Optimal. Leaf size=193 \[ -\frac {9 c^{5/2} (7 b B-11 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{13/2}}-\frac {9 c^2 (7 b B-11 A c)}{4 b^6 \sqrt {x}}+\frac {3 c (7 b B-11 A c)}{4 b^5 x^{3/2}}-\frac {9 (7 b B-11 A c)}{20 b^4 x^{5/2}}+\frac {9 (7 b B-11 A c)}{28 b^3 c x^{7/2}}-\frac {7 b B-11 A c}{4 b^2 c x^{7/2} (b+c x)}-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {9 c^2 (7 b B-11 A c)}{4 b^6 \sqrt {x}}-\frac {9 c^{5/2} (7 b B-11 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{13/2}}+\frac {3 c (7 b B-11 A c)}{4 b^5 x^{3/2}}-\frac {9 (7 b B-11 A c)}{20 b^4 x^{5/2}}-\frac {7 b B-11 A c}{4 b^2 c x^{7/2} (b+c x)}+\frac {9 (7 b B-11 A c)}{28 b^3 c x^{7/2}}-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{3/2} \left (b x+c x^2\right )^3} \, dx &=\int \frac {A+B x}{x^{9/2} (b+c x)^3} \, dx\\ &=-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2}-\frac {\left (\frac {7 b B}{2}-\frac {11 A c}{2}\right ) \int \frac {1}{x^{9/2} (b+c x)^2} \, dx}{2 b c}\\ &=-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2}-\frac {7 b B-11 A c}{4 b^2 c x^{7/2} (b+c x)}-\frac {(9 (7 b B-11 A c)) \int \frac {1}{x^{9/2} (b+c x)} \, dx}{8 b^2 c}\\ &=\frac {9 (7 b B-11 A c)}{28 b^3 c x^{7/2}}-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2}-\frac {7 b B-11 A c}{4 b^2 c x^{7/2} (b+c x)}+\frac {(9 (7 b B-11 A c)) \int \frac {1}{x^{7/2} (b+c x)} \, dx}{8 b^3}\\ &=\frac {9 (7 b B-11 A c)}{28 b^3 c x^{7/2}}-\frac {9 (7 b B-11 A c)}{20 b^4 x^{5/2}}-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2}-\frac {7 b B-11 A c}{4 b^2 c x^{7/2} (b+c x)}-\frac {(9 c (7 b B-11 A c)) \int \frac {1}{x^{5/2} (b+c x)} \, dx}{8 b^4}\\ &=\frac {9 (7 b B-11 A c)}{28 b^3 c x^{7/2}}-\frac {9 (7 b B-11 A c)}{20 b^4 x^{5/2}}+\frac {3 c (7 b B-11 A c)}{4 b^5 x^{3/2}}-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2}-\frac {7 b B-11 A c}{4 b^2 c x^{7/2} (b+c x)}+\frac {\left (9 c^2 (7 b B-11 A c)\right ) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{8 b^5}\\ &=\frac {9 (7 b B-11 A c)}{28 b^3 c x^{7/2}}-\frac {9 (7 b B-11 A c)}{20 b^4 x^{5/2}}+\frac {3 c (7 b B-11 A c)}{4 b^5 x^{3/2}}-\frac {9 c^2 (7 b B-11 A c)}{4 b^6 \sqrt {x}}-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2}-\frac {7 b B-11 A c}{4 b^2 c x^{7/2} (b+c x)}-\frac {\left (9 c^3 (7 b B-11 A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{8 b^6}\\ &=\frac {9 (7 b B-11 A c)}{28 b^3 c x^{7/2}}-\frac {9 (7 b B-11 A c)}{20 b^4 x^{5/2}}+\frac {3 c (7 b B-11 A c)}{4 b^5 x^{3/2}}-\frac {9 c^2 (7 b B-11 A c)}{4 b^6 \sqrt {x}}-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2}-\frac {7 b B-11 A c}{4 b^2 c x^{7/2} (b+c x)}-\frac {\left (9 c^3 (7 b B-11 