Optimal. Leaf size=194 \[ \frac {c^2 \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{9/2}}-\frac {c x \sqrt {c+d x^2} \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right )}{128 d^4}+\frac {x^3 \sqrt {c+d x^2} \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right )}{192 d^3}-\frac {b x^5 \sqrt {c+d x^2} (7 b c-16 a d)}{48 d^2}+\frac {b^2 x^7 \sqrt {c+d x^2}}{8 d} \]
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Rubi [A] time = 0.15, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {464, 459, 321, 217, 206} \begin {gather*} \frac {c^2 \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{9/2}}+\frac {x^3 \sqrt {c+d x^2} \left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right )}{192 d}-\frac {c x \sqrt {c+d x^2} \left (48 a^2 d^2+5 b c (7 b c-16 a d)\right )}{128 d^4}-\frac {b x^5 \sqrt {c+d x^2} (7 b c-16 a d)}{48 d^2}+\frac {b^2 x^7 \sqrt {c+d x^2}}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 459
Rule 464
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx &=\frac {b^2 x^7 \sqrt {c+d x^2}}{8 d}+\frac {\int \frac {x^4 \left (8 a^2 d-b (7 b c-16 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{8 d}\\ &=-\frac {b (7 b c-16 a d) x^5 \sqrt {c+d x^2}}{48 d^2}+\frac {b^2 x^7 \sqrt {c+d x^2}}{8 d}-\frac {1}{48} \left (-48 a^2-\frac {5 b c (7 b c-16 a d)}{d^2}\right ) \int \frac {x^4}{\sqrt {c+d x^2}} \, dx\\ &=\frac {\left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{192 d}-\frac {b (7 b c-16 a d) x^5 \sqrt {c+d x^2}}{48 d^2}+\frac {b^2 x^7 \sqrt {c+d x^2}}{8 d}-\frac {\left (c \left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2}} \, dx}{64 d}\\ &=-\frac {c \left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{128 d^2}+\frac {\left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{192 d}-\frac {b (7 b c-16 a d) x^5 \sqrt {c+d x^2}}{48 d^2}+\frac {b^2 x^7 \sqrt {c+d x^2}}{8 d}+\frac {\left (c^2 \left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{128 d^2}\\ &=-\frac {c \left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{128 d^2}+\frac {\left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{192 d}-\frac {b (7 b c-16 a d) x^5 \sqrt {c+d x^2}}{48 d^2}+\frac {b^2 x^7 \sqrt {c+d x^2}}{8 d}+\frac {\left (c^2 \left (48 a^2+\frac {5 b c (7 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^2}\\ &=-\frac {c \left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right ) x \sqrt {c+d x^2}}{128 d^2}+\frac {\left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right ) x^3 \sqrt {c+d x^2}}{192 d}-\frac {b (7 b c-16 a d) x^5 \sqrt {c+d x^2}}{48 d^2}+\frac {b^2 x^7 \sqrt {c+d x^2}}{8 d}+\frac {c^2 \left (48 a^2+\frac {5 b c (7 b c-16 a d)}{d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{128 d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 159, normalized size = 0.82 \begin {gather*} \frac {3 c^2 \left (48 a^2 d^2-80 a b c d+35 b^2 c^2\right ) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )+\sqrt {d} x \sqrt {c+d x^2} \left (48 a^2 d^2 \left (2 d x^2-3 c\right )+16 a b d \left (15 c^2-10 c d x^2+8 d^2 x^4\right )+b^2 \left (-105 c^3+70 c^2 d x^2-56 c d^2 x^4+48 d^3 x^6\right )\right )}{384 d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 173, normalized size = 0.89 \begin {gather*} \frac {\sqrt {c+d x^2} \left (-144 a^2 c d^2 x+96 a^2 d^3 x^3+240 a b c^2 d x-160 a b c d^2 x^3+128 a b d^3 x^5-105 b^2 c^3 x+70 b^2 c^2 d x^3-56 b^2 c d^2 x^5+48 b^2 d^3 x^7\right )}{384 d^4}+\frac {\left (-48 a^2 c^2 d^2+80 a b c^3 d-35 b^2 c^4\right ) \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{128 d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 344, normalized size = 1.77 \begin {gather*} \left [\frac {3 \, {\left (35 \, b^{2} c^{4} - 80 \, a b c^{3} d + 48 \, 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 (7 \, b^{2} c d^{3} - 16 \, a b d^{4}\right )} x^{5} + 2 \, {\left (35 \, b^{2} c^{2} d^{2} - 80 \, a b c d^{3} + 48 \, a^{2} d^{4}\right )} x^{3} - 3 \, {\left (35 \, b^{2} c^{3} d - 80 \, a b c^{2} d^{2} + 48 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{768 \, d^{5}}, -\frac {3 \, {\left (35 \, b^{2} c^{4} - 80 \, a b c^{3} d + 48 \, 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 (7 \, b^{2} c d^{3} - 16 \, a b d^{4}\right )} x^{5} + 2 \, {\left (35 \, b^{2} c^{2} d^{2} - 80 \, a b c d^{3} + 48 \, a^{2} d^{4}\right )} x^{3} - 3 \, {\left (35 \, b^{2} c^{3} d - 80 \, a b c^{2} d^{2} + 48 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{384 \, d^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 178, normalized size = 0.92 \begin {gather*} \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (\frac {6 \, b^{2} x^{2}}{d} - \frac {7 \, b^{2} c d^{5} - 16 \, a b d^{6}}{d^{7}}\right )} x^{2} + \frac {35 \, b^{2} c^{2} d^{4} - 80 \, a b c d^{5} + 48 \, a^{2} d^{6}}{d^{7}}\right )} x^{2} - \frac {3 \, {\left (35 \, b^{2} c^{3} d^{3} - 80 \, a b c^{2} d^{4} + 48 \, a^{2} c d^{5}\right )}}{d^{7}}\right )} \sqrt {d x^{2} + c} x - \frac {{\left (35 \, b^{2} c^{4} - 80 \, a b c^{3} d + 48 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{128 \, d^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 265, normalized size = 1.37 \begin {gather*} \frac {\sqrt {d \,x^{2}+c}\, b^{2} x^{7}}{8 d}+\frac {\sqrt {d \,x^{2}+c}\, a b \,x^{5}}{3 d}-\frac {7 \sqrt {d \,x^{2}+c}\, b^{2} c \,x^{5}}{48 d^{2}}+\frac {\sqrt {d \,x^{2}+c}\, a^{2} x^{3}}{4 d}-\frac {5 \sqrt {d \,x^{2}+c}\, a b c \,x^{3}}{12 d^{2}}+\frac {35 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} x^{3}}{192 d^{3}}+\frac {3 a^{2} c^{2} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{8 d^{\frac {5}{2}}}-\frac {5 a b \,c^{3} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{8 d^{\frac {7}{2}}}+\frac {35 b^{2} c^{4} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{128 d^{\frac {9}{2}}}-\frac {3 \sqrt {d \,x^{2}+c}\, a^{2} c x}{8 d^{2}}+\frac {5 \sqrt {d \,x^{2}+c}\, a b \,c^{2} x}{8 d^{3}}-\frac {35 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} x}{128 d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 243, normalized size = 1.25 \begin {gather*} \frac {\sqrt {d x^{2} + c} b^{2} x^{7}}{8 \, d} - \frac {7 \, \sqrt {d x^{2} + c} b^{2} c x^{5}}{48 \, d^{2}} + \frac {\sqrt {d x^{2} + c} a b x^{5}}{3 \, d} + \frac {35 \, \sqrt {d x^{2} + c} b^{2} c^{2} x^{3}}{192 \, d^{3}} - \frac {5 \, \sqrt {d x^{2} + c} a b c x^{3}}{12 \, d^{2}} + \frac {\sqrt {d x^{2} + c} a^{2} x^{3}}{4 \, d} - \frac {35 \, \sqrt {d x^{2} + c} b^{2} c^{3} x}{128 \, d^{4}} + \frac {5 \, \sqrt {d x^{2} + c} a b c^{2} x}{8 \, d^{3}} - \frac {3 \, \sqrt {d x^{2} + c} a^{2} c x}{8 \, d^{2}} + \frac {35 \, b^{2} c^{4} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{128 \, d^{\frac {9}{2}}} - \frac {5 \, a b c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, d^{\frac {7}{2}}} + \frac {3 \, a^{2} c^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{8 \, d^{\frac {5}{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^4\,{\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 = 28.18, size = 422, normalized size = 2.18 \begin {gather*} - \frac {3 a^{2} c^{\frac {3}{2}} x}{8 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a^{2} \sqrt {c} x^{3}}{8 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a^{2} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{8 d^{\frac {5}{2}}} + \frac {a^{2} x^{5}}{4 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 a b c^{\frac {5}{2}} x}{8 d^{3} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 a b c^{\frac {3}{2}} x^{3}}{24 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a b \sqrt {c} x^{5}}{12 d \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {5 a b c^{3} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{8 d^{\frac {7}{2}}} + \frac {a b x^{7}}{3 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {35 b^{2} c^{\frac {7}{2}} x}{128 d^{4} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {35 b^{2} c^{\frac {5}{2}} x^{3}}{384 d^{3} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {7 b^{2} c^{\frac {3}{2}} x^{5}}{192 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} \sqrt {c} x^{7}}{48 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {35 b^{2} c^{4} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{128 d^{\frac {9}{2}}} + \frac {b^{2} 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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