Optimal. Leaf size=119 \[ \frac {b (2 A b+3 a B) x \sqrt {a+b x^2}}{2 a}-\frac {(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {1}{2} \sqrt {b} (2 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Rubi [A]
time = 0.03, 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 = {464, 283, 201,
223, 212} \begin {gather*} -\frac {\left (a+b x^2\right )^{3/2} (3 a B+2 A b)}{3 a x}+\frac {b x \sqrt {a+b x^2} (3 a B+2 A b)}{2 a}+\frac {1}{2} \sqrt {b} (3 a B+2 A b) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 283
Rule 464
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^4} \, dx &=-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}-\frac {(-2 A b-3 a B) \int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx}{3 a}\\ &=-\frac {(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {(b (2 A b+3 a B)) \int \sqrt {a+b x^2} \, dx}{a}\\ &=\frac {b (2 A b+3 a B) x \sqrt {a+b x^2}}{2 a}-\frac {(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {1}{2} (b (2 A b+3 a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {b (2 A b+3 a B) x \sqrt {a+b x^2}}{2 a}-\frac {(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {1}{2} (b (2 A b+3 a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {b (2 A b+3 a B) x \sqrt {a+b x^2}}{2 a}-\frac {(2 A b+3 a B) \left (a+b x^2\right )^{3/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{5/2}}{3 a x^3}+\frac {1}{2} \sqrt {b} (2 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 84, normalized size = 0.71 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-2 a A-8 A b x^2-6 a B x^2+3 b B x^4\right )}{6 x^3}-\frac {1}{2} \sqrt {b} (2 A b+3 a B) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 180, normalized size = 1.51
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-3 b B \,x^{4}+8 A b \,x^{2}+6 B a \,x^{2}+2 A a \right )}{6 x^{3}}+A \,b^{\frac {3}{2}} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )+\frac {3 B \sqrt {b}\, \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a}{2}\) | \(86\) |
default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )\) | \(180\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 115, normalized size = 0.97 \begin {gather*} \frac {3}{2} \, \sqrt {b x^{2} + a} B b x + \frac {\sqrt {b x^{2} + a} A b^{2} x}{a} + \frac {3}{2} \, B a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) + A b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{x} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{3 \, a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.12, size = 166, normalized size = 1.39 \begin {gather*} \left [\frac {3 \, {\left (3 \, B a + 2 \, A b\right )} \sqrt {b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (3 \, B b x^{4} - 2 \, {\left (3 \, B a + 4 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, x^{3}}, -\frac {3 \, {\left (3 \, B a + 2 \, A b\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, B b x^{4} - 2 \, {\left (3 \, B a + 4 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt {b x^{2} + a}}{6 \, x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.21, size = 202, normalized size = 1.70 \begin {gather*} - \frac {A \sqrt {a} b}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {A b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} + A b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {A b^{2} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{\frac {3}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {B \sqrt {a} b x \sqrt {1 + \frac {b x^{2}}{a}}}{2} - \frac {B \sqrt {a} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs.
\(2 (99) = 198\).
time = 1.23, size = 207, normalized size = 1.74 \begin {gather*} \frac {1}{2} \, \sqrt {b x^{2} + a} B b x - \frac {1}{4} \, {\left (3 \, B a \sqrt {b} + 2 \, A b^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{2} \sqrt {b} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a b^{\frac {3}{2}} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{3} \sqrt {b} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{2} b^{\frac {3}{2}} + 3 \, B a^{4} \sqrt {b} + 4 \, A a^{3} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \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 {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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