Optimal. Leaf size=194 \[ \frac {x^3 \sqrt {a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{192 b^3}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right )}{128 b^{9/2}}+\frac {x \sqrt {a+b x^2} \left (-35 a^3 f+40 a^2 b e-48 a b^2 d+64 b^3 c\right )}{128 b^4}+\frac {x^5 \sqrt {a+b x^2} (8 b e-7 a f)}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b} \]
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Rubi [A] time = 0.21, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1809, 1267, 459, 321, 217, 206} \begin {gather*} \frac {x \sqrt {a+b x^2} \left (40 a^2 b e-35 a^3 f-48 a b^2 d+64 b^3 c\right )}{128 b^4}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (40 a^2 b e-35 a^3 f-48 a b^2 d+64 b^3 c\right )}{128 b^{9/2}}+\frac {x^3 \sqrt {a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{192 b^3}+\frac {x^5 \sqrt {a+b x^2} (8 b e-7 a f)}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 459
Rule 1267
Rule 1809
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx &=\frac {f x^7 \sqrt {a+b x^2}}{8 b}+\frac {\int \frac {x^2 \left (8 b c+8 b d x^2+(8 b e-7 a f) x^4\right )}{\sqrt {a+b x^2}} \, dx}{8 b}\\ &=\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}+\frac {\int \frac {x^2 \left (48 b^2 c+\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx}{48 b^2}\\ &=\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {1}{64} \left (-64 c+\frac {a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx\\ &=\frac {\left (64 c-\frac {a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) x \sqrt {a+b x^2}}{128 b}+\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {\left (a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b^4}\\ &=\frac {\left (64 c-\frac {a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) x \sqrt {a+b x^2}}{128 b}+\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {\left (a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b^4}\\ &=\frac {\left (64 c-\frac {a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) x \sqrt {a+b x^2}}{128 b}+\frac {\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt {a+b x^2}}{192 b^3}+\frac {(8 b e-7 a f) x^5 \sqrt {a+b x^2}}{48 b^2}+\frac {f x^7 \sqrt {a+b x^2}}{8 b}-\frac {a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 149, normalized size = 0.77 \begin {gather*} \frac {3 a \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (35 a^3 f-40 a^2 b e+48 a b^2 d-64 b^3 c\right )+\sqrt {b} x \sqrt {a+b x^2} \left (-105 a^3 f+10 a^2 b \left (12 e+7 f x^2\right )-8 a b^2 \left (18 d+10 e x^2+7 f x^4\right )+16 b^3 \left (12 c+6 d x^2+4 e x^4+3 f x^6\right )\right )}{384 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 167, normalized size = 0.86 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-105 a^3 f x+120 a^2 b e x+70 a^2 b f x^3-144 a b^2 d x-80 a b^2 e x^3-56 a b^2 f x^5+192 b^3 c x+96 b^3 d x^3+64 b^3 e x^5+48 b^3 f x^7\right )}{384 b^4}+\frac {\log \left (\sqrt {a+b x^2}-\sqrt {b} x\right ) \left (-35 a^4 f+40 a^3 b e-48 a^2 b^2 d+64 a b^3 c\right )}{128 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 329, normalized size = 1.70 \begin {gather*} \left [-\frac {3 \, {\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d + 40 \, a^{3} b e - 35 \, a^{4} f\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (48 \, b^{4} f x^{7} + 8 \, {\left (8 \, b^{4} e - 7 \, a b^{3} f\right )} x^{5} + 2 \, {\left (48 \, b^{4} d - 40 \, a b^{3} e + 35 \, a^{2} b^{2} f\right )} x^{3} + 3 \, {\left (64 \, b^{4} c - 48 \, a b^{3} d + 40 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{5}}, \frac {3 \, {\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d + 40 \, a^{3} b e - 35 \, a^{4} f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (48 \, b^{4} f x^{7} + 8 \, {\left (8 \, b^{4} e - 7 \, a b^{3} f\right )} x^{5} + 2 \, {\left (48 \, b^{4} d - 40 \, a b^{3} e + 35 \, a^{2} b^{2} f\right )} x^{3} + 3 \, {\left (64 \, b^{4} c - 48 \, a b^{3} d + 40 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 175, normalized size = 0.90 \begin {gather*} \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (\frac {6 \, f x^{2}}{b} - \frac {7 \, a b^{5} f - 8 \, b^{6} e}{b^{7}}\right )} x^{2} + \frac {48 \, b^{6} d + 35 \, a^{2} b^{4} f - 40 \, a b^{5} e}{b^{7}}\right )} x^{2} + \frac {3 \, {\left (64 \, b^{6} c - 48 \, a b^{5} d - 35 \, a^{3} b^{3} f + 40 \, a^{2} b^{4} e\right )}}{b^{7}}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d - 35 \, a^{4} f + 40 \, a^{3} b e\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 284, normalized size = 1.46 \begin {gather*} \frac {\sqrt {b \,x^{2}+a}\, f \,x^{7}}{8 b}-\frac {7 \sqrt {b \,x^{2}+a}\, a f \,x^{5}}{48 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, e \,x^{5}}{6 b}+\frac {35 \sqrt {b \,x^{2}+a}\, a^{2} f \,x^{3}}{192 b^{3}}-\frac {5 \sqrt {b \,x^{2}+a}\, a e \,x^{3}}{24 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, d \,x^{3}}{4 b}+\frac {35 a^{4} f \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {9}{2}}}-\frac {5 a^{3} e \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {7}{2}}}+\frac {3 a^{2} d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {5}{2}}}-\frac {a c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}-\frac {35 \sqrt {b \,x^{2}+a}\, a^{3} f x}{128 b^{4}}+\frac {5 \sqrt {b \,x^{2}+a}\, a^{2} e x}{16 b^{3}}-\frac {3 \sqrt {b \,x^{2}+a}\, a d x}{8 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, c x}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 255, normalized size = 1.31 \begin {gather*} \frac {\sqrt {b x^{2} + a} f x^{7}}{8 \, b} + \frac {\sqrt {b x^{2} + a} e x^{5}}{6 \, b} - \frac {7 \, \sqrt {b x^{2} + a} a f x^{5}}{48 \, b^{2}} + \frac {\sqrt {b x^{2} + a} d x^{3}}{4 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a e x^{3}}{24 \, b^{2}} + \frac {35 \, \sqrt {b x^{2} + a} a^{2} f x^{3}}{192 \, b^{3}} + \frac {\sqrt {b x^{2} + a} c x}{2 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a d x}{8 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a} a^{2} e x}{16 \, b^{3}} - \frac {35 \, \sqrt {b x^{2} + a} a^{3} f x}{128 \, b^{4}} - \frac {a c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {3 \, a^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {5 \, a^{3} e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {7}{2}}} + \frac {35 \, a^{4} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {9}{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 (f\,x^6+e\,x^4+d\,x^2+c\right )}{\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 32.56, size = 444, normalized size = 2.29 \begin {gather*} - \frac {35 a^{\frac {7}{2}} f x}{128 b^{4} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {5}{2}} e x}{16 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {35 a^{\frac {5}{2}} f x^{3}}{384 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {3}{2}} d x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {3}{2}} e x^{3}}{48 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {7 a^{\frac {3}{2}} f x^{5}}{192 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {\sqrt {a} c x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {\sqrt {a} d x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} e x^{5}}{24 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} f x^{7}}{48 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {35 a^{4} f \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {9}{2}}} - \frac {5 a^{3} e \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {7}{2}}} + \frac {3 a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} - \frac {a c \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} + \frac {d x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {e x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {f x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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