Optimal. Leaf size=114 \[ \frac {(2 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{7/2}}-\frac {x (4 A b-7 a B)}{3 b^3 \sqrt {a+b x^2}}+\frac {a x (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac {B x \sqrt {a+b x^2}}{2 b^3} \]
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Rubi [A] time = 0.09, antiderivative size = 114, 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 = {455, 1157, 388, 217, 206} \[ -\frac {x (4 A b-7 a B)}{3 b^3 \sqrt {a+b x^2}}+\frac {a x (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac {(2 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{7/2}}+\frac {B x \sqrt {a+b x^2}}{2 b^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 388
Rule 455
Rule 1157
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {a (A b-a B) x}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {a (A b-a B)-3 b (A b-a B) x^2-3 b^2 B x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{3 b^3}\\ &=\frac {a (A b-a B) x}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {(4 A b-7 a B) x}{3 b^3 \sqrt {a+b x^2}}+\frac {\int \frac {3 a (A b-2 a B)+3 a b B x^2}{\sqrt {a+b x^2}} \, dx}{3 a b^3}\\ &=\frac {a (A b-a B) x}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {(4 A b-7 a B) x}{3 b^3 \sqrt {a+b x^2}}+\frac {B x \sqrt {a+b x^2}}{2 b^3}+\frac {(2 A b-5 a B) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b^3}\\ &=\frac {a (A b-a B) x}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {(4 A b-7 a B) x}{3 b^3 \sqrt {a+b x^2}}+\frac {B x \sqrt {a+b x^2}}{2 b^3}+\frac {(2 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 )}{2 b^3}\\ &=\frac {a (A b-a B) x}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {(4 A b-7 a B) x}{3 b^3 \sqrt {a+b x^2}}+\frac {B x \sqrt {a+b x^2}}{2 b^3}+\frac {(2 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 116, normalized size = 1.02 \[ \frac {\sqrt {b} x \left (15 a^2 B+a \left (20 b B x^2-6 A b\right )+b^2 x^2 \left (3 B x^2-8 A\right )\right )-3 \sqrt {a} \left (a+b x^2\right ) \sqrt {\frac {b x^2}{a}+1} (5 a B-2 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6 b^{7/2} \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 333, normalized size = 2.92 \[ \left [-\frac {3 \, {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{4} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (3 \, B b^{3} x^{5} + 4 \, {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 3 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}, \frac {3 \, {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{4} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, B b^{3} x^{5} + 4 \, {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 3 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 112, normalized size = 0.98 \[ \frac {{\left ({\left (\frac {3 \, B x^{2}}{b} + \frac {4 \, {\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )}}{a b^{5}}\right )} x^{2} + \frac {3 \, {\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )}}{a b^{5}}\right )} x}{6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, B a - 2 \, A b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 134, normalized size = 1.18 \[ \frac {B \,x^{5}}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b}-\frac {A \,x^{3}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b}+\frac {5 B a \,x^{3}}{6 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}-\frac {A x}{\sqrt {b \,x^{2}+a}\, b^{2}}+\frac {5 B a x}{2 \sqrt {b \,x^{2}+a}\, b^{3}}+\frac {A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {5}{2}}}-\frac {5 B a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 160, normalized size = 1.40 \[ \frac {B x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {1}{3} \, A x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {5 \, B a x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{6 \, b} + \frac {5 \, B a x}{6 \, \sqrt {b x^{2} + a} b^{3}} - \frac {A x}{3 \, \sqrt {b x^{2} + a} b^{2}} - \frac {5 \, B a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {7}{2}}} + \frac {A \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} \]
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 \[ \int \frac {x^4\,\left (B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 20.57, size = 675, normalized size = 5.92 \[ A \left (\frac {3 a^{\frac {39}{2}} b^{11} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{\frac {37}{2}} b^{12} x^{2} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{19} b^{\frac {23}{2}} x}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {4 a^{18} b^{\frac {25}{2}} x^{3}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (- \frac {15 a^{\frac {81}{2}} b^{22} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {15 a^{\frac {79}{2}} b^{23} x^{2} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{40} b^{\frac {45}{2}} x}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {20 a^{39} b^{\frac {47}{2}} x^{3}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{38} b^{\frac {49}{2}} x^{5}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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