Optimal. Leaf size=91 \[ -\frac {2 b^2 \sqrt {a x+b x^2}}{x}-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {676, 634, 212}
\begin {gather*} 2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )-\frac {2 b^2 \sqrt {a x+b x^2}}{x}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 676
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^2\right )^{5/2}}{x^6} \, dx &=-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+b \int \frac {\left (a x+b x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+b^2 \int \frac {\sqrt {a x+b x^2}}{x^2} \, dx\\ &=-\frac {2 b^2 \sqrt {a x+b x^2}}{x}-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+b^3 \int \frac {1}{\sqrt {a x+b x^2}} \, dx\\ &=-\frac {2 b^2 \sqrt {a x+b x^2}}{x}-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )\\ &=-\frac {2 b^2 \sqrt {a x+b x^2}}{x}-\frac {2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac {2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 92, normalized size = 1.01 \begin {gather*} -\frac {2 \sqrt {x (a+b x)} \left (\sqrt {a+b x} \left (3 a^2+11 a b x+23 b^2 x^2\right )+15 b^{5/2} x^{5/2} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )\right )}{15 x^3 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(233\) vs.
\(2(75)=150\).
time = 0.40, size = 234, normalized size = 2.57
method | result | size |
risch | \(-\frac {2 \left (b x +a \right ) \left (23 b^{2} x^{2}+11 a b x +3 a^{2}\right )}{15 x^{2} \sqrt {x \left (b x +a \right )}}+b^{\frac {5}{2}} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )\) | \(68\) |
default | \(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{5 a \,x^{6}}+\frac {2 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{3 a \,x^{5}}+\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{a \,x^{4}}+\frac {6 b \left (\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{a \,x^{3}}-\frac {8 b \left (\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{3 a \,x^{2}}-\frac {10 b \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5}+\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2}\right )}{3 a}\right )}{a}\right )}{a}\right )}{3 a}\right )}{5 a}\) | \(234\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 134, normalized size = 1.47 \begin {gather*} b^{\frac {5}{2}} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {38 \, \sqrt {b x^{2} + a x} b^{2}}{15 \, x} - \frac {7 \, \sqrt {b x^{2} + a x} a b}{30 \, x^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} a^{2}}{10 \, x^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b}{3 \, x^{3}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{2 \, x^{4}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.47, size = 144, normalized size = 1.58 \begin {gather*} \left [\frac {15 \, b^{\frac {5}{2}} x^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (23 \, b^{2} x^{2} + 11 \, a b x + 3 \, a^{2}\right )} \sqrt {b x^{2} + a x}}{15 \, x^{3}}, -\frac {2 \, {\left (15 \, \sqrt {-b} b^{2} x^{3} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) + {\left (23 \, b^{2} x^{2} + 11 \, a b x + 3 \, a^{2}\right )} \sqrt {b x^{2} + a x}\right )}}{15 \, x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 175 vs.
\(2 (75) = 150\).
time = 0.68, size = 175, normalized size = 1.92 \begin {gather*} -b^{\frac {5}{2}} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right ) + \frac {2 \, {\left (45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a b^{2} + 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{2} b^{\frac {3}{2}} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{3} b + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{4} \sqrt {b} + 3 \, a^{5}\right )}}{15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5}} \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 (b\,x^2+a\,x\right )}^{5/2}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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