Optimal. Leaf size=148 \[ -\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}+\frac {8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt {a+b x^2}}+\frac {16 b^3 (8 A b-7 a B) x}{35 a^5 \sqrt {a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {464, 277, 197}
\begin {gather*} \frac {16 b^3 x (8 A b-7 a B)}{35 a^5 \sqrt {a+b x^2}}+\frac {8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {A}{7 a x^7 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 197
Rule 277
Rule 464
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}-\frac {(8 A b-7 a B) \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2}} \, dx}{7 a}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}+\frac {(6 b (8 A b-7 a B)) \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2}} \, dx}{35 a^2}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}-\frac {\left (8 b^2 (8 A b-7 a B)\right ) \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{35 a^3}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}+\frac {8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt {a+b x^2}}+\frac {\left (16 b^3 (8 A b-7 a B)\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a^4}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}+\frac {8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt {a+b x^2}}+\frac {16 b^3 (8 A b-7 a B) x}{35 a^5 \sqrt {a+b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.17, size = 105, normalized size = 0.71 \begin {gather*} \frac {128 A b^4 x^8+16 a b^3 x^6 \left (4 A-7 B x^2\right )-8 a^2 b^2 x^4 \left (2 A+7 B x^2\right )+2 a^3 b x^2 \left (4 A+7 B x^2\right )-a^4 \left (5 A+7 B x^2\right )}{35 a^5 x^7 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 194, normalized size = 1.31
method | result | size |
gosper | \(-\frac {-128 A \,b^{4} x^{8}+112 B a \,b^{3} x^{8}-64 A a \,b^{3} x^{6}+56 B \,a^{2} b^{2} x^{6}+16 A \,a^{2} b^{2} x^{4}-14 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+7 B \,a^{4} x^{2}+5 A \,a^{4}}{35 x^{7} \sqrt {b \,x^{2}+a}\, a^{5}}\) | \(107\) |
trager | \(-\frac {-128 A \,b^{4} x^{8}+112 B a \,b^{3} x^{8}-64 A a \,b^{3} x^{6}+56 B \,a^{2} b^{2} x^{6}+16 A \,a^{2} b^{2} x^{4}-14 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+7 B \,a^{4} x^{2}+5 A \,a^{4}}{35 x^{7} \sqrt {b \,x^{2}+a}\, a^{5}}\) | \(107\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-93 x^{6} A \,b^{3}+77 x^{6} B a \,b^{2}+29 A a \,b^{2} x^{4}-21 x^{4} B \,a^{2} b -13 x^{2} A \,a^{2} b +7 B \,a^{3} x^{2}+5 A \,a^{3}\right )}{35 a^{5} x^{7}}+\frac {x \left (A b -B a \right ) b^{3}}{\sqrt {b \,x^{2}+a}\, a^{5}}\) | \(109\) |
default | \(B \left (-\frac {1}{5 a \,x^{5} \sqrt {b \,x^{2}+a}}-\frac {6 b \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )}{5 a}\right )+A \left (-\frac {1}{7 a \,x^{7} \sqrt {b \,x^{2}+a}}-\frac {8 b \left (-\frac {1}{5 a \,x^{5} \sqrt {b \,x^{2}+a}}-\frac {6 b \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 176, normalized size = 1.19 \begin {gather*} -\frac {16 \, B b^{3} x}{5 \, \sqrt {b x^{2} + a} a^{4}} + \frac {128 \, A b^{4} x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {8 \, B b^{2}}{5 \, \sqrt {b x^{2} + a} a^{3} x} + \frac {64 \, A b^{3}}{35 \, \sqrt {b x^{2} + a} a^{4} x} + \frac {2 \, B b}{5 \, \sqrt {b x^{2} + a} a^{2} x^{3}} - \frac {16 \, A b^{2}}{35 \, \sqrt {b x^{2} + a} a^{3} x^{3}} - \frac {B}{5 \, \sqrt {b x^{2} + a} a x^{5}} + \frac {8 \, A b}{35 \, \sqrt {b x^{2} + a} a^{2} x^{5}} - \frac {A}{7 \, \sqrt {b x^{2} + a} a x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.48, size = 117, normalized size = 0.79 \begin {gather*} -\frac {{\left (16 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 8 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 5 \, A a^{4} - 2 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}} \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. 1030 vs.
\(2 (143) = 286\).
time = 6.74, size = 1030, normalized size = 6.96 \begin {gather*} A \left (- \frac {5 a^{7} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} - \frac {7 a^{6} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} - \frac {7 a^{5} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {35 a^{4} b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {280 a^{3} b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {560 a^{2} b^{\frac {43}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {448 a b^{\frac {45}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {128 b^{\frac {47}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}}\right ) + B \left (- \frac {a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {5 a^{3} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {30 a^{2} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {40 a b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {16 b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}}\right ) \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. 407 vs.
\(2 (128) = 256\).
time = 0.57, size = 407, normalized size = 2.75 \begin {gather*} -\frac {{\left (B a b^{3} - A b^{4}\right )} x}{\sqrt {b x^{2} + a} a^{5}} + \frac {2 \, {\left (35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a b^{\frac {5}{2}} - 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A b^{\frac {7}{2}} - 280 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{2} b^{\frac {5}{2}} + 280 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a b^{\frac {7}{2}} + 1015 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{3} b^{\frac {5}{2}} - 1015 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{2} b^{\frac {7}{2}} - 1680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{4} b^{\frac {5}{2}} + 2240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{3} b^{\frac {7}{2}} + 1337 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{5} b^{\frac {5}{2}} - 1673 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{4} b^{\frac {7}{2}} - 504 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{6} b^{\frac {5}{2}} + 616 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{5} b^{\frac {7}{2}} + 77 \, B a^{7} b^{\frac {5}{2}} - 93 \, A a^{6} b^{\frac {7}{2}}\right )}}{35 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.51, size = 148, normalized size = 1.00 \begin {gather*} -\frac {x^2\,\left (\frac {58\,A\,b^4-42\,B\,a\,b^3}{35\,a^5}-\frac {2\,b^3\,\left (93\,A\,b-77\,B\,a\right )}{35\,a^5}\right )-\frac {b^2\,\left (93\,A\,b-77\,B\,a\right )}{35\,a^4}}{x\,\sqrt {b\,x^2+a}}-\frac {\left (7\,B\,a^2-13\,A\,a\,b\right )\,\sqrt {b\,x^2+a}}{35\,a^4\,x^5}-\frac {A\,\sqrt {b\,x^2+a}}{7\,a^2\,x^7}-\frac {b\,\sqrt {b\,x^2+a}\,\left (29\,A\,b-21\,B\,a\right )}{35\,a^4\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________