Optimal. Leaf size=118 \[ \frac {a (6 A b-a B) x \sqrt {a+b x^2}}{16 b}+\frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {a^2 (6 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {396, 201, 223,
212} \begin {gather*} \frac {a^2 (6 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}}+\frac {x \left (a+b x^2\right )^{3/2} (6 A b-a B)}{24 b}+\frac {a x \sqrt {a+b x^2} (6 A b-a B)}{16 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 396
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx &=\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}-\frac {(-6 A b+a B) \int \left (a+b x^2\right )^{3/2} \, dx}{6 b}\\ &=\frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {(a (6 A b-a B)) \int \sqrt {a+b x^2} \, dx}{8 b}\\ &=\frac {a (6 A b-a B) x \sqrt {a+b x^2}}{16 b}+\frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {\left (a^2 (6 A b-a B)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{16 b}\\ &=\frac {a (6 A b-a B) x \sqrt {a+b x^2}}{16 b}+\frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {\left (a^2 (6 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b}\\ &=\frac {a (6 A b-a B) x \sqrt {a+b x^2}}{16 b}+\frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {a^2 (6 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 99, normalized size = 0.84 \begin {gather*} \frac {x \sqrt {a+b x^2} \left (30 a A b+3 a^2 B+12 A b^2 x^2+14 a b B x^2+8 b^2 B x^4\right )}{48 b}+\frac {a^2 (-6 A b+a B) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{16 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 130, normalized size = 1.10
method | result | size |
risch | \(\frac {x \left (8 b^{2} B \,x^{4}+12 A \,b^{2} x^{2}+14 B a b \,x^{2}+30 a b A +3 a^{2} B \right ) \sqrt {b \,x^{2}+a}}{48 b}+\frac {3 a^{2} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) A}{8 \sqrt {b}}-\frac {a^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) B}{16 b^{\frac {3}{2}}}\) | \(105\) |
default | \(B \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+A \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 116, normalized size = 0.98 \begin {gather*} \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A x + \frac {3}{8} \, \sqrt {b x^{2} + a} A a x + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x}{6 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a x}{24 \, b} - \frac {\sqrt {b x^{2} + a} B a^{2} x}{16 \, b} - \frac {B a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} + \frac {3 \, A a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.52, size = 207, normalized size = 1.75 \begin {gather*} \left [-\frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8 \, B b^{3} x^{5} + 2 \, {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x^{3} + 3 \, {\left (B a^{2} b + 10 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, b^{2}}, \frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (8 \, B b^{3} x^{5} + 2 \, {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x^{3} + 3 \, {\left (B a^{2} b + 10 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs.
\(2 (102) = 204\).
time = 10.28, size = 253, normalized size = 2.14 \begin {gather*} \frac {A a^{\frac {3}{2}} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {A a^{\frac {3}{2}} x}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 A \sqrt {a} b x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 A a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 \sqrt {b}} + \frac {A b^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {B a^{\frac {5}{2}} x}{16 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 B a^{\frac {3}{2}} x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {11 B \sqrt {a} b x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {3}{2}}} + \frac {B b^{2} x^{7}}{6 \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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Giac [A]
time = 1.12, size = 102, normalized size = 0.86 \begin {gather*} \frac {1}{48} \, {\left (2 \, {\left (4 \, B b x^{2} + \frac {7 \, B a b^{4} + 6 \, A b^{5}}{b^{4}}\right )} x^{2} + \frac {3 \, {\left (B a^{2} b^{3} + 10 \, A a b^{4}\right )}}{b^{4}}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (B a^{3} - 6 \, A a^{2} b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {3}{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 \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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