Optimal. Leaf size=158 \[ \frac {\sqrt {x} (A b-a B)}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\sqrt {x} (a B+3 A b)}{4 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {\sqrt {x} (A b-a B)}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\sqrt {x} (a B+3 A b)}{4 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 78
Rule 205
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {x} \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}{\sqrt {x} \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) \sqrt {x}}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((3 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )^2} \, dx}{4 a \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(3 A b+a B) \sqrt {x}}{4 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((3 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^2 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(3 A b+a B) \sqrt {x}}{4 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((3 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^2 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {(3 A b+a B) \sqrt {x}}{4 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) \sqrt {x}}{2 a b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(3 A b+a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 106, normalized size = 0.67 \[ \frac {\sqrt {a} \sqrt {b} \sqrt {x} \left (a^2 (-B)+a b (5 A+B x)+3 A b^2 x\right )+(a+b x)^2 (a B+3 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2} (a+b x) \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.75, size = 291, normalized size = 1.84 \[ \left [-\frac {{\left (B a^{3} + 3 \, A a^{2} b + {\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (B a^{3} b - 5 \, A a^{2} b^{2} - {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}}, -\frac {{\left (B a^{3} + 3 \, A a^{2} b + {\left (B a b^{2} + 3 \, A b^{3}\right )} x^{2} + 2 \, {\left (B a^{2} b + 3 \, A a b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (B a^{3} b - 5 \, A a^{2} b^{2} - {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} x\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 98, normalized size = 0.62 \[ \frac {{\left (B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2} b \mathrm {sgn}\left (b x + a\right )} + \frac {B a b x^{\frac {3}{2}} + 3 \, A b^{2} x^{\frac {3}{2}} - B a^{2} \sqrt {x} + 5 \, A a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{2} b \mathrm {sgn}\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 194, normalized size = 1.23 \[ \frac {\left (3 A \,b^{3} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+B a \,b^{2} x^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+6 A a \,b^{2} x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+2 B \,a^{2} b x \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3 A \,a^{2} b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+B \,a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+3 \sqrt {a b}\, A \,b^{2} x^{\frac {3}{2}}+\sqrt {a b}\, B a b \,x^{\frac {3}{2}}+5 \sqrt {a b}\, A a b \sqrt {x}-\sqrt {a b}\, B \,a^{2} \sqrt {x}\right ) \left (b x +a \right )}{4 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.71, size = 234, normalized size = 1.48 \[ \frac {12 \, {\left (B a^{3} b + 5 \, A a^{2} b^{2}\right )} x^{\frac {5}{2}} - {\left ({\left (B a b^{3} + A b^{4}\right )} x^{2} - 3 \, {\left (B a^{2} b^{2} + 5 \, A a b^{3}\right )} x\right )} x^{\frac {5}{2}} + {\left (9 \, {\left (B a^{3} b + A a^{2} b^{2}\right )} x^{2} + 17 \, {\left (B a^{4} + 5 \, A a^{3} b\right )} x\right )} \sqrt {x} + \frac {16 \, {\left (A a^{3} b x^{2} + 3 \, A a^{4} x\right )}}{\sqrt {x}}}{24 \, {\left (a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} + 3 \, a^{6} b x + a^{7}\right )}} + \frac {{\left (B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2} b} + \frac {{\left (B a b + A b^{2}\right )} x^{\frac {3}{2}} - 6 \, {\left (B a^{2} + 3 \, A a b\right )} \sqrt {x}}{24 \, a^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,x}{\sqrt {x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B x}{\sqrt {x} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________