Optimal. Leaf size=89 \[ \frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {15}{4} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {49, 52, 65, 223,
212} \begin {gather*} \frac {15}{4} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}+\frac {15}{4} a b \sqrt {x} \sqrt {a+b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx &=-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+(5 b) \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx\\ &=\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {1}{4} (15 a b) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx\\ &=\frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {1}{8} \left (15 a^2 b\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx\\ &=\frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {1}{4} \left (15 a^2 b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {1}{4} \left (15 a^2 b\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )\\ &=\frac {15}{4} a b \sqrt {x} \sqrt {a+b x}+\frac {5}{2} b \sqrt {x} (a+b x)^{3/2}-\frac {2 (a+b x)^{5/2}}{\sqrt {x}}+\frac {15}{4} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 73, normalized size = 0.82 \begin {gather*} \frac {\sqrt {a+b x} \left (-8 a^2+9 a b x+2 b^2 x^2\right )}{4 \sqrt {x}}-\frac {15}{4} a^2 \sqrt {b} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 5.69, size = 91, normalized size = 1.02 \begin {gather*} \frac {-8 a^3 \left (a+b x\right )+15 a^{\frac {7}{2}} \sqrt {b} \sqrt {x} \text {ArcSinh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {a+b x}{a}\right )^{\frac {3}{2}}+a b x \left (a+11 b x\right ) \left (a+b x\right )+2 b^3 x^3 \left (a+b x\right )}{4 a^{\frac {3}{2}} \sqrt {x} \left (\frac {a+b x}{a}\right )^{\frac {3}{2}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 84, normalized size = 0.94
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-2 x^{2} b^{2}-9 a b x +8 a^{2}\right )}{4 \sqrt {x}}+\frac {15 a^{2} \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {x^{2} b +a x}\right ) \sqrt {x \left (b x +a \right )}}{8 \sqrt {x}\, \sqrt {b x +a}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 125, normalized size = 1.40 \begin {gather*} -\frac {15}{8} \, a^{2} \sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) - \frac {2 \, \sqrt {b x + a} a^{2}}{\sqrt {x}} - \frac {\frac {7 \, \sqrt {b x + a} a^{2} b^{2}}{\sqrt {x}} - \frac {9 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b}{x^{\frac {3}{2}}}}{4 \, {\left (b^{2} - \frac {2 \, {\left (b x + a\right )} b}{x} + \frac {{\left (b x + a\right )}^{2}}{x^{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 137, normalized size = 1.54 \begin {gather*} \left [\frac {15 \, a^{2} \sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{8 \, x}, -\frac {15 \, a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{4 \, x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.95, size = 126, normalized size = 1.42 \begin {gather*} - \frac {2 a^{\frac {5}{2}}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + \frac {a^{\frac {3}{2}} b \sqrt {x}}{4 \sqrt {1 + \frac {b x}{a}}} + \frac {11 \sqrt {a} b^{2} x^{\frac {3}{2}}}{4 \sqrt {1 + \frac {b x}{a}}} + \frac {15 a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4} + \frac {b^{3} x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 10.50, size = 155, normalized size = 1.74 \begin {gather*} \frac {b b^{2} \left (\frac {2 \left (\left (\frac {1}{4} \sqrt {a+b x} \sqrt {a+b x}+\frac {5}{8} a\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {15}{8} a^{2}\right ) \sqrt {a+b x} \sqrt {-a b+b \left (a+b x\right )}}{-a b+b \left (a+b x\right )}-\frac {30 a^{2} \ln \left |\sqrt {-a b+b \left (a+b x\right )}-\sqrt {b} \sqrt {a+b x}\right |}{8 \sqrt {b}}\right )}{\left |b\right | b} \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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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