Optimal. Leaf size=311 \[ \frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (9 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (9 A b-a B)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \begin {gather*} \frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (9 A b-a B)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (a+b x) (9 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
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^{3/2} \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^{3/2} \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 \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \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 \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (7 b (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \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 \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )^2} \, dx}{192 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 (9 A b-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^4 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (9 A b-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^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 (9 A b-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^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 (9 A b-a B)}{192 a^4 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b \sqrt {x} (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {9 A b-a B}{24 a^2 b \sqrt {x} (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 (9 A b-a B)}{96 a^3 b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 (9 A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 79, normalized size = 0.25 \begin {gather*} \frac {a^4 (A b-a B)-(a+b x)^4 (9 A b-a B) \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};-\frac {b x}{a}\right )}{4 a^5 b \sqrt {x} (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 23.47, size = 161, normalized size = 0.52 \begin {gather*} \frac {(a+b x) \left (\frac {35 (a B-9 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{11/2} \sqrt {b}}+\frac {-384 a^4 A+279 a^4 B x-2511 a^3 A b x+511 a^3 b B x^2-4599 a^2 A b^2 x^2+385 a^2 b^2 B x^3-3465 a A b^3 x^3+105 a b^3 B x^4-945 A b^4 x^4}{192 a^5 \sqrt {x} (a+b x)^4}\right )}{\sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 559, normalized size = 1.80 \begin {gather*} \left [\frac {105 \, {\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \, {\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \, {\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \, {\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (384 \, A a^{5} b - 105 \, {\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \, {\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \, {\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \, {\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{384 \, {\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}, -\frac {105 \, {\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \, {\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \, {\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \, {\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + {\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (384 \, A a^{5} b - 105 \, {\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \, {\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \, {\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \, {\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{192 \, {\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 158, normalized size = 0.51 \begin {gather*} \frac {35 \, {\left (B a - 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, A}{a^{5} \sqrt {x} \mathrm {sgn}\left (b x + a\right )} + \frac {105 \, B a b^{3} x^{\frac {7}{2}} - 561 \, A b^{4} x^{\frac {7}{2}} + 385 \, B a^{2} b^{2} x^{\frac {5}{2}} - 1929 \, A a b^{3} x^{\frac {5}{2}} + 511 \, B a^{3} b x^{\frac {3}{2}} - 2295 \, A a^{2} b^{2} x^{\frac {3}{2}} + 279 \, B a^{4} \sqrt {x} - 975 \, A a^{3} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x + a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 374, normalized size = 1.20 \begin {gather*} -\frac {\left (945 A \,b^{5} x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-105 B a \,b^{4} x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3780 A a \,b^{4} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-420 B \,a^{2} b^{3} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+5670 A \,a^{2} b^{3} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-630 B \,a^{3} b^{2} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+945 \sqrt {a b}\, A \,b^{4} x^{4}-105 \sqrt {a b}\, B a \,b^{3} x^{4}+3780 A \,a^{3} b^{2} x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-420 B \,a^{4} b \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3465 \sqrt {a b}\, A a \,b^{3} x^{3}-385 \sqrt {a b}\, B \,a^{2} b^{2} x^{3}+945 A \,a^{4} b \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-105 B \,a^{5} \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+4599 \sqrt {a b}\, A \,a^{2} b^{2} x^{2}-511 \sqrt {a b}\, B \,a^{3} b \,x^{2}+2511 \sqrt {a b}\, A \,a^{3} b x -279 \sqrt {a b}\, B \,a^{4} x +384 \sqrt {a b}\, A \,a^{4}\right ) \left (b x +a \right )}{192 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{5} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.93, size = 427, normalized size = 1.37 \begin {gather*} -\frac {35 \, {\left ({\left (B a b^{6} + 9 \, A b^{7}\right )} x^{2} - 27 \, {\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x\right )} x^{\frac {9}{2}} + 70 \, {\left ({\left (B a^{2} b^{5} + 9 \, A a b^{6}\right )} x^{2} - 81 \, {\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x\right )} x^{\frac {7}{2}} - 140 \, {\left (2 \, {\left (B a^{3} b^{4} + 9 \, A a^{2} b^{5}\right )} x^{2} + 99 \, {\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x\right )} x^{\frac {5}{2}} - 14 \, {\left (85 \, {\left (B a^{4} b^{3} + 9 \, A a^{3} b^{4}\right )} x^{2} + 1251 \, {\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x\right )} x^{\frac {3}{2}} - {\left (1771 \, {\left (B a^{5} b^{2} + 9 \, A a^{4} b^{3}\right )} x^{2} + 11835 \, {\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt {x} - \frac {1280 \, {\left ({\left (B a^{6} b + 9 \, A a^{5} b^{2}\right )} x^{2} + 3 \, {\left (B a^{7} - 11 \, A a^{6} b\right )} x\right )}}{\sqrt {x}} - \frac {3840 \, {\left (A a^{6} b x^{2} - A a^{7} x\right )}}{x^{\frac {3}{2}}}}{1920 \, {\left (a^{7} b^{5} x^{5} + 5 \, a^{8} b^{4} x^{4} + 10 \, a^{9} b^{3} x^{3} + 10 \, a^{10} b^{2} x^{2} + 5 \, a^{11} b x + a^{12}\right )}} + \frac {35 \, {\left (B a - 9 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{5}} + \frac {7 \, {\left ({\left (B a b + 9 \, A b^{2}\right )} x^{\frac {3}{2}} - 30 \, {\left (B a^{2} - 9 \, A a b\right )} \sqrt {x}\right )}}{384 \, a^{7}} \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 = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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