b x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {\left (a b c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 a b+2 b^2 x}\\ &=\frac {a x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{2 (a+b x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {\left (a b c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a b+2 b^2 x}\\ &=\frac {a x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{2 (a+b x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{3 d (a+b x)}+\frac {a c \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d} (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 85, normalized size = 0.57 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (\sqrt {c+d x^2} \left (3 a d x+2 b \left (c+d x^2\right )\right )+3 a c \sqrt {d} \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )\right )}{6 d (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 87, normalized size = 0.59 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (\frac {\sqrt {c+d x^2} \left (3 a d x+2 b c+2 b d x^2\right )}{6 d}-\frac {a c \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{2 \sqrt {d}}\right )}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 128, normalized size = 0.86 \begin {gather*} \left [\frac {3 \, a c \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (2 \, b d x^{2} + 3 \, a d x + 2 \, b c\right )} \sqrt {d x^{2} + c}}{12 \, d}, -\frac {3 \, a c \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (2 \, b d x^{2} + 3 \, a d x + 2 \, b c\right )} \sqrt {d x^{2} + c}}{6 \, d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 79, normalized size = 0.53 \begin {gather*} -\frac {a c \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{2 \, \sqrt {d}} + \frac {1}{6} \, \sqrt {d x^{2} + c} {\left ({\left (2 \, b x \mathrm {sgn}\left (b x + a\right ) + 3 \, a \mathrm {sgn}\left (b x + a\right )\right )} x + \frac {2 \, b 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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maple [C] time = 0.01, size = 65, normalized size = 0.44 \begin {gather*} \frac {\left (3 a c d \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+3 \sqrt {d \,x^{2}+c}\, a \,d^{\frac {3}{2}} x +2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b \sqrt {d}\right ) \mathrm {csgn}\left (b x +a \right )}{6 d^{\frac {3}{2}}} \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 \sqrt {d x^{2} + c} \sqrt {{\left (b x + a\right )}^{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 = -0.01 \begin {gather*} \int \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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \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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