Optimal. Leaf size=84 \[ -\frac {A \left (a+b x^2\right )^{3/2}}{7 a x^7}+\frac {(4 A b-7 a B) \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}-\frac {2 b (4 A b-7 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {464, 277, 270}
\begin {gather*} -\frac {2 b \left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{105 a^3 x^3}+\frac {\left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{35 a^2 x^5}-\frac {A \left (a+b x^2\right )^{3/2}}{7 a x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 277
Rule 464
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^8} \, dx &=-\frac {A \left (a+b x^2\right )^{3/2}}{7 a x^7}-\frac {(4 A b-7 a B) \int \frac {\sqrt {a+b x^2}}{x^6} \, dx}{7 a}\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{7 a x^7}+\frac {(4 A b-7 a B) \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}+\frac {(2 b (4 A b-7 a B)) \int \frac {\sqrt {a+b x^2}}{x^4} \, dx}{35 a^2}\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{7 a x^7}+\frac {(4 A b-7 a B) \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}-\frac {2 b (4 A b-7 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 62, normalized size = 0.74 \begin {gather*} \frac {\left (a+b x^2\right )^{3/2} \left (-15 a^2 A+12 a A b x^2-21 a^2 B x^2-8 A b^2 x^4+14 a b B x^4\right )}{105 a^3 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 102, normalized size = 1.21
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (8 A \,b^{2} x^{4}-14 B a b \,x^{4}-12 a A b \,x^{2}+21 B \,a^{2} x^{2}+15 a^{2} A \right )}{105 a^{3} x^{7}}\) | \(59\) |
trager | \(-\frac {\left (8 x^{6} A \,b^{3}-14 x^{6} B a \,b^{2}-4 A a \,b^{2} x^{4}+7 x^{4} B \,a^{2} b +3 x^{2} A \,a^{2} b +21 B \,a^{3} x^{2}+15 A \,a^{3}\right ) \sqrt {b \,x^{2}+a}}{105 a^{3} x^{7}}\) | \(83\) |
risch | \(-\frac {\left (8 x^{6} A \,b^{3}-14 x^{6} B a \,b^{2}-4 A a \,b^{2} x^{4}+7 x^{4} B \,a^{2} b +3 x^{2} A \,a^{2} b +21 B \,a^{3} x^{2}+15 A \,a^{3}\right ) \sqrt {b \,x^{2}+a}}{105 a^{3} x^{7}}\) | \(83\) |
default | \(B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 96, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{15 \, a^{2} x^{3}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{105 \, a^{3} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{5 \, a x^{5}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{35 \, a^{2} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{7 \, a x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.51, size = 81, normalized size = 0.96 \begin {gather*} \frac {{\left (2 \, {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{6} - {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{4} - 15 \, A a^{3} - 3 \, {\left (7 \, B a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, a^{3} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 442 vs.
\(2 (78) = 156\).
time = 1.62, size = 442, normalized size = 5.26 \begin {gather*} - \frac {15 A a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {33 A a^{4} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {17 A a^{3} b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {3 A a^{2} b^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {12 A a b^{\frac {17}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {8 A b^{\frac {19}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {B b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a x^{2}} + \frac {2 B b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 288 vs.
\(2 (72) = 144\).
time = 0.88, size = 288, normalized size = 3.43 \begin {gather*} \frac {4 \, {\left (105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B b^{\frac {5}{2}} - 175 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {5}{2}} + 280 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {7}{2}} + 70 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {5}{2}} + 140 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {7}{2}} - 42 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {5}{2}} + 84 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {7}{2}} + 49 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {5}{2}} - 28 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {7}{2}} - 7 \, B a^{5} b^{\frac {5}{2}} + 4 \, A a^{4} b^{\frac {7}{2}}\right )}}{105 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.55, size = 132, normalized size = 1.57 \begin {gather*} \frac {4\,A\,b^2\,\sqrt {b\,x^2+a}}{105\,a^2\,x^3}-\frac {B\,\sqrt {b\,x^2+a}}{5\,x^5}-\frac {A\,b\,\sqrt {b\,x^2+a}}{35\,a\,x^5}-\frac {B\,b\,\sqrt {b\,x^2+a}}{15\,a\,x^3}-\frac {A\,\sqrt {b\,x^2+a}}{7\,x^7}-\frac {8\,A\,b^3\,\sqrt {b\,x^2+a}}{105\,a^3\,x}+\frac {2\,B\,b^2\,\sqrt {b\,x^2+a}}{15\,a^2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________