Optimal. Leaf size=75 \[ -\frac {8 a \sqrt {a x^2+b x^3}}{15 b^2}+\frac {16 a^2 \sqrt {a x^2+b x^3}}{15 b^3 x}+\frac {2 x \sqrt {a x^2+b x^3}}{5 b} \]
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Rubi [A]
time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2041, 1602}
\begin {gather*} \frac {16 a^2 \sqrt {a x^2+b x^3}}{15 b^3 x}-\frac {8 a \sqrt {a x^2+b x^3}}{15 b^2}+\frac {2 x \sqrt {a x^2+b x^3}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 1602
Rule 2041
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a x^2+b x^3}} \, dx &=\frac {2 x \sqrt {a x^2+b x^3}}{5 b}-\frac {(4 a) \int \frac {x^2}{\sqrt {a x^2+b x^3}} \, dx}{5 b}\\ &=-\frac {8 a \sqrt {a x^2+b x^3}}{15 b^2}+\frac {2 x \sqrt {a x^2+b x^3}}{5 b}+\frac {\left (8 a^2\right ) \int \frac {x}{\sqrt {a x^2+b x^3}} \, dx}{15 b^2}\\ &=-\frac {8 a \sqrt {a x^2+b x^3}}{15 b^2}+\frac {16 a^2 \sqrt {a x^2+b x^3}}{15 b^3 x}+\frac {2 x \sqrt {a x^2+b x^3}}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 42, normalized size = 0.56 \begin {gather*} \frac {2 \sqrt {x^2 (a+b x)} \left (8 a^2-4 a b x+3 b^2 x^2\right )}{15 b^3 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 44, normalized size = 0.59
method | result | size |
trager | \(\frac {2 \left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right ) \sqrt {b \,x^{3}+a \,x^{2}}}{15 b^{3} x}\) | \(41\) |
risch | \(\frac {2 x \left (b x +a \right ) \left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right )}{15 \sqrt {x^{2} \left (b x +a \right )}\, b^{3}}\) | \(42\) |
gosper | \(\frac {2 \left (b x +a \right ) \left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right ) x}{15 b^{3} \sqrt {b \,x^{3}+a \,x^{2}}}\) | \(44\) |
default | \(\frac {2 \left (b x +a \right ) \left (3 b^{2} x^{2}-4 a b x +8 a^{2}\right ) x}{15 b^{3} \sqrt {b \,x^{3}+a \,x^{2}}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 42, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (3 \, b^{3} x^{3} - a b^{2} x^{2} + 4 \, a^{2} b x + 8 \, a^{3}\right )}}{15 \, \sqrt {b x + a} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.17, size = 40, normalized size = 0.53 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} x^{2} - 4 \, a b x + 8 \, a^{2}\right )} \sqrt {b x^{3} + a x^{2}}}{15 \, b^{3} x} \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 {x^{3}}{\sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.78, size = 52, normalized size = 0.69 \begin {gather*} -\frac {16 \, a^{\frac {5}{2}} \mathrm {sgn}\left (x\right )}{15 \, b^{3}} + \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )}}{15 \, b^{3} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.20, size = 40, normalized size = 0.53 \begin {gather*} \frac {2\,\sqrt {b\,x^3+a\,x^2}\,\left (8\,a^2-4\,a\,b\,x+3\,b^2\,x^2\right )}{15\,b^3\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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