Optimal. Leaf size=147 \[ \frac {5 \sqrt {b} (7 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 c^{9/2}}-\frac {5 \sqrt {x} (7 b B-3 A c)}{4 c^4}+\frac {5 x^{3/2} (7 b B-3 A c)}{12 b c^3}-\frac {x^{5/2} (7 b B-3 A c)}{4 b c^2 (b+c x)}-\frac {x^{7/2} (b B-A c)}{2 b c (b+c x)^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {781, 78, 47, 50, 63, 205} \begin {gather*} -\frac {x^{5/2} (7 b B-3 A c)}{4 b c^2 (b+c x)}+\frac {5 x^{3/2} (7 b B-3 A c)}{12 b c^3}-\frac {5 \sqrt {x} (7 b B-3 A c)}{4 c^4}+\frac {5 \sqrt {b} (7 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 c^{9/2}}-\frac {x^{7/2} (b B-A c)}{2 b c (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{11/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx &=\int \frac {x^{5/2} (A+B x)}{(b+c x)^3} \, dx\\ &=-\frac {(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac {\left (-\frac {7 b B}{2}+\frac {3 A c}{2}\right ) \int \frac {x^{5/2}}{(b+c x)^2} \, dx}{2 b c}\\ &=-\frac {(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac {(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}+\frac {(5 (7 b B-3 A c)) \int \frac {x^{3/2}}{b+c x} \, dx}{8 b c^2}\\ &=\frac {5 (7 b B-3 A c) x^{3/2}}{12 b c^3}-\frac {(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac {(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}-\frac {(5 (7 b B-3 A c)) \int \frac {\sqrt {x}}{b+c x} \, dx}{8 c^3}\\ &=-\frac {5 (7 b B-3 A c) \sqrt {x}}{4 c^4}+\frac {5 (7 b B-3 A c) x^{3/2}}{12 b c^3}-\frac {(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac {(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}+\frac {(5 b (7 b B-3 A c)) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{8 c^4}\\ &=-\frac {5 (7 b B-3 A c) \sqrt {x}}{4 c^4}+\frac {5 (7 b B-3 A c) x^{3/2}}{12 b c^3}-\frac {(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac {(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}+\frac {(5 b (7 b B-3 A c)) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{4 c^4}\\ &=-\frac {5 (7 b B-3 A c) \sqrt {x}}{4 c^4}+\frac {5 (7 b B-3 A c) x^{3/2}}{12 b c^3}-\frac {(b B-A c) x^{7/2}}{2 b c (b+c x)^2}-\frac {(7 b B-3 A c) x^{5/2}}{4 b c^2 (b+c x)}+\frac {5 \sqrt {b} (7 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 c^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 61, normalized size = 0.41 \begin {gather*} \frac {x^{7/2} \left (\frac {7 b^2 (A c-b B)}{(b+c x)^2}+(7 b B-3 A c) \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};-\frac {c x}{b}\right )\right )}{14 b^3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.20, size = 122, normalized size = 0.83 \begin {gather*} \frac {5 \left (7 b^{3/2} B-3 A \sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 c^{9/2}}+\frac {\sqrt {x} \left (45 A b^2 c+75 A b c^2 x+24 A c^3 x^2-105 b^3 B-175 b^2 B c x-56 b B c^2 x^2+8 B c^3 x^3\right )}{12 c^4 (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 349, normalized size = 2.37 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{3} - 3 \, A b^{2} c + {\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} + 2 \, {\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x - 2 \, c \sqrt {x} \sqrt {-\frac {b}{c}} - b}{c x + b}\right ) - 2 \, {\left (8 \, B c^{3} x^{3} - 105 \, B b^{3} + 45 \, A b^{2} c - 8 \, {\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} - 25 \, {\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt {x}}{24 \, {\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}}, \frac {15 \, {\left (7 \, B b^{3} - 3 \, A b^{2} c + {\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} + 2 \, {\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c \sqrt {x} \sqrt {\frac {b}{c}}}{b}\right ) + {\left (8 \, B c^{3} x^{3} - 105 \, B b^{3} + 45 \, A b^{2} c - 8 \, {\left (7 \, B b c^{2} - 3 \, A c^{3}\right )} x^{2} - 25 \, {\left (7 \, B b^{2} c - 3 \, A b c^{2}\right )} x\right )} \sqrt {x}}{12 \, {\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 119, normalized size = 0.81 \begin {gather*} \frac {5 \, {\left (7 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} c^{4}} - \frac {13 \, B b^{2} c x^{\frac {3}{2}} - 9 \, A b c^{2} x^{\frac {3}{2}} + 11 \, B b^{3} \sqrt {x} - 7 \, A b^{2} c \sqrt {x}}{4 \, {\left (c x + b\right )}^{2} c^{4}} + \frac {2 \, {\left (B c^{6} x^{\frac {3}{2}} - 9 \, B b c^{5} \sqrt {x} + 3 \, A c^{6} \sqrt {x}\right )}}{3 \, c^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 152, normalized size = 1.03 \begin {gather*} \frac {9 A b \,x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} c^{2}}-\frac {13 B \,b^{2} x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} c^{3}}+\frac {7 A \,b^{2} \sqrt {x}}{4 \left (c x +b \right )^{2} c^{3}}-\frac {11 B \,b^{3} \sqrt {x}}{4 \left (c x +b \right )^{2} c^{4}}-\frac {15 A b \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, c^{3}}+\frac {35 B \,b^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, c^{4}}+\frac {2 B \,x^{\frac {3}{2}}}{3 c^{3}}+\frac {2 A \sqrt {x}}{c^{3}}-\frac {6 B b \sqrt {x}}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.19, size = 124, normalized size = 0.84 \begin {gather*} -\frac {{\left (13 \, B b^{2} c - 9 \, A b c^{2}\right )} x^{\frac {3}{2}} + {\left (11 \, B b^{3} - 7 \, A b^{2} c\right )} \sqrt {x}}{4 \, {\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}} + \frac {5 \, {\left (7 \, B b^{2} - 3 \, A b c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} c^{4}} + \frac {2 \, {\left (B c x^{\frac {3}{2}} - 3 \, {\left (3 \, B b - A c\right )} \sqrt {x}\right )}}{3 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 143, normalized size = 0.97 \begin {gather*} \frac {x^{3/2}\,\left (\frac {9\,A\,b\,c^2}{4}-\frac {13\,B\,b^2\,c}{4}\right )-\sqrt {x}\,\left (\frac {11\,B\,b^3}{4}-\frac {7\,A\,b^2\,c}{4}\right )}{b^2\,c^4+2\,b\,c^5\,x+c^6\,x^2}+\sqrt {x}\,\left (\frac {2\,A}{c^3}-\frac {6\,B\,b}{c^4}\right )+\frac {2\,B\,x^{3/2}}{3\,c^3}+\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c}\,\sqrt {x}\,\left (3\,A\,c-7\,B\,b\right )}{7\,B\,b^2-3\,A\,b\,c}\right )\,\left (3\,A\,c-7\,B\,b\right )}{4\,c^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________