Optimal. Leaf size=146 \[ \frac {\left (8 a^2 d^2+b c (5 b c-12 a d)\right ) x \sqrt {c+d x^2}}{16 d^3}-\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}-\frac {c \left (8 a^2 d^2+b c (5 b c-12 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{7/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {475, 470, 327,
223, 212} \begin {gather*} \frac {x \sqrt {c+d x^2} \left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right )}{16 d}-\frac {c \left (8 a^2 d^2+b c (5 b c-12 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{7/2}}-\frac {b x^3 \sqrt {c+d x^2} (5 b c-12 a d)}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 327
Rule 470
Rule 475
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx &=\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}+\frac {\int \frac {x^2 \left (6 a^2 d-b (5 b c-12 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{6 d}\\ &=-\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}+\frac {1}{8} \left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx\\ &=\frac {\left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{16 d}-\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}-\frac {\left (c \left (5 b^2 c^2-12 a b c d+8 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{16 d^3}\\ &=\frac {\left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{16 d}-\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}-\frac {\left (c \left (5 b^2 c^2-12 a b c d+8 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{16 d^3}\\ &=\frac {\left (8 a^2+\frac {b c (5 b c-12 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{16 d}-\frac {b (5 b c-12 a d) x^3 \sqrt {c+d x^2}}{24 d^2}+\frac {b^2 x^5 \sqrt {c+d x^2}}{6 d}-\frac {c \left (5 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 d^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 124, normalized size = 0.85 \begin {gather*} \frac {\sqrt {d} x \sqrt {c+d x^2} \left (24 a^2 d^2+12 a b d \left (-3 c+2 d x^2\right )+b^2 \left (15 c^2-10 c d x^2+8 d^2 x^4\right )\right )+3 c \left (5 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{48 d^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 200, normalized size = 1.37
method | result | size |
risch | \(\frac {x \left (8 b^{2} x^{4} d^{2}+24 a b \,d^{2} x^{2}-10 b^{2} c d \,x^{2}+24 a^{2} d^{2}-36 a b c d +15 b^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}}{48 d^{3}}-\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a^{2}}{2 d^{\frac {3}{2}}}+\frac {3 c^{2} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) a b}{4 d^{\frac {5}{2}}}-\frac {5 c^{3} \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right ) b^{2}}{16 d^{\frac {7}{2}}}\) | \(149\) |
default | \(b^{2} \left (\frac {x^{5} \sqrt {d \,x^{2}+c}}{6 d}-\frac {5 c \left (\frac {x^{3} \sqrt {d \,x^{2}+c}}{4 d}-\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2 d}-\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 d^{\frac {3}{2}}}\right )}{4 d}\right )}{6 d}\right )+2 a b \left (\frac {x^{3} \sqrt {d \,x^{2}+c}}{4 d}-\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2 d}-\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 d^{\frac {3}{2}}}\right )}{4 d}\right )+a^{2} \left (\frac {x \sqrt {d \,x^{2}+c}}{2 d}-\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 d^{\frac {3}{2}}}\right )\) | \(200\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 175, normalized size = 1.20 \begin {gather*} \frac {\sqrt {d x^{2} + c} b^{2} x^{5}}{6 \, d} - \frac {5 \, \sqrt {d x^{2} + c} b^{2} c x^{3}}{24 \, d^{2}} + \frac {\sqrt {d x^{2} + c} a b x^{3}}{2 \, d} + \frac {5 \, \sqrt {d x^{2} + c} b^{2} c^{2} x}{16 \, d^{3}} - \frac {3 \, \sqrt {d x^{2} + c} a b c x}{4 \, d^{2}} + \frac {\sqrt {d x^{2} + c} a^{2} x}{2 \, d} - \frac {5 \, b^{2} c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{16 \, d^{\frac {7}{2}}} + \frac {3 \, a b c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{4 \, d^{\frac {5}{2}}} - \frac {a^{2} c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{2 \, d^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.41, size = 267, normalized size = 1.83 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{2} c^{3} - 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (8 \, b^{2} d^{3} x^{5} - 2 \, {\left (5 \, b^{2} c d^{2} - 12 \, a b d^{3}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{96 \, d^{4}}, \frac {3 \, {\left (5 \, b^{2} c^{3} - 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (8 \, b^{2} d^{3} x^{5} - 2 \, {\left (5 \, b^{2} c d^{2} - 12 \, a b d^{3}\right )} x^{3} + 3 \, {\left (5 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{48 \, d^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 301 vs.
\(2 (138) = 276\).
time = 12.53, size = 301, normalized size = 2.06 \begin {gather*} \frac {a^{2} \sqrt {c} x \sqrt {1 + \frac {d x^{2}}{c}}}{2 d} - \frac {a^{2} c \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{2 d^{\frac {3}{2}}} - \frac {3 a b c^{\frac {3}{2}} x}{4 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a b \sqrt {c} x^{3}}{4 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{4 d^{\frac {5}{2}}} + \frac {a b x^{5}}{2 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 b^{2} c^{\frac {5}{2}} x}{16 d^{3} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 b^{2} c^{\frac {3}{2}} x^{3}}{48 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} \sqrt {c} x^{5}}{24 d \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {5 b^{2} c^{3} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{16 d^{\frac {7}{2}}} + \frac {b^{2} x^{7}}{6 \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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Giac [A]
time = 0.77, size = 135, normalized size = 0.92 \begin {gather*} \frac {1}{48} \, {\left (2 \, {\left (\frac {4 \, b^{2} x^{2}}{d} - \frac {5 \, b^{2} c d^{3} - 12 \, a b d^{4}}{d^{5}}\right )} x^{2} + \frac {3 \, {\left (5 \, b^{2} c^{2} d^{2} - 12 \, a b c d^{3} + 8 \, a^{2} d^{4}\right )}}{d^{5}}\right )} \sqrt {d x^{2} + c} x + \frac {{\left (5 \, b^{2} c^{3} - 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{16 \, d^{\frac {7}{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 \frac {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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