Optimal. Leaf size=88 \[ -\frac {(3 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{5/2}}+\frac {\sqrt {x} (3 b B-A c)}{b c^2}-\frac {x^{3/2} (b B-A c)}{b c (b+c x)} \]
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Rubi [A] time = 0.04, antiderivative size = 88, 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, 50, 63, 205} \begin {gather*} \frac {\sqrt {x} (3 b B-A c)}{b c^2}-\frac {(3 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{5/2}}-\frac {x^{3/2} (b B-A c)}{b c (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx &=\int \frac {\sqrt {x} (A+B x)}{(b+c x)^2} \, dx\\ &=-\frac {(b B-A c) x^{3/2}}{b c (b+c x)}-\frac {\left (-\frac {3 b B}{2}+\frac {A c}{2}\right ) \int \frac {\sqrt {x}}{b+c x} \, dx}{b c}\\ &=\frac {(3 b B-A c) \sqrt {x}}{b c^2}-\frac {(b B-A c) x^{3/2}}{b c (b+c x)}-\frac {(3 b B-A c) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{2 c^2}\\ &=\frac {(3 b B-A c) \sqrt {x}}{b c^2}-\frac {(b B-A c) x^{3/2}}{b c (b+c x)}-\frac {(3 b B-A c) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=\frac {(3 b B-A c) \sqrt {x}}{b c^2}-\frac {(b B-A c) x^{3/2}}{b c (b+c x)}-\frac {(3 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 69, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x} (-A c+3 b B+2 B c x)}{c^2 (b+c x)}-\frac {(3 b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 67, normalized size = 0.76 \begin {gather*} \frac {(A c-3 b B) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} c^{5/2}}+\frac {\sqrt {x} (-A c+3 b B+2 B c x)}{c^2 (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 198, normalized size = 2.25 \begin {gather*} \left [\frac {{\left (3 \, B b^{2} - A b c + {\left (3 \, B b c - A c^{2}\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 \, B b c^{2} x + 3 \, B b^{2} c - A b c^{2}\right )} \sqrt {x}}{2 \, {\left (b c^{4} x + b^{2} c^{3}\right )}}, \frac {{\left (3 \, B b^{2} - A b c + {\left (3 \, B b c - A c^{2}\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right ) + {\left (2 \, B b c^{2} x + 3 \, B b^{2} c - A b c^{2}\right )} \sqrt {x}}{b c^{4} x + b^{2} c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 65, normalized size = 0.74 \begin {gather*} \frac {2 \, B \sqrt {x}}{c^{2}} - \frac {{\left (3 \, B b - A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} + \frac {B b \sqrt {x} - A c \sqrt {x}}{{\left (c x + b\right )} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 87, normalized size = 0.99 \begin {gather*} \frac {A \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c}-\frac {3 B b \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{2}}-\frac {A \sqrt {x}}{\left (c x +b \right ) c}+\frac {B b \sqrt {x}}{\left (c x +b \right ) c^{2}}+\frac {2 B \sqrt {x}}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 65, normalized size = 0.74 \begin {gather*} \frac {{\left (B b - A c\right )} \sqrt {x}}{c^{3} x + b c^{2}} + \frac {2 \, B \sqrt {x}}{c^{2}} - \frac {{\left (3 \, B b - A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 62, normalized size = 0.70 \begin {gather*} \frac {2\,B\,\sqrt {x}}{c^2}-\frac {\sqrt {x}\,\left (A\,c-B\,b\right )}{x\,c^3+b\,c^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (A\,c-3\,B\,b\right )}{\sqrt {b}\,c^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 107.25, size = 782, normalized size = 8.89 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{c^{2}} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{b^{2}} & \text {for}\: c = 0 \\- \frac {2 i A \sqrt {b} c^{2} \sqrt {x} \sqrt {\frac {1}{c}}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} + \frac {A b c \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} - \frac {A b c \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} + \frac {A c^{2} x \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} - \frac {A c^{2} x \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} + \frac {6 i B b^{\frac {3}{2}} c \sqrt {x} \sqrt {\frac {1}{c}}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} + \frac {4 i B \sqrt {b} c^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{c}}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} - \frac {3 B b^{2} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} + \frac {3 B b^{2} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} - \frac {3 B b c x \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} + \frac {3 B b c x \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {3}{2}} c^{3} \sqrt {\frac {1}{c}} + 2 i \sqrt {b} c^{4} x \sqrt {\frac {1}{c}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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