Optimal. Leaf size=212 \[ -\frac {a c^2 \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2} (8 b c-15 a d x)}{60 d^2 (a+b x)}-\frac {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)} \]
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Rubi [A] time = 0.12, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1001, 833, 780, 195, 217, 206} \begin {gather*} -\frac {a 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 {\sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2} (8 b c-15 a d x)}{60 d^2 (a+b x)}-\frac {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 780
Rule 833
Rule 1001
Rubi steps
\begin {align*} \int x^2 \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^2 \left (2 a b+2 b^2 x\right ) \sqrt {c+d x^2} \, dx}{2 a b+2 b^2 x}\\ &=\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x \left (-4 b^2 c+10 a b d x\right ) \sqrt {c+d x^2} \, dx}{5 d \left (2 a b+2 b^2 x\right )}\\ &=\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {(8 b c-15 a d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{60 d^2 (a+b x)}-\frac {\left (a b 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 {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {(8 b c-15 a d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{60 d^2 (a+b x)}-\frac {\left (a b 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 {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {(8 b c-15 a d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{60 d^2 (a+b x)}-\frac {\left (a b c^2 \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 )}{4 d \left (2 a b+2 b^2 x\right )}\\ &=-\frac {a c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {b x^2 \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{5 d (a+b x)}-\frac {(8 b c-15 a d x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{60 d^2 (a+b x)}-\frac {a 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 = 0.15, size = 129, normalized size = 0.61 \begin {gather*} \frac {\sqrt {(a+b x)^2} \sqrt {c+d x^2} \left (\sqrt {\frac {d x^2}{c}+1} \left (15 a d x \left (c+2 d x^2\right )+8 b \left (-2 c^2+c d x^2+3 d^2 x^4\right )\right )-15 a c^{3/2} \sqrt {d} \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )\right )}{120 d^2 (a+b x) \sqrt {\frac {d x^2}{c}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 111, normalized size = 0.52 \begin {gather*} \frac {\sqrt {(a+b x)^2} \left (\frac {\sqrt {c+d x^2} \left (15 a c d x+30 a d^2 x^3-16 b c^2+8 b c d x^2+24 b d^2 x^4\right )}{120 d^2}+\frac {a c^2 \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{8 d^{3/2}}\right )}{a+b x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 175, normalized size = 0.83 \begin {gather*} \left [\frac {15 \, a c^{2} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (24 \, b d^{2} x^{4} + 30 \, a d^{2} x^{3} + 8 \, b c d x^{2} + 15 \, a c d x - 16 \, b c^{2}\right )} \sqrt {d x^{2} + c}}{240 \, d^{2}}, \frac {15 \, a c^{2} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (24 \, b d^{2} x^{4} + 30 \, a d^{2} x^{3} + 8 \, b c d x^{2} + 15 \, a c d x - 16 \, b c^{2}\right )} \sqrt {d x^{2} + c}}{120 \, d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 117, normalized size = 0.55 \begin {gather*} \frac {a 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}{120} \, \sqrt {d x^{2} + c} {\left ({\left (2 \, {\left (3 \, {\left (4 \, b x \mathrm {sgn}\left (b x + a\right ) + 5 \, a \mathrm {sgn}\left (b x + a\right )\right )} x + \frac {4 \, b c \mathrm {sgn}\left (b x + a\right )}{d}\right )} x + \frac {15 \, a c \mathrm {sgn}\left (b x + a\right )}{d}\right )} x - \frac {16 \, b c^{2} \mathrm {sgn}\left (b x + a\right )}{d^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 103, normalized size = 0.49 \begin {gather*} -\frac {\left (15 a \,c^{2} d \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+15 \sqrt {d \,x^{2}+c}\, a c \,d^{\frac {3}{2}} x -24 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b \,d^{\frac {3}{2}} x^{2}-30 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a \,d^{\frac {3}{2}} x +16 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b c \sqrt {d}\right ) \mathrm {csgn}\left (b x +a \right )}{120 d^{\frac {5}{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}} x^{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 = -0.00 \begin {gather*} \int x^2\,\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 x^{2} \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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