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{4 b^6}\\ &=\frac {9 (7 b B-11 A c)}{28 b^3 c x^{7/2}}-\frac {9 (7 b B-11 A c)}{20 b^4 x^{5/2}}+\frac {3 c (7 b B-11 A c)}{4 b^5 x^{3/2}}-\frac {9 c^2 (7 b B-11 A c)}{4 b^6 \sqrt {x}}-\frac {b B-A c}{2 b c x^{7/2} (b+c x)^2}-\frac {7 b B-11 A c}{4 b^2 c x^{7/2} (b+c x)}-\frac {9 c^{5/2} (7 b B-11 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{13/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 61, normalized size = 0.32 \begin {gather*} \frac {\frac {7 b^2 (A c-b B)}{(b+c x)^2}+(7 b B-11 A c) \, _2F_1\left (-\frac {7}{2},2;-\frac {5}{2};-\frac {c x}{b}\right )}{14 b^3 c x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.22, size = 173, normalized size = 0.90 \begin {gather*} \frac {-40 A b^5+88 A b^4 c x-264 A b^3 c^2 x^2+1848 A b^2 c^3 x^3+5775 A b c^4 x^4+3465 A c^5 x^5-56 b^5 B x+168 b^4 B c x^2-1176 b^3 B c^2 x^3-3675 b^2 B c^3 x^4-2205 b B c^4 x^5}{140 b^6 x^{7/2} (b+c x)^2}-\frac {9 \left (7 b B c^{5/2}-11 A c^{7/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{13/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 490, normalized size = 2.54 \begin {gather*} \left [-\frac {315 \, {\left ({\left (7 \, B b c^{4} - 11 \, A c^{5}\right )} x^{6} + 2 \, {\left (7 \, B b^{2} c^{3} - 11 \, A b c^{4}\right )} x^{5} + {\left (7 \, B b^{3} c^{2} - 11 \, A b^{2} c^{3}\right )} x^{4}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x + 2 \, b \sqrt {x} \sqrt {-\frac {c}{b}} - b}{c x + b}\right ) + 2 \, {\left (40 \, A b^{5} + 315 \, {\left (7 \, B b c^{4} - 11 \, A c^{5}\right )} x^{5} + 525 \, {\left (7 \, B b^{2} c^{3} - 11 \, A b c^{4}\right )} x^{4} + 168 \, {\left (7 \, B b^{3} c^{2} - 11 \, A b^{2} c^{3}\right )} x^{3} - 24 \, {\left (7 \, B b^{4} c - 11 \, A b^{3} c^{2}\right )} x^{2} + 8 \, {\left (7 \, B b^{5} - 11 \, A b^{4} c\right )} x\right )} \sqrt {x}}{280 \, {\left (b^{6} c^{2} x^{6} + 2 \, b^{7} c x^{5} + b^{8} x^{4}\right )}}, \frac {315 \, {\left ({\left (7 \, B b c^{4} - 11 \, A c^{5}\right )} x^{6} + 2 \, {\left (7 \, B b^{2} c^{3} - 11 \, A b c^{4}\right )} x^{5} + {\left (7 \, B b^{3} c^{2} - 11 \, A b^{2} c^{3}\right )} x^{4}\right )} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) - {\left (40 \, A b^{5} + 315 \, {\left (7 \, B b c^{4} - 11 \, A c^{5}\right )} x^{5} + 525 \, {\left (7 \, B b^{2} c^{3} - 11 \, A b c^{4}\right )} x^{4} + 168 \, {\left (7 \, B b^{3} c^{2} - 11 \, A b^{2} c^{3}\right )} x^{3} - 24 \, {\left (7 \, B b^{4} c - 11 \, A b^{3} c^{2}\right )} x^{2} + 8 \, {\left (7 \, B b^{5} - 11 \, A b^{4} c\right )} x\right )} \sqrt {x}}{140 \, {\left (b^{6} c^{2} x^{6} + 2 \, b^{7} c x^{5} + b^{8} x^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 