Optimal. Leaf size=209 \[ \frac {A b-a B}{2 a b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 (a+b x) (5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 (a+b x) (5 A b-a B)}{4 a^3 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 6, 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}{2 a b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 (a+b x) (5 A b-a B)}{4 a^3 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 A b-a B}{4 a^2 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 (a+b x) (5 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \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 )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{x^{3/2} \left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {A b-a B}{2 a b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((5 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}{4 a \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 A b-a B}{4 a^2 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 (5 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}{8 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 A b-a B}{4 a^2 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 (5 A b-a B) (a+b x)}{4 a^3 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (3 (5 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}{8 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 A b-a B}{4 a^2 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 (5 A b-a B) (a+b x)}{4 a^3 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (3 (5 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 )}{4 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {5 A b-a B}{4 a^2 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b \sqrt {x} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 (5 A b-a B) (a+b x)}{4 a^3 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 (5 A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \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 = 77, normalized size = 0.37 \begin {gather*} \frac {a^2 (A b-a B)+(a+b x)^2 (a B-5 A b) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {b x}{a}\right )}{2 a^3 b \sqrt {x} (a+b x) \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 17.13, size = 113, normalized size = 0.54 \begin {gather*} \frac {(a+b x) \left (\frac {3 (a B-5 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b}}+\frac {-8 a^2 A+5 a^2 B x-25 a A b x+3 a b B x^2-15 A b^2 x^2}{4 a^3 \sqrt {x} (a+b x)^2}\right )}{\sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 331, normalized size = 1.58 \begin {gather*} \left [\frac {3 \, {\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 5 \, A a^{2} 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 (8 \, A a^{3} b - 3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{8 \, {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )}}, -\frac {3 \, {\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (8 \, A a^{3} b - 3 \, {\left (B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{2} - 5 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{4 \, {\left (a^{4} b^{3} x^{3} + 2 \, a^{5} b^{2} x^{2} + a^{6} b x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 110, normalized size = 0.53 \begin {gather*} \frac {3 \, {\left (B a - 5 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, A}{a^{3} \sqrt {x} \mathrm {sgn}\left (b x + a\right )} + \frac {3 \, B a b x^{\frac {3}{2}} - 7 \, A b^{2} x^{\frac {3}{2}} + 5 \, B a^{2} \sqrt {x} - 9 \, A a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{3} \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 = 214, normalized size = 1.02 \begin {gather*} -\frac {\left (15 A \,b^{3} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-3 B a \,b^{2} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+30 A a \,b^{2} x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-6 B \,a^{2} b \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+15 A \,a^{2} b \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-3 B \,a^{3} \sqrt {x}\, \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+15 \sqrt {a b}\, A \,b^{2} x^{2}-3 \sqrt {a b}\, B a b \,x^{2}+25 \sqrt {a b}\, A a b x -5 \sqrt {a b}\, B \,a^{2} x +8 \sqrt {a b}\, A \,a^{2}\right ) \left (b x +a \right )}{4 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} a^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.65, size = 280, normalized size = 1.34 \begin {gather*} \frac {60 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{\frac {5}{2}} - {\left ({\left (B a b^{4} + 5 \, A b^{5}\right )} x^{2} - 15 \, {\left (B a^{2} b^{3} - 7 \, A a b^{4}\right )} x\right )} x^{\frac {5}{2}} + {\left (9 \, {\left (B a^{3} b^{2} + 5 \, A a^{2} b^{3}\right )} x^{2} + 85 \, {\left (B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {x} + \frac {16 \, {\left ({\left (B a^{4} b + 5 \, A a^{3} b^{2}\right )} x^{2} + 3 \, {\left (B a^{5} - 7 \, A a^{4} b\right )} x\right )}}{\sqrt {x}} + \frac {48 \, {\left (A a^{4} b x^{2} - A a^{5} x\right )}}{x^{\frac {3}{2}}}}{24 \, {\left (a^{5} b^{3} x^{3} + 3 \, a^{6} b^{2} x^{2} + 3 \, a^{7} b x + a^{8}\right )}} + \frac {3 \, {\left (B a - 5 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{3}} + \frac {{\left (B a b + 5 \, A b^{2}\right )} x^{\frac {3}{2}} - 18 \, {\left (B a^{2} - 5 \, A a b\right )} \sqrt {x}}{24 \, a^{5}} \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 )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{x^{\frac {3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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