Optimal. Leaf size=404 \[ \frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b^{3/2} (a+b x) (13 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b (a+b x) (13 A b-5 a B)}{64 a^7 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (a+b x) (13 A b-5 a B)}{64 a^6 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (a+b x) (13 A b-5 a B)}{320 a^5 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 78, 51, 63, 205} \[ -\frac {231 b (a+b x) (13 A b-5 a B)}{64 a^7 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (a+b x) (13 A b-5 a B)}{64 a^6 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (a+b x) (13 A b-5 a B)}{320 a^5 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b^{3/2} (a+b x) (13 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 205
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{x^{7/2} \left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (13 A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{7/2} \left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (11 b (13 A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{7/2} \left (a b+b^2 x\right )^3} \, dx}{48 a^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (33 (13 A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{7/2} \left (a b+b^2 x\right )^2} \, dx}{64 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 (13 A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{7/2} \left (a b+b^2 x\right )} \, dx}{128 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (13 A b-5 a B) (a+b x)}{320 a^5 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 (13 A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{5/2} \left (a b+b^2 x\right )} \, dx}{128 a^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (13 A b-5 a B) (a+b x)}{320 a^5 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (13 A b-5 a B) (a+b x)}{64 a^6 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (231 b (13 A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )} \, dx}{128 a^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (13 A b-5 a B) (a+b x)}{320 a^5 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (13 A b-5 a B) (a+b x)}{64 a^6 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b (13 A b-5 a B) (a+b x)}{64 a^7 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 b^2 (13 A b-5 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{128 a^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (13 A b-5 a B) (a+b x)}{320 a^5 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (13 A b-5 a B) (a+b x)}{64 a^6 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b (13 A b-5 a B) (a+b x)}{64 a^7 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (231 b^2 (13 A b-5 a B) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{64 a^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {33 (13 A b-5 a B)}{64 a^4 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {13 A b-5 a B}{24 a^2 b x^{5/2} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {11 (13 A b-5 a B)}{96 a^3 b x^{5/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 (13 A b-5 a B) (a+b x)}{320 a^5 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {77 (13 A b-5 a B) (a+b x)}{64 a^6 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b (13 A b-5 a B) (a+b x)}{64 a^7 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b^{3/2} (13 A b-5 a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 80, normalized size = 0.20 \[ \frac {5 a^4 (A b-a B)-(a+b x)^4 (13 A b-5 a B) \, _2F_1\left (-\frac {5}{2},4;-\frac {3}{2};-\frac {b x}{a}\right )}{20 a^5 b x^{5/2} (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 673, normalized size = 1.67 \[ \left [-\frac {3465 \, {\left ({\left (5 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 4 \, {\left (5 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 6 \, {\left (5 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 4 \, {\left (5 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + {\left (5 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (384 \, A a^{6} - 3465 \, {\left (5 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} - 12705 \, {\left (5 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} - 16863 \, {\left (5 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} - 9207 \, {\left (5 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} - 1408 \, {\left (5 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (5 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{7} b^{4} x^{7} + 4 \, a^{8} b^{3} x^{6} + 6 \, a^{9} b^{2} x^{5} + 4 \, a^{10} b x^{4} + a^{11} x^{3}\right )}}, -\frac {3465 \, {\left ({\left (5 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 4 \, {\left (5 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 6 \, {\left (5 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 4 \, {\left (5 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + {\left (5 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (384 \, A a^{6} - 3465 \, {\left (5 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} - 12705 \, {\left (5 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} - 16863 \, {\left (5 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} - 9207 \, {\left (5 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} - 1408 \, {\left (5 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (5 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{960 \, {\left (a^{7} b^{4} x^{7} + 4 \, a^{8} b^{3} x^{6} + 6 \, a^{9} b^{2} x^{5} + 4 \, a^{10} b x^{4} + a^{11} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 207, normalized size = 0.51 \[ \frac {231 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (75 \, B a b x^{2} - 225 \, A b^{2} x^{2} - 5 \, B a^{2} x + 25 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{7} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right )} + \frac {1545 \, B a b^{5} x^{\frac {7}{2}} - 3249 \, A b^{6} x^{\frac {7}{2}} + 5153 \, B a^{2} b^{4} x^{\frac {5}{2}} - 10633 \, A a b^{5} x^{\frac {5}{2}} + 5855 \, B a^{3} b^{3} x^{\frac {3}{2}} - 11767 \, A a^{2} b^{4} x^{\frac {3}{2}} + 2295 \, B a^{4} b^{2} \sqrt {x} - 4431 \, A a^{3} b^{3} \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{7} \mathrm {sgn}\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 449, normalized size = 1.11 \[ -\frac {\left (45045 A \,b^{7} x^{\frac {13}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-17325 B a \,b^{6} x^{\frac {13}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+180180 A a \,b^{6} x^{\frac {11}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-69300 B \,a^{2} b^{5} x^{\frac {11}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+270270 A \,a^{2} b^{5} x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-103950 B \,a^{3} b^{4} x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+45045 \sqrt {a b}\, A \,b^{6} x^{6}-17325 \sqrt {a b}\, B a \,b^{5} x^{6}+180180 A \,a^{3} b^{4} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-69300 B \,a^{4} b^{3} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+165165 \sqrt {a b}\, A a \,b^{5} x^{5}-63525 \sqrt {a b}\, B \,a^{2} b^{4} x^{5}+45045 A \,a^{4} b^{3} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-17325 B \,a^{5} b^{2} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+219219 \sqrt {a b}\, A \,a^{2} b^{4} x^{4}-84315 \sqrt {a b}\, B \,a^{3} b^{3} x^{4}+119691 \sqrt {a b}\, A \,a^{3} b^{3} x^{3}-46035 \sqrt {a b}\, B \,a^{4} b^{2} x^{3}+18304 \sqrt {a b}\, A \,a^{4} b^{2} x^{2}-7040 \sqrt {a b}\, B \,a^{5} b \,x^{2}-1664 \sqrt {a b}\, A \,a^{5} b x +640 \sqrt {a b}\, B \,a^{6} x +384 \sqrt {a b}\, A \,a^{6}\right ) \left (b x +a \right )}{960 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{7} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.12, size = 550, normalized size = 1.36 \[ \frac {1155 \, {\left ({\left (3 \, B a b^{8} - 13 \, A b^{9}\right )} x^{2} + 39 \, {\left (B a^{2} b^{7} - 3 \, A a b^{8}\right )} x\right )} x^{\frac {9}{2}} + 2310 \, {\left ({\left (3 \, B a^{2} b^{7} - 13 \, A a b^{8}\right )} x^{2} + 117 \, {\left (B a^{3} b^{6} - 3 \, A a^{2} b^{7}\right )} x\right )} x^{\frac {7}{2}} - 4620 \, {\left (2 \, {\left (3 \, B a^{3} b^{6} - 13 \, A a^{2} b^{7}\right )} x^{2} - 143 \, {\left (B a^{4} b^{5} - 3 \, A a^{3} b^{6}\right )} x\right )} x^{\frac {5}{2}} - 462 \, {\left (85 \, {\left (3 \, B a^{4} b^{5} - 13 \, A a^{3} b^{6}\right )} x^{2} - 1807 \, {\left (B a^{5} b^{4} - 3 \, A a^{4} b^{5}\right )} x\right )} x^{\frac {3}{2}} - 33 \, {\left (1771 \, {\left (3 \, B a^{5} b^{4} - 13 \, A a^{4} b^{5}\right )} x^{2} - 17095 \, {\left (B a^{6} b^{3} - 3 \, A a^{5} b^{4}\right )} x\right )} \sqrt {x} - \frac {14080 \, {\left (3 \, {\left (3 \, B a^{6} b^{3} - 13 \, A a^{5} b^{4}\right )} x^{2} - 13 \, {\left (B a^{7} b^{2} - 3 \, A a^{6} b^{3}\right )} x\right )}}{\sqrt {x}} - \frac {1280 \, {\left (11 \, {\left (3 \, B a^{7} b^{2} - 13 \, A a^{6} b^{3}\right )} x^{2} - 13 \, {\left (B a^{8} b - 3 \, A a^{7} b^{2}\right )} x\right )}}{x^{\frac {3}{2}}} - \frac {1280 \, {\left ({\left (3 \, B a^{8} b - 13 \, A a^{7} b^{2}\right )} x^{2} + {\left (B a^{9} - 3 \, A a^{8} b\right )} x\right )}}{x^{\frac {5}{2}}} - \frac {256 \, {\left (5 \, A a^{8} b x^{2} + 3 \, A a^{9} x\right )}}{x^{\frac {7}{2}}}}{1920 \, {\left (a^{9} b^{5} x^{5} + 5 \, a^{10} b^{4} x^{4} + 10 \, a^{11} b^{3} x^{3} + 10 \, a^{12} b^{2} x^{2} + 5 \, a^{13} b x + a^{14}\right )}} + \frac {231 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{7}} - \frac {77 \, {\left ({\left (3 \, B a b^{3} - 13 \, A b^{4}\right )} x^{\frac {3}{2}} + 6 \, {\left (5 \, B a^{2} b^{2} - 13 \, A a b^{3}\right )} \sqrt {x}\right )}}{128 \, a^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{x^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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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