Optimal. Leaf size=85 \[ \frac {(b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2} \sqrt {c}}+\frac {b B-3 A c}{b^2 c \sqrt {x}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {781, 78, 51, 63, 205} \begin {gather*} \frac {b B-3 A c}{b^2 c \sqrt {x}}+\frac {(b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2} \sqrt {c}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^2} \, dx &=\int \frac {A+B x}{x^{3/2} (b+c x)^2} \, dx\\ &=-\frac {b B-A c}{b c \sqrt {x} (b+c x)}-\frac {\left (\frac {b B}{2}-\frac {3 A c}{2}\right ) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{b c}\\ &=\frac {b B-3 A c}{b^2 c \sqrt {x}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)}+\frac {(b B-3 A c) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{2 b^2}\\ &=\frac {b B-3 A c}{b^2 c \sqrt {x}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)}+\frac {(b B-3 A c) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {b B-3 A c}{b^2 c \sqrt {x}}-\frac {b B-A c}{b c \sqrt {x} (b+c x)}+\frac {(b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2} \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 59, normalized size = 0.69 \begin {gather*} \frac {(b+c x) (b B-3 A c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {c x}{b}\right )+b (A c-b B)}{b^2 c \sqrt {x} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.10, size = 67, normalized size = 0.79 \begin {gather*} \frac {(b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2} \sqrt {c}}+\frac {-2 A b-3 A c x+b B x}{b^2 \sqrt {x} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 215, normalized size = 2.53 \begin {gather*} \left [\frac {{\left ({\left (B b c - 3 \, A c^{2}\right )} x^{2} + {\left (B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {-b c} \log \left (\frac {c x - b + 2 \, \sqrt {-b c} \sqrt {x}}{c x + b}\right ) - 2 \, {\left (2 \, A b^{2} c - {\left (B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt {x}}{2 \, {\left (b^{3} c^{2} x^{2} + b^{4} c x\right )}}, -\frac {{\left ({\left (B b c - 3 \, A c^{2}\right )} x^{2} + {\left (B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right ) + {\left (2 \, A b^{2} c - {\left (B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt {x}}{b^{3} c^{2} x^{2} + b^{4} c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 60, normalized size = 0.71 \begin {gather*} \frac {{\left (B b - 3 \, A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{2}} + \frac {B b x - 3 \, A c x - 2 \, A b}{{\left (c x^{\frac {3}{2}} + b \sqrt {x}\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 87, normalized size = 1.02 \begin {gather*} -\frac {3 A c \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{2}}+\frac {B \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b}-\frac {A c \sqrt {x}}{\left (c x +b \right ) b^{2}}+\frac {B \sqrt {x}}{\left (c x +b \right ) b}-\frac {2 A}{b^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 65, normalized size = 0.76 \begin {gather*} -\frac {2 \, A b - {\left (B b - 3 \, A c\right )} x}{b^{2} c x^{\frac {3}{2}} + b^{3} \sqrt {x}} + \frac {{\left (B b - 3 \, A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.08, size = 65, normalized size = 0.76 \begin {gather*} -\frac {\frac {2\,A}{b}+\frac {x\,\left (3\,A\,c-B\,b\right )}{b^2}}{b\,\sqrt {x}+c\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (3\,A\,c-B\,b\right )}{b^{5/2}\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 33.16, size = 884, normalized size = 10.40 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{c^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{b^{2}} & \text {for}\: c = 0 \\- \frac {4 i A b^{\frac {3}{2}} c \sqrt {\frac {1}{c}}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} - \frac {6 i A \sqrt {b} c^{2} x \sqrt {\frac {1}{c}}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} - \frac {3 A b c \sqrt {x} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} + \frac {3 A b c \sqrt {x} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} - \frac {3 A c^{2} x^{\frac {3}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} + \frac {3 A c^{2} x^{\frac {3}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} + \frac {2 i B b^{\frac {3}{2}} c x \sqrt {\frac {1}{c}}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} + \frac {B b^{2} \sqrt {x} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} - \frac {B b^{2} \sqrt {x} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} + \frac {B b c x^{\frac {3}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} - \frac {B b c x^{\frac {3}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {7}{2}} c \sqrt {x} \sqrt {\frac {1}{c}} + 2 i b^{\frac {5}{2}} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________