Optimal. Leaf size=94 \[ -\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{5/2}}+\frac {2 A \sqrt {x}}{b^2 \sqrt {b x+c x^2}}-\frac {2 x^{3/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {788, 666, 660, 207} \begin {gather*} \frac {2 A \sqrt {x}}{b^2 \sqrt {b x+c x^2}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{5/2}}-\frac {2 x^{3/2} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 207
Rule 660
Rule 666
Rule 788
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) x^{3/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {A \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{b}\\ &=-\frac {2 (b B-A c) x^{3/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 A \sqrt {x}}{b^2 \sqrt {b x+c x^2}}+\frac {A \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{b^2}\\ &=-\frac {2 (b B-A c) x^{3/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 A \sqrt {x}}{b^2 \sqrt {b x+c x^2}}+\frac {(2 A) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{b^2}\\ &=-\frac {2 (b B-A c) x^{3/2}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 A \sqrt {x}}{b^2 \sqrt {b x+c x^2}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 62, normalized size = 0.66 \begin {gather*} \frac {2 x^{3/2} \left (b (A c-b B)+3 A c (b+c x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c x}{b}+1\right )\right )}{3 b^2 c (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.26, size = 88, normalized size = 0.94 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (4 A b c+3 A c^2 x+b^2 (-B)\right )}{3 b^2 c \sqrt {x} (b+c x)^2}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 262, normalized size = 2.79 \begin {gather*} \left [\frac {3 \, {\left (A c^{3} x^{3} + 2 \, A b c^{2} x^{2} + A b^{2} c x\right )} \sqrt {b} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (3 \, A b c^{2} x - B b^{3} + 4 \, A b^{2} c\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{3 \, {\left (b^{3} c^{3} x^{3} + 2 \, b^{4} c^{2} x^{2} + b^{5} c x\right )}}, \frac {2 \, {\left (3 \, {\left (A c^{3} x^{3} + 2 \, A b c^{2} x^{2} + A b^{2} c x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (3 \, A b c^{2} x - B b^{3} + 4 \, A b^{2} c\right )} \sqrt {c x^{2} + b x} \sqrt {x}\right )}}{3 \, {\left (b^{3} c^{3} x^{3} + 2 \, b^{4} c^{2} x^{2} + b^{5} c x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 110, normalized size = 1.17 \begin {gather*} \frac {2 \, A \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} - \frac {2 \, {\left (3 \, A \sqrt {b} c \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) - B \sqrt {-b} b + 4 \, A \sqrt {-b} c\right )}}{3 \, \sqrt {-b} b^{\frac {5}{2}} c} - \frac {2 \, {\left (B b^{2} - 3 \, {\left (c x + b\right )} A c - A b c\right )}}{3 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 101, normalized size = 1.07 \begin {gather*} -\frac {2 \sqrt {\left (c x +b \right ) x}\, \left (3 \sqrt {c x +b}\, A \,c^{2} x \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-3 A \sqrt {b}\, c^{2} x +3 \sqrt {c x +b}\, A b c \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-4 A \,b^{\frac {3}{2}} c +B \,b^{\frac {5}{2}}\right )}{3 \left (c x +b \right )^{2} b^{\frac {5}{2}} c \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} x^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{3/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {3}{2}} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________