Optimal. Leaf size=146 \[ -\frac {(2 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{7/2}}+\frac {\sqrt {x} (2 b B-5 A c)}{b^3 \sqrt {b x+c x^2}}+\frac {2 b B-5 A c}{3 b^2 c \sqrt {x} \sqrt {b x+c x^2}}-\frac {2 \sqrt {x} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {788, 672, 666, 660, 207} \begin {gather*} \frac {\sqrt {x} (2 b B-5 A c)}{b^3 \sqrt {b x+c x^2}}+\frac {2 b B-5 A c}{3 b^2 c \sqrt {x} \sqrt {b x+c x^2}}-\frac {(2 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{7/2}}-\frac {2 \sqrt {x} (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 666
Rule 672
Rule 788
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b B-A c) \sqrt {x}}{3 b c \left (b x+c x^2\right )^{3/2}}-\frac {\left (2 \left (\frac {1}{2} (-b B+A c)-\frac {3}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b c}\\ &=-\frac {2 (b B-A c) \sqrt {x}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 b B-5 A c}{3 b^2 c \sqrt {x} \sqrt {b x+c x^2}}+\frac {(2 b B-5 A c) \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{2 b^2}\\ &=-\frac {2 (b B-A c) \sqrt {x}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 b B-5 A c}{3 b^2 c \sqrt {x} \sqrt {b x+c x^2}}+\frac {(2 b B-5 A c) \sqrt {x}}{b^3 \sqrt {b x+c x^2}}+\frac {(2 b B-5 A c) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{2 b^3}\\ &=-\frac {2 (b B-A c) \sqrt {x}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 b B-5 A c}{3 b^2 c \sqrt {x} \sqrt {b x+c x^2}}+\frac {(2 b B-5 A c) \sqrt {x}}{b^3 \sqrt {b x+c x^2}}+\frac {(2 b B-5 A c) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{b^3}\\ &=-\frac {2 (b B-A c) \sqrt {x}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 b B-5 A c}{3 b^2 c \sqrt {x} \sqrt {b x+c x^2}}+\frac {(2 b B-5 A c) \sqrt {x}}{b^3 \sqrt {b x+c x^2}}-\frac {(2 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 55, normalized size = 0.38 \begin {gather*} \frac {\sqrt {x} \left (x (2 b B-5 A c) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c x}{b}+1\right )-3 A b\right )}{3 b^2 (x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.73, size = 110, normalized size = 0.75 \begin {gather*} \frac {(5 A c-2 b B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{b^{7/2}}+\frac {\sqrt {b x+c x^2} \left (-3 A b^2-20 A b c x-15 A c^2 x^2+8 b^2 B x+6 b B c x^2\right )}{3 b^3 x^{3/2} (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 367, normalized size = 2.51 \begin {gather*} \left [-\frac {3 \, {\left ({\left (2 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} + 2 \, {\left (2 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} + {\left (2 \, B b^{3} - 5 \, A b^{2} c\right )} x^{2}\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^{3} - 3 \, {\left (2 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} - 4 \, {\left (2 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{6 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}}, \frac {3 \, {\left ({\left (2 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} + 2 \, {\left (2 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} + {\left (2 \, B b^{3} - 5 \, A b^{2} c\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (3 \, A b^{3} - 3 \, {\left (2 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} - 4 \, {\left (2 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{3 \, {\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 90, normalized size = 0.62 \begin {gather*} \frac {{\left (2 \, B b - 5 \, A c\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} - \frac {\sqrt {c x + b} A}{b^{3} x} + \frac {2 \, {\left (3 \, {\left (c x + b\right )} B b + B b^{2} - 6 \, {\left (c x + b\right )} A c - A b c\right )}}{3 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 175, normalized size = 1.20 \begin {gather*} \frac {\sqrt {\left (c x +b \right ) x}\, \left (15 \sqrt {c x +b}\, A \,c^{2} x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-6 \sqrt {c x +b}\, B b c \,x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-15 A \sqrt {b}\, c^{2} x^{2}+6 B \,b^{\frac {3}{2}} c \,x^{2}+15 \sqrt {c x +b}\, A b c x \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-6 \sqrt {c x +b}\, B \,b^{2} x \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-20 A \,b^{\frac {3}{2}} c x +8 B \,b^{\frac {5}{2}} x -3 A \,b^{\frac {5}{2}}\right )}{3 \left (c x +b \right )^{2} b^{\frac {7}{2}} x^{\frac {3}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (B x + A\right )} \sqrt {x}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x}\,\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.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x} \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.
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