Optimal. Leaf size=64 \[ \frac {(A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} c^{3/2}}-\frac {\sqrt {x} (b B-A c)}{b c (b+c x)} \]
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Rubi [A] time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {781, 78, 63, 205} \begin {gather*} \frac {(A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} c^{3/2}}-\frac {\sqrt {x} (b B-A c)}{b c (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{\left (b x+c x^2\right )^2} \, dx &=\int \frac {A+B x}{\sqrt {x} (b+c x)^2} \, dx\\ &=-\frac {(b B-A c) \sqrt {x}}{b c (b+c x)}+\frac {(b B+A c) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{2 b c}\\ &=-\frac {(b B-A c) \sqrt {x}}{b c (b+c x)}+\frac {(b B+A c) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{b c}\\ &=-\frac {(b B-A c) \sqrt {x}}{b c (b+c x)}+\frac {(b B+A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.98 \begin {gather*} \frac {(A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} c^{3/2}}+\frac {\sqrt {x} (A c-b B)}{b c (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 63, normalized size = 0.98 \begin {gather*} \frac {(A c+b B) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2} c^{3/2}}+\frac {\sqrt {x} (A c-b B)}{b c (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 177, normalized size = 2.77 \begin {gather*} \left [-\frac {{\left (B b^{2} + A b c + {\left (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 (B b^{2} c - A b c^{2}\right )} \sqrt {x}}{2 \, {\left (b^{2} c^{3} x + b^{3} c^{2}\right )}}, -\frac {{\left (B b^{2} + A b c + {\left (B b c + A c^{2}\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c}}{c \sqrt {x}}\right ) + {\left (B b^{2} c - A b c^{2}\right )} \sqrt {x}}{b^{2} c^{3} x + b^{3} c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 60, normalized size = 0.94 \begin {gather*} \frac {{\left (B b + A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b c} - \frac {B b \sqrt {x} - A c \sqrt {x}}{{\left (c x + b\right )} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 69, normalized size = 1.08 \begin {gather*} \frac {A \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b}+\frac {B \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c}+\frac {\left (A c -b B \right ) \sqrt {x}}{\left (c x +b \right ) b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 58, normalized size = 0.91 \begin {gather*} -\frac {{\left (B b - A c\right )} \sqrt {x}}{b c^{2} x + b^{2} c} + \frac {{\left (B b + A c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 51, normalized size = 0.80 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (A\,c+B\,b\right )}{b^{3/2}\,c^{3/2}}+\frac {\sqrt {x}\,\left (A\,c-B\,b\right )}{b\,c\,\left (b+c\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 55.27, size = 716, normalized size = 11.19 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{c^{2}} & \text {for}\: b = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {3}{2}}}{3}}{b^{2}} & \text {for}\: c = 0 \\\frac {2 i A \sqrt {b} c^{2} \sqrt {x} \sqrt {\frac {1}{c}}}{2 i b^{\frac {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} 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 {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} 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 {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} 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 {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} 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 {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} x \sqrt {\frac {1}{c}}} - \frac {2 i B b^{\frac {3}{2}} c \sqrt {x} \sqrt {\frac {1}{c}}}{2 i b^{\frac {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} x \sqrt {\frac {1}{c}}} + \frac {B b^{2} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} x \sqrt {\frac {1}{c}}} - \frac {B b^{2} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} x \sqrt {\frac {1}{c}}} + \frac {B b c x \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} x \sqrt {\frac {1}{c}}} - \frac {B b c x \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{2 i b^{\frac {5}{2}} c^{2} \sqrt {\frac {1}{c}} + 2 i b^{\frac {3}{2}} c^{3} 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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