Optimal. Leaf size=293 \[ \frac {3 (7 a B+A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (7 a B+A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (7 a B+A b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (7 a B+A b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}-\frac {x^{3/2} (7 a B+A b)}{16 a b^2 \left (a+b x^2\right )}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.22, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {457, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {3 (7 a B+A b) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (7 a B+A b) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (7 a B+A b) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (7 a B+A b) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}-\frac {x^{3/2} (7 a B+A b)}{16 a b^2 \left (a+b x^2\right )}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 288
Rule 297
Rule 329
Rule 457
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}+\frac {\left (\frac {A b}{2}+\frac {7 a B}{2}\right ) \int \frac {x^{5/2}}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 (A b+7 a B)) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{32 a b^2}\\ &=\frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 (A b+7 a B)) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a b^2}\\ &=\frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {(3 (A b+7 a B)) \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a b^{5/2}}+\frac {(3 (A b+7 a B)) \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a b^{5/2}}\\ &=\frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {(3 (A b+7 a B)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{64 a b^3}+\frac {(3 (A b+7 a B)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{64 a b^3}+\frac {(3 (A b+7 a B)) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}+\frac {(3 (A b+7 a B)) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}\\ &=\frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}+\frac {3 (A b+7 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (A b+7 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}+\frac {(3 (A b+7 a B)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}-\frac {(3 (A b+7 a B)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}\\ &=\frac {(A b-a B) x^{7/2}}{4 a b \left (a+b x^2\right )^2}-\frac {(A b+7 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {3 (A b+7 a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (A b+7 a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (A b+7 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (A b+7 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{5/4} b^{11/4}}\\ \end {align*}
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Mathematica [C] time = 0.21, size = 137, normalized size = 0.47 \[ \frac {2 b^{3/4} x^{3/2} (A b-2 a B) \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-\frac {b x^2}{a}\right )+2 b^{3/4} x^{3/2} (a B-A b) \, _2F_1\left (\frac {3}{4},3;\frac {7}{4};-\frac {b x^2}{a}\right )+3 (-a)^{7/4} B \left (\tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{-a}}\right )+\tanh ^{-1}\left (\frac {a \sqrt [4]{b} \sqrt {x}}{(-a)^{5/4}}\right )\right )}{3 a^2 b^{11/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 990, normalized size = 3.38 \[ -\frac {12 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (117649 \, B^{6} a^{6} + 100842 \, A B^{5} a^{5} b + 36015 \, A^{2} B^{4} a^{4} b^{2} + 6860 \, A^{3} B^{3} a^{3} b^{3} + 735 \, A^{4} B^{2} a^{2} b^{4} + 42 \, A^{5} B a b^{5} + A^{6} b^{6}\right )} x - {\left (2401 \, B^{4} a^{7} b^{5} + 1372 \, A B^{3} a^{6} b^{6} + 294 \, A^{2} B^{2} a^{5} b^{7} + 28 \, A^{3} B a^{4} b^{8} + A^{4} a^{3} b^{9}\right )} \sqrt {-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}}} a b^{3} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {1}{4}} - {\left (343 \, B^{3} a^{4} b^{3} + 147 \, A B^{2} a^{3} b^{4} + 21 \, A^{2} B a^{2} b^{5} + A^{3} a b^{6}\right )} \sqrt {x} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {1}{4}}}{2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}\right ) - 3 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {1}{4}} \log \left (27 \, a^{4} b^{8} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} a^{3} + 147 \, A B^{2} a^{2} b + 21 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) + 3 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {1}{4}} \log \left (-27 \, a^{4} b^{8} \left (-\frac {2401 \, B^{4} a^{4} + 1372 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 28 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{5} b^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} a^{3} + 147 \, A B^{2} a^{2} b + 21 \, A^{2} B a b^{2} + A^{3} b^{3}\right )} \sqrt {x}\right ) + 4 \, {\left ({\left (11 \, B a b - 3 \, A b^{2}\right )} x^{3} + {\left (7 \, B a^{2} + A a b\right )} x\right )} \sqrt {x}}{64 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 293, normalized size = 1.00 \[ -\frac {11 \, B a b x^{\frac {7}{2}} - 3 \, A b^{2} x^{\frac {7}{2}} + 7 \, B a^{2} x^{\frac {3}{2}} + A a b x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a b^{2}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{5}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{5}} - \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{5}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 325, normalized size = 1.11 \[ \frac {3 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {3 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {3 \sqrt {2}\, A \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \,b^{2}}+\frac {21 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {21 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {21 \sqrt {2}\, B \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{3}}+\frac {\frac {\left (3 A b -11 B a \right ) x^{\frac {7}{2}}}{16 a b}-\frac {\left (A b +7 B a \right ) x^{\frac {3}{2}}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.37, size = 251, normalized size = 0.86 \[ -\frac {{\left (11 \, B a b - 3 \, A b^{2}\right )} x^{\frac {7}{2}} + {\left (7 \, B a^{2} + A a b\right )} x^{\frac {3}{2}}}{16 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} + \frac {3 \, {\left (7 \, B a + A b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 122, normalized size = 0.42 \[ \frac {3\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+7\,B\,a\right )}{32\,{\left (-a\right )}^{5/4}\,b^{11/4}}-\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+7\,B\,a\right )}{32\,{\left (-a\right )}^{5/4}\,b^{11/4}}-\frac {\frac {x^{3/2}\,\left (A\,b+7\,B\,a\right )}{16\,b^2}-\frac {x^{7/2}\,\left (3\,A\,b-11\,B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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