Optimal. Leaf size=123 \[ -\frac {3 (A b-5 a B) \sqrt {x}}{4 a b^3}+\frac {(A b-a B) x^{5/2}}{2 a b (a+b x)^2}+\frac {(A b-5 a B) x^{3/2}}{4 a b^2 (a+b x)}+\frac {3 (A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{7/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 43, 52, 65,
211} \begin {gather*} \frac {3 (A b-5 a B) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{7/2}}-\frac {3 \sqrt {x} (A b-5 a B)}{4 a b^3}+\frac {x^{3/2} (A b-5 a B)}{4 a b^2 (a+b x)}+\frac {x^{5/2} (A b-a B)}{2 a b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 211
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{(a+b x)^3} \, dx &=\frac {(A b-a B) x^{5/2}}{2 a b (a+b x)^2}-\frac {\left (\frac {A b}{2}-\frac {5 a B}{2}\right ) \int \frac {x^{3/2}}{(a+b x)^2} \, dx}{2 a b}\\ &=\frac {(A b-a B) x^{5/2}}{2 a b (a+b x)^2}+\frac {(A b-5 a B) x^{3/2}}{4 a b^2 (a+b x)}-\frac {(3 (A b-5 a B)) \int \frac {\sqrt {x}}{a+b x} \, dx}{8 a b^2}\\ &=-\frac {3 (A b-5 a B) \sqrt {x}}{4 a b^3}+\frac {(A b-a B) x^{5/2}}{2 a b (a+b x)^2}+\frac {(A b-5 a B) x^{3/2}}{4 a b^2 (a+b x)}+\frac {(3 (A b-5 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 b^3}\\ &=-\frac {3 (A b-5 a B) \sqrt {x}}{4 a b^3}+\frac {(A b-a B) x^{5/2}}{2 a b (a+b x)^2}+\frac {(A b-5 a B) x^{3/2}}{4 a b^2 (a+b x)}+\frac {(3 (A b-5 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=-\frac {3 (A b-5 a B) \sqrt {x}}{4 a b^3}+\frac {(A b-a B) x^{5/2}}{2 a b (a+b x)^2}+\frac {(A b-5 a B) x^{3/2}}{4 a b^2 (a+b x)}+\frac {3 (A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 91, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x} \left (15 a^2 B+b^2 x (-5 A+8 B x)+a (-3 A b+25 b B x)\right )}{4 b^3 (a+b x)^2}+\frac {3 (A b-5 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 \sqrt {a} b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 83, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (\left (-\frac {5}{8} b^{2} A +\frac {9}{8} a b B \right ) x^{\frac {3}{2}}-\frac {a \left (3 A b -7 B a \right ) \sqrt {x}}{8}\right )}{\left (b x +a \right )^{2}}+\frac {3 \left (A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}}{b^{3}}\) | \(83\) |
default | \(\frac {2 B \sqrt {x}}{b^{3}}+\frac {\frac {2 \left (\left (-\frac {5}{8} b^{2} A +\frac {9}{8} a b B \right ) x^{\frac {3}{2}}-\frac {a \left (3 A b -7 B a \right ) \sqrt {x}}{8}\right )}{\left (b x +a \right )^{2}}+\frac {3 \left (A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}}{b^{3}}\) | \(83\) |
risch | \(\frac {2 B \sqrt {x}}{b^{3}}-\frac {5 x^{\frac {3}{2}} A}{4 b \left (b x +a \right )^{2}}+\frac {9 x^{\frac {3}{2}} a B}{4 b^{2} \left (b x +a \right )^{2}}-\frac {3 A \sqrt {x}\, a}{4 b^{2} \left (b x +a \right )^{2}}+\frac {7 B \sqrt {x}\, a^{2}}{4 b^{3} \left (b x +a \right )^{2}}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) A}{4 b^{2} \sqrt {a b}}-\frac {15 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) B a}{4 b^{3} \sqrt {a b}}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 99, normalized size = 0.80 \begin {gather*} \frac {{\left (9 \, B a b - 5 \, A b^{2}\right )} x^{\frac {3}{2}} + {\left (7 \, B a^{2} - 3 \, A a b\right )} \sqrt {x}}{4 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {2 \, B \sqrt {x}}{b^{3}} - \frac {3 \, {\left (5 \, B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 319, normalized size = 2.59 \begin {gather*} \left [\frac {3 \, {\left (5 \, B a^{3} - A a^{2} b + {\left (5 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (8 \, B a b^{3} x^{2} + 15 \, B a^{3} b - 3 \, A a^{2} b^{2} + 5 \, {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt {x}}{8 \, {\left (a b^{6} x^{2} + 2 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}, \frac {3 \, {\left (5 \, B a^{3} - A a^{2} b + {\left (5 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (8 \, B a b^{3} x^{2} + 15 \, B a^{3} b - 3 \, A a^{2} b^{2} + 5 \, {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt {x}}{4 \, {\left (a b^{6} x^{2} + 2 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1333 vs.
\(2 (116) = 232\).
time = 17.88, size = 1333, normalized size = 10.84 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{a^{3}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{b^{3}} & \text {for}\: a = 0 \\\frac {3 A a^{2} b \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 A a^{2} b \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {6 A a b^{2} \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {6 A a b^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {6 A a b^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {10 A b^{3} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {3 A b^{3} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {3 A b^{3} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {15 B a^{3} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {15 B a^{3} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {30 B a^{2} b \sqrt {x} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {30 B a^{2} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {30 B a^{2} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {50 B a b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} - \frac {15 B a b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {15 B a b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} + \frac {16 B b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{8 a^{2} b^{4} \sqrt {- \frac {a}{b}} + 16 a b^{5} x \sqrt {- \frac {a}{b}} + 8 b^{6} x^{2} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.35, size = 87, normalized size = 0.71 \begin {gather*} \frac {2 \, B \sqrt {x}}{b^{3}} - \frac {3 \, {\left (5 \, B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{3}} + \frac {9 \, B a b x^{\frac {3}{2}} - 5 \, A b^{2} x^{\frac {3}{2}} + 7 \, B a^{2} \sqrt {x} - 3 \, A a b \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 96, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x}\,\left (\frac {7\,B\,a^2}{4}-\frac {3\,A\,a\,b}{4}\right )-x^{3/2}\,\left (\frac {5\,A\,b^2}{4}-\frac {9\,B\,a\,b}{4}\right )}{a^2\,b^3+2\,a\,b^4\,x+b^5\,x^2}+\frac {2\,B\,\sqrt {x}}{b^3}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b-5\,B\,a\right )}{4\,\sqrt {a}\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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