Optimal. Leaf size=92 \[ \frac {15}{4} a b \sqrt {a x+b x^2}+\frac {5 b \left (a x+b x^2\right )^{3/2}}{2 x}-\frac {2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac {15}{4} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {676, 678, 634,
212} \begin {gather*} \frac {15}{4} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )+\frac {5 b \left (a x+b x^2\right )^{3/2}}{2 x}+\frac {15}{4} a b \sqrt {a x+b x^2}-\frac {2 \left (a x+b x^2\right )^{5/2}}{x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 676
Rule 678
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^2\right )^{5/2}}{x^4} \, dx &=-\frac {2 \left (a x+b x^2\right )^{5/2}}{x^3}+(5 b) \int \frac {\left (a x+b x^2\right )^{3/2}}{x^2} \, dx\\ &=\frac {5 b \left (a x+b x^2\right )^{3/2}}{2 x}-\frac {2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac {1}{4} (15 a b) \int \frac {\sqrt {a x+b x^2}}{x} \, dx\\ &=\frac {15}{4} a b \sqrt {a x+b x^2}+\frac {5 b \left (a x+b x^2\right )^{3/2}}{2 x}-\frac {2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac {1}{8} \left (15 a^2 b\right ) \int \frac {1}{\sqrt {a x+b x^2}} \, dx\\ &=\frac {15}{4} a b \sqrt {a x+b x^2}+\frac {5 b \left (a x+b x^2\right )^{3/2}}{2 x}-\frac {2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac {1}{4} \left (15 a^2 b\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a x+b x^2}}\right )\\ &=\frac {15}{4} a b \sqrt {a x+b x^2}+\frac {5 b \left (a x+b x^2\right )^{3/2}}{2 x}-\frac {2 \left (a x+b x^2\right )^{5/2}}{x^3}+\frac {15}{4} a^2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 92, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+b x} \left (\sqrt {a+b x} \left (-8 a^2+9 a b x+2 b^2 x^2\right )-15 a^2 \sqrt {b} \sqrt {x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )\right )}{4 \sqrt {x (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. \(181\) vs.
\(2(74)=148\).
time = 0.40, size = 182, normalized size = 1.98
method | result | size |
risch | \(-\frac {\left (b x +a \right ) \left (-2 b^{2} x^{2}-9 a b x +8 a^{2}\right )}{4 \sqrt {x \left (b x +a \right )}}+\frac {15 a^{2} \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8}\) | \(69\) |
default | \(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{a \,x^{4}}+\frac {6 b \left (\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{a \,x^{3}}-\frac {8 b \left (\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{3 a \,x^{2}}-\frac {10 b \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5}+\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2}\right )}{3 a}\right )}{a}\right )}{a}\) | \(182\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 84, normalized size = 0.91 \begin {gather*} \frac {15}{8} \, a^{2} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {15 \, \sqrt {b x^{2} + a x} a^{2}}{4 \, x} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{4 \, x^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{2 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.18, size = 144, normalized size = 1.57 \begin {gather*} \left [\frac {15 \, a^{2} \sqrt {b} x \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt {b x^{2} + a x}}{8 \, x}, -\frac {15 \, a^{2} \sqrt {-b} x \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x}\right ) - {\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt {b x^{2} + a x}}{4 \, x}\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^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 89, normalized size = 0.97 \begin {gather*} -\frac {15}{8} \, a^{2} \sqrt {b} \log \left ({\left | -2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} - a \right |}\right ) + \frac {2 \, a^{3}}{\sqrt {b} x - \sqrt {b x^{2} + a x}} + \frac {1}{4} \, {\left (2 \, b^{2} x + 9 \, a b\right )} \sqrt {b x^{2} + a x} \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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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