159, normalized size = 0.82 \begin {gather*} -\frac {9 \, {\left (7 \, B b c^{3} - 11 \, A c^{4}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} b^{6}} - \frac {15 \, B b c^{4} x^{\frac {3}{2}} - 19 \, A c^{5} x^{\frac {3}{2}} + 17 \, B b^{2} c^{3} \sqrt {x} - 21 \, A b c^{4} \sqrt {x}}{4 \, {\left (c x + b\right )}^{2} b^{6}} - \frac {2 \, {\left (210 \, B b c^{2} x^{3} - 350 \, A c^{3} x^{3} - 35 \, B b^{2} c x^{2} + 70 \, A b c^{2} x^{2} + 7 \, B b^{3} x - 21 \, A b^{2} c x + 5 \, A b^{3}\right )}}{35 \, b^{6} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 202, normalized size = 1.05 \begin {gather*} \frac {19 A \,c^{5} x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} b^{6}}-\frac {15 B \,c^{4} x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} b^{5}}+\frac {21 A \,c^{4} \sqrt {x}}{4 \left (c x +b \right )^{2} b^{5}}-\frac {17 B \,c^{3} \sqrt {x}}{4 \left (c x +b \right )^{2} b^{4}}+\frac {99 A \,c^{4} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, b^{6}}-\frac {63 B \,c^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, b^{5}}+\frac {20 A \,c^{3}}{b^{6} \sqrt {x}}-\frac {12 B \,c^{2}}{b^{5} \sqrt {x}}-\frac {4 A \,c^{2}}{b^{5} x^{\frac {3}{2}}}+\frac {2 B c}{b^{4} x^{\frac {3}{2}}}+\frac {6 A c}{5 b^{4} x^{\frac {5}{2}}}-\frac {2 B}{5 b^{3} x^{\frac {5}{2}}}-\frac {2 A}{7 b^{3} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 178, normalized size = 0.92 \begin {gather*} -\frac {40 \, A b^{5} + 315 \, {\left (7 \, B b c^{4} - 11 \, A c^{5}\right )} x^{5} + 525 \, {\left (7 \, B b^{2} c^{3} - 11 \, A b c^{4}\right )} x^{4} + 168 \, {\left (7 \, B b^{3} c^{2} - 11 \, A b^{2} c^{3}\right )} x^{3} - 24 \, {\left (7 \, B b^{4} c - 11 \, A b^{3} c^{2}\right )} x^{2} + 8 \, {\left (7 \, B b^{5} - 11 \, A b^{4} c\right )} x}{140 \, {\left (b^{6} c^{2} x^{\frac {11}{2}} + 2 \, b^{7} c x^{\frac {9}{2}} + b^{8} x^{\frac {7}{2}}\right )}} - \frac {9 \, {\left (7 \, B b c^{3} - 11 \, A c^{4}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.18, size = 154, normalized size = 0.80 \begin {gather*} \frac {\frac {2\,x\,\left (11\,A\,c-7\,B\,b\right )}{35\,b^2}-\frac {2\,A}{7\,b}+\frac {6\,c^2\,x^3\,\left (11\,A\,c-7\,B\,b\right )}{5\,b^4}+\frac {15\,c^3\,x^4\,\left (11\,A\,c-7\,B\,b\right )}{4\,b^5}+\frac {9\,c^4\,x^5\,\left (11\,A\,c-7\,B\,b\right )}{4\,b^6}-\frac {6\,c\,x^2\,\left (11\,A\,c-7\,B\,b\right )}{35\,b^3}}{b^2\,x^{7/2}+c^2\,x^{11/2}+2\,b\,c\,x^{9/2}}+\frac {9\,c^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (11\,A\,c-7\,B\,b\right )}{4\,b^{13/2}} \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]
________________________________________________________________________________________