Optimal. Leaf size=216 \[ \frac {c^4 (10 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{128 b^{5/2}}-\frac {c^3 \sqrt {b x+c x^2} (10 b B-3 A c)}{128 b^2 x^{3/2}}-\frac {c^2 \sqrt {b x+c x^2} (10 b B-3 A c)}{64 b x^{5/2}}-\frac {c \left (b x+c x^2\right )^{3/2} (10 b B-3 A c)}{48 b x^{9/2}}-\frac {\left (b x+c x^2\right )^{5/2} (10 b B-3 A c)}{40 b x^{13/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {792, 662, 672, 660, 207} \[ -\frac {c^3 \sqrt {b x+c x^2} (10 b B-3 A c)}{128 b^2 x^{3/2}}+\frac {c^4 (10 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{128 b^{5/2}}-\frac {c^2 \sqrt {b x+c x^2} (10 b B-3 A c)}{64 b x^{5/2}}-\frac {c \left (b x+c x^2\right )^{3/2} (10 b B-3 A c)}{48 b x^{9/2}}-\frac {\left (b x+c x^2\right )^{5/2} (10 b B-3 A c)}{40 b x^{13/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 662
Rule 672
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{17/2}} \, dx &=-\frac {A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}}+\frac {\left (-\frac {17}{2} (-b B+A c)+\frac {7}{2} (-b B+2 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^{15/2}} \, dx}{5 b}\\ &=-\frac {(10 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{40 b x^{13/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}}+\frac {(c (10 b B-3 A c)) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx}{16 b}\\ &=-\frac {c (10 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{48 b x^{9/2}}-\frac {(10 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{40 b x^{13/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}}+\frac {\left (c^2 (10 b B-3 A c)\right ) \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx}{32 b}\\ &=-\frac {c^2 (10 b B-3 A c) \sqrt {b x+c x^2}}{64 b x^{5/2}}-\frac {c (10 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{48 b x^{9/2}}-\frac {(10 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{40 b x^{13/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}}+\frac {\left (c^3 (10 b B-3 A c)\right ) \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx}{128 b}\\ &=-\frac {c^2 (10 b B-3 A c) \sqrt {b x+c x^2}}{64 b x^{5/2}}-\frac {c^3 (10 b B-3 A c) \sqrt {b x+c x^2}}{128 b^2 x^{3/2}}-\frac {c (10 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{48 b x^{9/2}}-\frac {(10 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{40 b x^{13/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}}-\frac {\left (c^4 (10 b B-3 A c)\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{256 b^2}\\ &=-\frac {c^2 (10 b B-3 A c) \sqrt {b x+c x^2}}{64 b x^{5/2}}-\frac {c^3 (10 b B-3 A c) \sqrt {b x+c x^2}}{128 b^2 x^{3/2}}-\frac {c (10 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{48 b x^{9/2}}-\frac {(10 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{40 b x^{13/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}}-\frac {\left (c^4 (10 b B-3 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{128 b^2}\\ &=-\frac {c^2 (10 b B-3 A c) \sqrt {b x+c x^2}}{64 b x^{5/2}}-\frac {c^3 (10 b B-3 A c) \sqrt {b x+c x^2}}{128 b^2 x^{3/2}}-\frac {c (10 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{48 b x^{9/2}}-\frac {(10 b B-3 A c) \left (b x+c x^2\right )^{5/2}}{40 b x^{13/2}}-\frac {A \left (b x+c x^2\right )^{7/2}}{5 b x^{17/2}}+\frac {c^4 (10 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{128 b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 69, normalized size = 0.32 \[ -\frac {(b+c x)^3 \sqrt {x (b+c x)} \left (7 A b^5+c^4 x^5 (10 b B-3 A c) \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {c x}{b}+1\right )\right )}{35 b^6 x^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 336, normalized size = 1.56 \[ \left [-\frac {15 \, {\left (10 \, B b c^{4} - 3 \, A c^{5}\right )} \sqrt {b} x^{6} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (384 \, A b^{5} + 15 \, {\left (10 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{4} + 10 \, {\left (118 \, B b^{3} c^{2} + 3 \, A b^{2} c^{3}\right )} x^{3} + 8 \, {\left (170 \, B b^{4} c + 93 \, A b^{3} c^{2}\right )} x^{2} + 48 \, {\left (10 \, B b^{5} + 21 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{3840 \, b^{3} x^{6}}, -\frac {15 \, {\left (10 \, B b c^{4} - 3 \, A c^{5}\right )} \sqrt {-b} x^{6} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (384 \, A b^{5} + 15 \, {\left (10 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{4} + 10 \, {\left (118 \, B b^{3} c^{2} + 3 \, A b^{2} c^{3}\right )} x^{3} + 8 \, {\left (170 \, B b^{4} c + 93 \, A b^{3} c^{2}\right )} x^{2} + 48 \, {\left (10 \, B b^{5} + 21 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{1920 \, b^{3} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 208, normalized size = 0.96 \[ -\frac {\frac {15 \, {\left (10 \, B b c^{5} - 3 \, A c^{6}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {150 \, {\left (c x + b\right )}^{\frac {9}{2}} B b c^{5} + 580 \, {\left (c x + b\right )}^{\frac {7}{2}} B b^{2} c^{5} - 1280 \, {\left (c x + b\right )}^{\frac {5}{2}} B b^{3} c^{5} + 700 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{4} c^{5} - 150 \, \sqrt {c x + b} B b^{5} c^{5} - 45 \, {\left (c x + b\right )}^{\frac {9}{2}} A c^{6} + 210 \, {\left (c x + b\right )}^{\frac {7}{2}} A b c^{6} + 384 \, {\left (c x + b\right )}^{\frac {5}{2}} A b^{2} c^{6} - 210 \, {\left (c x + b\right )}^{\frac {3}{2}} A b^{3} c^{6} + 45 \, \sqrt {c x + b} A b^{4} c^{6}}{b^{2} c^{5} x^{5}}}{1920 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 223, normalized size = 1.03 \[ -\frac {\sqrt {\left (c x +b \right ) x}\, \left (45 A \,c^{5} x^{5} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-150 B b \,c^{4} x^{5} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-45 \sqrt {c x +b}\, A \sqrt {b}\, c^{4} x^{4}+150 \sqrt {c x +b}\, B \,b^{\frac {3}{2}} c^{3} x^{4}+30 \sqrt {c x +b}\, A \,b^{\frac {3}{2}} c^{3} x^{3}+1180 \sqrt {c x +b}\, B \,b^{\frac {5}{2}} c^{2} x^{3}+744 \sqrt {c x +b}\, A \,b^{\frac {5}{2}} c^{2} x^{2}+1360 \sqrt {c x +b}\, B \,b^{\frac {7}{2}} c \,x^{2}+1008 \sqrt {c x +b}\, A \,b^{\frac {7}{2}} c x +480 \sqrt {c x +b}\, B \,b^{\frac {9}{2}} x +384 \sqrt {c x +b}\, A \,b^{\frac {9}{2}}\right )}{1920 \sqrt {c x +b}\, b^{\frac {5}{2}} x^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{x^{\frac {17}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{x^{17/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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