Optimal. Leaf size=238 \[ \frac {2 a^2 (A b-a B) \sqrt {x} (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a (A b-a B) x^{3/2} (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a^{5/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {784, 81, 52, 65,
211} \begin {gather*} \frac {2 x^{5/2} (a+b x) (A b-a B)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 a^2 \sqrt {x} (a+b x) (A b-a B)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a x^{3/2} (a+b x) (A b-a B)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a^{5/2} (a+b x) (A b-a B) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 211
Rule 784
Rubi steps
\begin {align*} \int \frac {x^{5/2} (A+B x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {x^{5/2} (A+B x)}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (\frac {7 A b^2}{2}-\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {x^{5/2}}{a b+b^2 x} \, dx}{7 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 a \left (\frac {7 A b^2}{2}-\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {x^{3/2}}{a b+b^2 x} \, dx}{7 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 a (A b-a B) x^{3/2} (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 a^2 \left (\frac {7 A b^2}{2}-\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {x}}{a b+b^2 x} \, dx}{7 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 a^2 (A b-a B) \sqrt {x} (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a (A b-a B) x^{3/2} (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (2 a^3 \left (\frac {7 A b^2}{2}-\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{7 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 a^2 (A b-a B) \sqrt {x} (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a (A b-a B) x^{3/2} (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (4 a^3 \left (\frac {7 A b^2}{2}-\frac {7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{7 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 a^2 (A b-a B) \sqrt {x} (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a (A b-a B) x^{3/2} (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (A b-a B) x^{5/2} (a+b x)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{7/2} (a+b x)}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 a^{5/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 120, normalized size = 0.50 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {b} \sqrt {x} \left (-105 a^3 B+35 a^2 b (3 A+B x)-7 a b^2 x (5 A+3 B x)+3 b^3 x^2 (7 A+5 B x)\right )+105 a^{5/2} (-A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{105 b^{9/2} \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 163, normalized size = 0.68
method | result | size |
risch | \(\frac {2 \left (15 B \,x^{3} b^{3}+21 A \,b^{3} x^{2}-21 B a \,b^{2} x^{2}-35 A a \,b^{2} x +35 B \,a^{2} b x +105 A \,a^{2} b -105 B \,a^{3}\right ) \sqrt {x}\, \sqrt {\left (b x +a \right )^{2}}}{105 b^{4} \left (b x +a \right )}+\frac {\left (-\frac {2 a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) A}{b^{3} \sqrt {a b}}+\frac {2 a^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) B}{b^{4} \sqrt {a b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(152\) |
default | \(\frac {2 \left (b x +a \right ) \left (15 B \,x^{\frac {7}{2}} \sqrt {a b}\, b^{3}+21 A \,x^{\frac {5}{2}} \sqrt {a b}\, b^{3}-21 B \,x^{\frac {5}{2}} \sqrt {a b}\, a \,b^{2}-35 A \,x^{\frac {3}{2}} \sqrt {a b}\, a \,b^{2}+35 B \,x^{\frac {3}{2}} \sqrt {a b}\, a^{2} b +105 A \sqrt {x}\, \sqrt {a b}\, a^{2} b -105 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{3} b -105 B \sqrt {x}\, \sqrt {a b}\, a^{3}+105 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{4}\right )}{105 \sqrt {\left (b x +a \right )^{2}}\, b^{4} \sqrt {a b}}\) | \(163\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 203, normalized size = 0.85 \begin {gather*} \frac {6 \, {\left (5 \, B b^{3} x^{2} + 7 \, B a b^{2} x\right )} x^{\frac {5}{2}} - 2 \, {\left (3 \, {\left (9 \, B a b^{2} - 7 \, A b^{3}\right )} x^{2} + 7 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} x^{\frac {3}{2}} - 7 \, {\left (3 \, {\left (9 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 5 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {x}}{105 \, {\left (b^{4} x + a b^{3}\right )}} + \frac {2 \, {\left (B a^{4} - A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {{\left (9 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{\frac {3}{2}} - 6 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {x}}{3 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.28, size = 229, normalized size = 0.96 \begin {gather*} \left [-\frac {105 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {x}}{105 \, b^{4}}, \frac {2 \, {\left (105 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (15 \, B b^{3} x^{3} - 105 \, B a^{3} + 105 \, A a^{2} b - 21 \, {\left (B a b^{2} - A b^{3}\right )} x^{2} + 35 \, {\left (B a^{2} b - A a b^{2}\right )} x\right )} \sqrt {x}\right )}}{105 \, b^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 169, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (B a^{4} \mathrm {sgn}\left (b x + a\right ) - A a^{3} b \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {2 \, {\left (15 \, B b^{6} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) - 21 \, B a b^{5} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 21 \, A b^{6} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 35 \, B a^{2} b^{4} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) - 35 \, A a b^{5} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) - 105 \, B a^{3} b^{3} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 105 \, A a^{2} b^{4} \sqrt {x} \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, b^{7}} \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 = -0.00 \begin {gather*} \int \frac {x^{5/2}\,\left (A+B\,x\right )}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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