Optimal. Leaf size=119 \[ -\frac {3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}}+\frac {3 x \sqrt {a+b x^2} (4 A b-5 a B)}{8 b^3}-\frac {x^3 (4 A b-5 a B)}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {459, 288, 321, 217, 206} \begin {gather*} -\frac {x^3 (4 A b-5 a B)}{4 b^2 \sqrt {a+b x^2}}+\frac {3 x \sqrt {a+b x^2} (4 A b-5 a B)}{8 b^3}-\frac {3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 288
Rule 321
Rule 459
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {B x^5}{4 b \sqrt {a+b x^2}}-\frac {(-4 A b+5 a B) \int \frac {x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{4 b}\\ &=-\frac {(4 A b-5 a B) x^3}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}}+\frac {(3 (4 A b-5 a B)) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{4 b^2}\\ &=-\frac {(4 A b-5 a B) x^3}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}}+\frac {3 (4 A b-5 a B) x \sqrt {a+b x^2}}{8 b^3}-\frac {(3 a (4 A b-5 a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^3}\\ &=-\frac {(4 A b-5 a B) x^3}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}}+\frac {3 (4 A b-5 a B) x \sqrt {a+b x^2}}{8 b^3}-\frac {(3 a (4 A b-5 a B)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^3}\\ &=-\frac {(4 A b-5 a B) x^3}{4 b^2 \sqrt {a+b x^2}}+\frac {B x^5}{4 b \sqrt {a+b x^2}}+\frac {3 (4 A b-5 a B) x \sqrt {a+b x^2}}{8 b^3}-\frac {3 a (4 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 108, normalized size = 0.91 \begin {gather*} \frac {3 a^{3/2} \sqrt {\frac {b x^2}{a}+1} (5 a B-4 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\sqrt {b} x \left (-15 a^2 B+a b \left (12 A-5 B x^2\right )+2 b^2 x^2 \left (2 A+B x^2\right )\right )}{8 b^{7/2} \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 101, normalized size = 0.85 \begin {gather*} \frac {-15 a^2 B x+12 a A b x-5 a b B x^3+4 A b^2 x^3+2 b^2 B x^5}{8 b^3 \sqrt {a+b x^2}}-\frac {3 \left (5 a^2 B-4 a A b\right ) \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{8 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 274, normalized size = 2.30 \begin {gather*} \left [-\frac {3 \, {\left (5 \, B a^{3} - 4 \, A a^{2} b + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (2 \, B b^{3} x^{5} - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, -\frac {3 \, {\left (5 \, B a^{3} - 4 \, A a^{2} b + {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B b^{3} x^{5} - {\left (5 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \, {\left (5 \, B a^{2} b - 4 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, {\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 104, normalized size = 0.87 \begin {gather*} \frac {{\left ({\left (\frac {2 \, B x^{2}}{b} - \frac {5 \, B a b^{3} - 4 \, A b^{4}}{b^{5}}\right )} x^{2} - \frac {3 \, {\left (5 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )}}{b^{5}}\right )} x}{8 \, \sqrt {b x^{2} + a}} - \frac {3 \, {\left (5 \, B a^{2} - 4 \, A a b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 141, normalized size = 1.18 \begin {gather*} \frac {B \,x^{5}}{4 \sqrt {b \,x^{2}+a}\, b}+\frac {A \,x^{3}}{2 \sqrt {b \,x^{2}+a}\, b}-\frac {5 B a \,x^{3}}{8 \sqrt {b \,x^{2}+a}\, b^{2}}+\frac {3 A a x}{2 \sqrt {b \,x^{2}+a}\, b^{2}}-\frac {15 B \,a^{2} x}{8 \sqrt {b \,x^{2}+a}\, b^{3}}-\frac {3 A a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {5}{2}}}+\frac {15 B \,a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 126, normalized size = 1.06 \begin {gather*} \frac {B x^{5}}{4 \, \sqrt {b x^{2} + a} b} - \frac {5 \, B a x^{3}}{8 \, \sqrt {b x^{2} + a} b^{2}} + \frac {A x^{3}}{2 \, \sqrt {b x^{2} + a} b} - \frac {15 \, B a^{2} x}{8 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, A a x}{2 \, \sqrt {b x^{2} + a} b^{2}} + \frac {15 \, B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {7}{2}}} - \frac {3 \, A a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\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 )}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.63, size = 177, normalized size = 1.49 \begin {gather*} A \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (- \frac {15 a^{\frac {3}{2}} x}{8 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 \sqrt {a} x^{3}}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {7}{2}}} + \frac {x^{5}}{4 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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