Optimal. Leaf size=155 \[ \frac {a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{7/2}}-\frac {a^2 x \sqrt {a+b x^2} (8 A b-5 a B)}{128 b^3}+\frac {a x^3 \sqrt {a+b x^2} (8 A b-5 a B)}{192 b^2}+\frac {x^5 \sqrt {a+b x^2} (8 A b-5 a B)}{48 b}+\frac {B x^5 \left (a+b x^2\right )^{3/2}}{8 b} \]
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Rubi [A] time = 0.07, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {459, 279, 321, 217, 206} \[ -\frac {a^2 x \sqrt {a+b x^2} (8 A b-5 a B)}{128 b^3}+\frac {a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{7/2}}+\frac {a x^3 \sqrt {a+b x^2} (8 A b-5 a B)}{192 b^2}+\frac {x^5 \sqrt {a+b x^2} (8 A b-5 a B)}{48 b}+\frac {B x^5 \left (a+b x^2\right )^{3/2}}{8 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 279
Rule 321
Rule 459
Rubi steps
\begin {align*} \int x^4 \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx &=\frac {B x^5 \left (a+b x^2\right )^{3/2}}{8 b}-\frac {(-8 A b+5 a B) \int x^4 \sqrt {a+b x^2} \, dx}{8 b}\\ &=\frac {(8 A b-5 a B) x^5 \sqrt {a+b x^2}}{48 b}+\frac {B x^5 \left (a+b x^2\right )^{3/2}}{8 b}+\frac {(a (8 A b-5 a B)) \int \frac {x^4}{\sqrt {a+b x^2}} \, dx}{48 b}\\ &=\frac {a (8 A b-5 a B) x^3 \sqrt {a+b x^2}}{192 b^2}+\frac {(8 A b-5 a B) x^5 \sqrt {a+b x^2}}{48 b}+\frac {B x^5 \left (a+b x^2\right )^{3/2}}{8 b}-\frac {\left (a^2 (8 A b-5 a B)\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{64 b^2}\\ &=-\frac {a^2 (8 A b-5 a B) x \sqrt {a+b x^2}}{128 b^3}+\frac {a (8 A b-5 a B) x^3 \sqrt {a+b x^2}}{192 b^2}+\frac {(8 A b-5 a B) x^5 \sqrt {a+b x^2}}{48 b}+\frac {B x^5 \left (a+b x^2\right )^{3/2}}{8 b}+\frac {\left (a^3 (8 A b-5 a B)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b^3}\\ &=-\frac {a^2 (8 A b-5 a B) x \sqrt {a+b x^2}}{128 b^3}+\frac {a (8 A b-5 a B) x^3 \sqrt {a+b x^2}}{192 b^2}+\frac {(8 A b-5 a B) x^5 \sqrt {a+b x^2}}{48 b}+\frac {B x^5 \left (a+b x^2\right )^{3/2}}{8 b}+\frac {\left (a^3 (8 A b-5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b^3}\\ &=-\frac {a^2 (8 A b-5 a B) x \sqrt {a+b x^2}}{128 b^3}+\frac {a (8 A b-5 a B) x^3 \sqrt {a+b x^2}}{192 b^2}+\frac {(8 A b-5 a B) x^5 \sqrt {a+b x^2}}{48 b}+\frac {B x^5 \left (a+b x^2\right )^{3/2}}{8 b}+\frac {a^3 (8 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 130, normalized size = 0.84 \[ \frac {\sqrt {a+b x^2} \left (\sqrt {b} x \left (15 a^3 B-2 a^2 b \left (12 A+5 B x^2\right )+8 a b^2 x^2 \left (2 A+B x^2\right )+16 b^3 x^4 \left (4 A+3 B x^2\right )\right )-\frac {3 a^{5/2} (5 a B-8 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}\right )}{384 b^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 257, normalized size = 1.66 \[ \left [-\frac {3 \, {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (48 \, B b^{4} x^{7} + 8 \, {\left (B a b^{3} + 8 \, A b^{4}\right )} x^{5} - 2 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} + 3 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{768 \, b^{4}}, \frac {3 \, {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (48 \, B b^{4} x^{7} + 8 \, {\left (B a b^{3} + 8 \, A b^{4}\right )} x^{5} - 2 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} + 3 \, {\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{384 \, b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 132, normalized size = 0.85 \[ \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, B x^{2} + \frac {B a b^{5} + 8 \, A b^{6}}{b^{6}}\right )} x^{2} - \frac {5 \, B a^{2} b^{4} - 8 \, A a b^{5}}{b^{6}}\right )} x^{2} + \frac {3 \, {\left (5 \, B a^{3} b^{3} - 8 \, A a^{2} b^{4}\right )}}{b^{6}}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 181, normalized size = 1.17 \[ \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,x^{5}}{8 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,x^{3}}{6 b}-\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B a \,x^{3}}{48 b^{2}}+\frac {A \,a^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {5}{2}}}-\frac {5 B \,a^{4} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {7}{2}}}+\frac {\sqrt {b \,x^{2}+a}\, A \,a^{2} x}{16 b^{2}}-\frac {5 \sqrt {b \,x^{2}+a}\, B \,a^{3} x}{128 b^{3}}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A a x}{8 b^{2}}+\frac {5 \left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,a^{2} x}{64 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 166, normalized size = 1.07 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B x^{5}}{8 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a x^{3}}{48 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A x^{3}}{6 \, b} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} x}{64 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a} B a^{3} x}{128 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A a x}{8 \, b^{2}} + \frac {\sqrt {b x^{2} + a} A a^{2} x}{16 \, b^{2}} - \frac {5 \, B a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {7}{2}}} + \frac {A a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {5}{2}}} \]
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 \[ \int x^4\,\left (B\,x^2+A\right )\,\sqrt {b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.65, size = 286, normalized size = 1.85 \[ - \frac {A a^{\frac {5}{2}} x}{16 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A a^{\frac {3}{2}} x^{3}}{48 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 A \sqrt {a} x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {A a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {5}{2}}} + \frac {A b x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 B a^{\frac {7}{2}} x}{128 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 B a^{\frac {5}{2}} x^{3}}{384 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{\frac {3}{2}} x^{5}}{192 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {7 B \sqrt {a} x^{7}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 B a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {7}{2}}} + \frac {B b x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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