Optimal. Leaf size=149 \[ -\frac {5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}}+\frac {5 x \sqrt {a+b x^2} (4 A b-7 a B)}{8 b^4}-\frac {5 x^3 (4 A b-7 a B)}{12 b^3 \sqrt {a+b x^2}}-\frac {x^5 (4 A b-7 a B)}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 149, 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, 288, 321, 217, 206} \begin {gather*} -\frac {x^5 (4 A b-7 a B)}{12 b^2 \left (a+b x^2\right )^{3/2}}-\frac {5 x^3 (4 A b-7 a B)}{12 b^3 \sqrt {a+b x^2}}+\frac {5 x \sqrt {a+b x^2} (4 A b-7 a B)}{8 b^4}-\frac {5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}} \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^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {(-4 A b+7 a B) \int \frac {x^6}{\left (a+b x^2\right )^{5/2}} \, dx}{4 b}\\ &=-\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}+\frac {(5 (4 A b-7 a B)) \int \frac {x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{12 b^2}\\ &=-\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {5 (4 A b-7 a B) x^3}{12 b^3 \sqrt {a+b x^2}}+\frac {(5 (4 A b-7 a B)) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{4 b^3}\\ &=-\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {5 (4 A b-7 a B) x^3}{12 b^3 \sqrt {a+b x^2}}+\frac {5 (4 A b-7 a B) x \sqrt {a+b x^2}}{8 b^4}-\frac {(5 a (4 A b-7 a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^4}\\ &=-\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {5 (4 A b-7 a B) x^3}{12 b^3 \sqrt {a+b x^2}}+\frac {5 (4 A b-7 a B) x \sqrt {a+b x^2}}{8 b^4}-\frac {(5 a (4 A b-7 a B)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^4}\\ &=-\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {5 (4 A b-7 a B) x^3}{12 b^3 \sqrt {a+b x^2}}+\frac {5 (4 A b-7 a B) x \sqrt {a+b x^2}}{8 b^4}-\frac {5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 139, normalized size = 0.93 \begin {gather*} \frac {x \left (-105 a^3 B+20 a^2 b \left (3 A-7 B x^2\right )+a b^2 x^2 \left (80 A-21 B x^2\right )+6 b^3 x^4 \left (2 A+B x^2\right )\right )}{24 b^4 \left (a+b x^2\right )^{3/2}}+\frac {5 \sqrt {a} \sqrt {a+b x^2} (7 a B-4 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2} \sqrt {\frac {b x^2}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 125, normalized size = 0.84 \begin {gather*} \frac {-105 a^3 B x+60 a^2 A b x-140 a^2 b B x^3+80 a A b^2 x^3-21 a b^2 B x^5+12 A b^3 x^5+6 b^3 B x^7}{24 b^4 \left (a+b x^2\right )^{3/2}}-\frac {5 \left (7 a^2 B-4 a A b\right ) \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{8 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 392, normalized size = 2.63 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} 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 (6 \, B b^{4} x^{7} - 3 \, {\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{5} - 20 \, {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}, -\frac {15 \, {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (6 \, B b^{4} x^{7} - 3 \, {\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{5} - 20 \, {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{24 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 148, normalized size = 0.99 \begin {gather*} \frac {{\left ({\left (3 \, {\left (\frac {2 \, B x^{2}}{b} - \frac {7 \, B a^{2} b^{5} - 4 \, A a b^{6}}{a b^{7}}\right )} x^{2} - \frac {20 \, {\left (7 \, B a^{3} b^{4} - 4 \, A a^{2} b^{5}\right )}}{a b^{7}}\right )} x^{2} - \frac {15 \, {\left (7 \, B a^{4} b^{3} - 4 \, A a^{3} b^{4}\right )}}{a b^{7}}\right )} x}{24 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (7 \, B a^{2} - 4 \, A a b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\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 = 181, normalized size = 1.21 \begin {gather*} \frac {B \,x^{7}}{4 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b}+\frac {A \,x^{5}}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b}-\frac {7 B a \,x^{5}}{8 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}+\frac {5 A a \,x^{3}}{6 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2}}-\frac {35 B \,a^{2} x^{3}}{24 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}+\frac {5 A a x}{2 \sqrt {b \,x^{2}+a}\, b^{3}}-\frac {35 B \,a^{2} x}{8 \sqrt {b \,x^{2}+a}\, b^{4}}-\frac {5 A a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {7}{2}}}+\frac {35 B \,a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 210, normalized size = 1.41 \begin {gather*} \frac {B x^{7}}{4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {7 \, B a x^{5}}{8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {A x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {35 \, B a^{2} 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 )}}{24 \, b^{2}} + \frac {5 \, A 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 {35 \, B a^{2} x}{24 \, \sqrt {b x^{2} + a} b^{4}} + \frac {5 \, A a x}{6 \, \sqrt {b x^{2} + a} b^{3}} + \frac {35 \, B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {9}{2}}} - \frac {5 \, A a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {7}{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^6\,\left (B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 38.82, size = 804, normalized size = 5.40 \begin {gather*} A \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 ) + B \left (\frac {105 a^{\frac {157}{2}} b^{41} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {105 a^{\frac {155}{2}} b^{42} x^{2} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {105 a^{78} b^{\frac {83}{2}} x}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {140 a^{77} b^{\frac {85}{2}} x^{3}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {21 a^{76} b^{\frac {87}{2}} x^{5}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {6 a^{75} b^{\frac {89}{2}} x^{7}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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