Optimal. Leaf size=98 \[ -9 a^{7/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+9 a^3 b \sqrt {a+b x}+3 a^2 b (a+b x)^{3/2}-\frac {(a+b x)^{9/2}}{x}+\frac {9}{7} b (a+b x)^{7/2}+\frac {9}{5} a b (a+b x)^{5/2} \]
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Rubi [A] time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {47, 50, 63, 208} \begin {gather*} 3 a^2 b (a+b x)^{3/2}+9 a^3 b \sqrt {a+b x}-9 a^{7/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-\frac {(a+b x)^{9/2}}{x}+\frac {9}{7} b (a+b x)^{7/2}+\frac {9}{5} a b (a+b x)^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^{9/2}}{x^2} \, dx &=-\frac {(a+b x)^{9/2}}{x}+\frac {1}{2} (9 b) \int \frac {(a+b x)^{7/2}}{x} \, dx\\ &=\frac {9}{7} b (a+b x)^{7/2}-\frac {(a+b x)^{9/2}}{x}+\frac {1}{2} (9 a b) \int \frac {(a+b x)^{5/2}}{x} \, dx\\ &=\frac {9}{5} a b (a+b x)^{5/2}+\frac {9}{7} b (a+b x)^{7/2}-\frac {(a+b x)^{9/2}}{x}+\frac {1}{2} \left (9 a^2 b\right ) \int \frac {(a+b x)^{3/2}}{x} \, dx\\ &=3 a^2 b (a+b x)^{3/2}+\frac {9}{5} a b (a+b x)^{5/2}+\frac {9}{7} b (a+b x)^{7/2}-\frac {(a+b x)^{9/2}}{x}+\frac {1}{2} \left (9 a^3 b\right ) \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=9 a^3 b \sqrt {a+b x}+3 a^2 b (a+b x)^{3/2}+\frac {9}{5} a b (a+b x)^{5/2}+\frac {9}{7} b (a+b x)^{7/2}-\frac {(a+b x)^{9/2}}{x}+\frac {1}{2} \left (9 a^4 b\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=9 a^3 b \sqrt {a+b x}+3 a^2 b (a+b x)^{3/2}+\frac {9}{5} a b (a+b x)^{5/2}+\frac {9}{7} b (a+b x)^{7/2}-\frac {(a+b x)^{9/2}}{x}+\left (9 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )\\ &=9 a^3 b \sqrt {a+b x}+3 a^2 b (a+b x)^{3/2}+\frac {9}{5} a b (a+b x)^{5/2}+\frac {9}{7} b (a+b x)^{7/2}-\frac {(a+b x)^{9/2}}{x}-9 a^{7/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 33, normalized size = 0.34 \begin {gather*} \frac {2 b (a+b x)^{11/2} \, _2F_1\left (2,\frac {11}{2};\frac {13}{2};\frac {b x}{a}+1\right )}{11 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 88, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a+b x} \left (-315 a^4+210 a^3 (a+b x)+42 a^2 (a+b x)^2+18 a (a+b x)^3+10 (a+b x)^4\right )}{35 x}-9 a^{7/2} b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 172, normalized size = 1.76 \begin {gather*} \left [\frac {315 \, a^{\frac {7}{2}} b x \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (10 \, b^{4} x^{4} + 58 \, a b^{3} x^{3} + 156 \, a^{2} b^{2} x^{2} + 388 \, a^{3} b x - 35 \, a^{4}\right )} \sqrt {b x + a}}{70 \, x}, \frac {315 \, \sqrt {-a} a^{3} b x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (10 \, b^{4} x^{4} + 58 \, a b^{3} x^{3} + 156 \, a^{2} b^{2} x^{2} + 388 \, a^{3} b x - 35 \, a^{4}\right )} \sqrt {b x + a}}{35 \, x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.43, size = 104, normalized size = 1.06 \begin {gather*} \frac {\frac {315 \, a^{4} b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 10 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{2} + 28 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{2} + 70 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{2} + 280 \, \sqrt {b x + a} a^{3} b^{2} - \frac {35 \, \sqrt {b x + a} a^{4} b}{x}}{35 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.86 \begin {gather*} 2 \left (\left (-\frac {9 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}-\frac {\sqrt {b x +a}}{2 b x}\right ) a^{4}+4 \sqrt {b x +a}\, a^{3}+\left (b x +a \right )^{\frac {3}{2}} a^{2}+\frac {2 \left (b x +a \right )^{\frac {5}{2}} a}{5}+\frac {\left (b x +a \right )^{\frac {7}{2}}}{7}\right ) b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 97, normalized size = 0.99 \begin {gather*} \frac {9}{2} \, a^{\frac {7}{2}} b \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2}{7} \, {\left (b x + a\right )}^{\frac {7}{2}} b + \frac {4}{5} \, {\left (b x + a\right )}^{\frac {5}{2}} a b + 2 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b + 8 \, \sqrt {b x + a} a^{3} b - \frac {\sqrt {b x + a} a^{4}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 84, normalized size = 0.86 \begin {gather*} \frac {2\,b\,{\left (a+b\,x\right )}^{7/2}}{7}-\frac {a^4\,\sqrt {a+b\,x}}{x}+\frac {4\,a\,b\,{\left (a+b\,x\right )}^{5/2}}{5}+8\,a^3\,b\,\sqrt {a+b\,x}+2\,a^2\,b\,{\left (a+b\,x\right )}^{3/2}+a^{7/2}\,b\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,9{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.92, size = 150, normalized size = 1.53 \begin {gather*} - \frac {a^{\frac {9}{2}} \sqrt {1 + \frac {b x}{a}}}{x} + \frac {388 a^{\frac {7}{2}} b \sqrt {1 + \frac {b x}{a}}}{35} + \frac {9 a^{\frac {7}{2}} b \log {\left (\frac {b x}{a} \right )}}{2} - 9 a^{\frac {7}{2}} b \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )} + \frac {156 a^{\frac {5}{2}} b^{2} x \sqrt {1 + \frac {b x}{a}}}{35} + \frac {58 a^{\frac {3}{2}} b^{3} x^{2} \sqrt {1 + \frac {b x}{a}}}{35} + \frac {2 \sqrt {a} b^{4} x^{3} \sqrt {1 + \frac {b x}{a}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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