Optimal. Leaf size=191 \[ -\frac {c^2 \left (16 a^2 d^2+b c (5 b c-16 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{7/2}}+\frac {x^3 \sqrt {c+d x^2} \left (16 a^2 d^2+b c (5 b c-16 a d)\right )}{64 d^2}+\frac {c x \sqrt {c+d x^2} \left (16 a^2 d^2+b c (5 b c-16 a d)\right )}{128 d^3}-\frac {b x^3 \left (c+d x^2\right )^{3/2} (5 b c-16 a d)}{48 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d} \]
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Rubi [A] time = 0.19, antiderivative size = 188, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {464, 459, 279, 321, 217, 206} \begin {gather*} -\frac {c^2 \left (16 a^2 d^2+b c (5 b c-16 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{7/2}}+\frac {1}{64} x^3 \sqrt {c+d x^2} \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right )+\frac {c x \sqrt {c+d x^2} \left (16 a^2 d^2+b c (5 b c-16 a d)\right )}{128 d^3}-\frac {b x^3 \left (c+d x^2\right )^{3/2} (5 b c-16 a d)}{48 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 279
Rule 321
Rule 459
Rule 464
Rubi steps
\begin {align*} \int x^2 \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx &=\frac {b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d}+\frac {\int x^2 \sqrt {c+d x^2} \left (8 a^2 d-b (5 b c-16 a d) x^2\right ) \, dx}{8 d}\\ &=-\frac {b (5 b c-16 a d) x^3 \left (c+d x^2\right )^{3/2}}{48 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d}+\frac {1}{16} \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right ) \int x^2 \sqrt {c+d x^2} \, dx\\ &=\frac {1}{64} \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}-\frac {b (5 b c-16 a d) x^3 \left (c+d x^2\right )^{3/2}}{48 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d}+\frac {1}{64} \left (c \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx\\ &=\frac {c \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{128 d}+\frac {1}{64} \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}-\frac {b (5 b c-16 a d) x^3 \left (c+d x^2\right )^{3/2}}{48 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d}-\frac {\left (c^2 \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{128 d}\\ &=\frac {c \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{128 d}+\frac {1}{64} \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}-\frac {b (5 b c-16 a d) x^3 \left (c+d x^2\right )^{3/2}}{48 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d}-\frac {\left (c^2 \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{128 d}\\ &=\frac {c \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{128 d}+\frac {1}{64} \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}-\frac {b (5 b c-16 a d) x^3 \left (c+d x^2\right )^{3/2}}{48 d^2}+\frac {b^2 x^5 \left (c+d x^2\right )^{3/2}}{8 d}-\frac {c^2 \left (16 a^2+\frac {b c (5 b c-16 a d)}{d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 157, normalized size = 0.82 \begin {gather*} \frac {\sqrt {d} x \sqrt {c+d x^2} \left (48 a^2 d^2 \left (c+2 d x^2\right )+16 a b d \left (-3 c^2+2 c d x^2+8 d^2 x^4\right )+b^2 \left (15 c^3-10 c^2 d x^2+8 c d^2 x^4+48 d^3 x^6\right )\right )-3 c^2 \left (16 a^2 d^2-16 a b c d+5 b^2 c^2\right ) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{384 d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 173, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c+d x^2} \left (48 a^2 c d^2 x+96 a^2 d^3 x^3-48 a b c^2 d x+32 a b c d^2 x^3+128 a b d^3 x^5+15 b^2 c^3 x-10 b^2 c^2 d x^3+8 b^2 c d^2 x^5+48 b^2 d^3 x^7\right )}{384 d^3}+\frac {\left (16 a^2 c^2 d^2-16 a b c^3 d+5 b^2 c^4\right ) \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{128 d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.66, size = 341, normalized size = 1.79 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{2} c^{4} - 16 \, a b c^{3} d + 16 \, a^{2} c^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (48 \, b^{2} d^{4} x^{7} + 8 \, {\left (b^{2} c d^{3} + 16 \, a b d^{4}\right )} x^{5} - 2 \, {\left (5 \, b^{2} c^{2} d^{2} - 16 \, a b c d^{3} - 48 \, a^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{3} d - 16 \, a b c^{2} d^{2} + 16 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{768 \, d^{4}}, \frac {3 \, {\left (5 \, b^{2} c^{4} - 16 \, a b c^{3} d + 16 \, a^{2} c^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (48 \, b^{2} d^{4} x^{7} + 8 \, {\left (b^{2} c d^{3} + 16 \, a b d^{4}\right )} x^{5} - 2 \, {\left (5 \, b^{2} c^{2} d^{2} - 16 \, a b c d^{3} - 48 \, a^{2} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{3} d - 16 \, a b c^{2} d^{2} + 16 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{384 \, d^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 174, normalized size = 0.91 \begin {gather*} \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, b^{2} x^{2} + \frac {b^{2} c d^{5} + 16 \, a b d^{6}}{d^{6}}\right )} x^{2} - \frac {5 \, b^{2} c^{2} d^{4} - 16 \, a b c d^{5} - 48 \, a^{2} d^{6}}{d^{6}}\right )} x^{2} + \frac {3 \, {\left (5 \, b^{2} c^{3} d^{3} - 16 \, a b c^{2} d^{4} + 16 \, a^{2} c d^{5}\right )}}{d^{6}}\right )} \sqrt {d x^{2} + c} x + \frac {{\left (5 \, b^{2} c^{4} - 16 \, a b c^{3} d + 16 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{128 \, d^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 259, normalized size = 1.36 \begin {gather*} \frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} x^{5}}{8 d}+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a b \,x^{3}}{3 d}-\frac {5 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} c \,x^{3}}{48 d^{2}}-\frac {a^{2} c^{2} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{8 d^{\frac {3}{2}}}+\frac {a b \,c^{3} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{8 d^{\frac {5}{2}}}-\frac {5 b^{2} c^{4} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{128 d^{\frac {7}{2}}}-\frac {\sqrt {d \,x^{2}+c}\, a^{2} c x}{8 d}+\frac {\sqrt {d \,x^{2}+c}\, a b \,c^{2} x}{8 d^{2}}-\frac {5 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} x}{128 d^{3}}+\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2} x}{4 d}-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a b c x}{4 d^{2}}+\frac {5 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2} c^{2} x}{64 d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 237, normalized size = 1.24 \begin {gather*} \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} x^{5}}{8 \, d} - \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c x^{3}}{48 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b x^{3}}{3 \, d} + \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{2} x}{64 \, d^{3}} - \frac {5 \, \sqrt {d x^{2} + c} b^{2} c^{3} x}{128 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c x}{4 \, d^{2}} + \frac {\sqrt {d x^{2} + c} a b c^{2} x}{8 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} x}{4 \, d} - \frac {\sqrt {d x^{2} + c} a^{2} c x}{8 \, d} - \frac {5 \, b^{2} c^{4} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {7}{2}}} + \frac {a b c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, d^{\frac {5}{2}}} - \frac {a^{2} c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, d^{\frac {3}{2}}} \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^2\,{\left (b\,x^2+a\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 [B] time = 21.91, size = 411, normalized size = 2.15 \begin {gather*} \frac {a^{2} c^{\frac {3}{2}} x}{8 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a^{2} \sqrt {c} x^{3}}{8 \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a^{2} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{8 d^{\frac {3}{2}}} + \frac {a^{2} d x^{5}}{4 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a b c^{\frac {5}{2}} x}{8 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a b c^{\frac {3}{2}} x^{3}}{24 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 a b \sqrt {c} x^{5}}{12 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {a b c^{3} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{8 d^{\frac {5}{2}}} + \frac {a b d x^{7}}{3 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 b^{2} c^{\frac {7}{2}} x}{128 d^{3} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 b^{2} c^{\frac {5}{2}} x^{3}}{384 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{\frac {3}{2}} x^{5}}{192 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {7 b^{2} \sqrt {c} x^{7}}{48 \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {5 b^{2} c^{4} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{128 d^{\frac {7}{2}}} + \frac {b^{2} d x^{9}}{8 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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