Optimal. Leaf size=116 \[ -\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}-\frac {16 b^2 \sqrt {a x^2+b x^3}}{35 a^3 x^{5/2}}+\frac {32 b^3 \sqrt {a x^2+b x^3}}{35 a^4 x^{3/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2041, 2039}
\begin {gather*} \frac {32 b^3 \sqrt {a x^2+b x^3}}{35 a^4 x^{3/2}}-\frac {16 b^2 \sqrt {a x^2+b x^3}}{35 a^3 x^{5/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \sqrt {a x^2+b x^3}} \, dx &=-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}}-\frac {(6 b) \int \frac {1}{x^{5/2} \sqrt {a x^2+b x^3}} \, dx}{7 a}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}+\frac {\left (24 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {a x^2+b x^3}} \, dx}{35 a^2}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}-\frac {16 b^2 \sqrt {a x^2+b x^3}}{35 a^3 x^{5/2}}-\frac {\left (16 b^3\right ) \int \frac {1}{\sqrt {x} \sqrt {a x^2+b x^3}} \, dx}{35 a^3}\\ &=-\frac {2 \sqrt {a x^2+b x^3}}{7 a x^{9/2}}+\frac {12 b \sqrt {a x^2+b x^3}}{35 a^2 x^{7/2}}-\frac {16 b^2 \sqrt {a x^2+b x^3}}{35 a^3 x^{5/2}}+\frac {32 b^3 \sqrt {a x^2+b x^3}}{35 a^4 x^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 55, normalized size = 0.47 \begin {gather*} \frac {2 \sqrt {x^2 (a+b x)} \left (-5 a^3+6 a^2 b x-8 a b^2 x^2+16 b^3 x^3\right )}{35 a^4 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 57, normalized size = 0.49
method | result | size |
risch | \(-\frac {2 \left (b x +a \right ) \left (-16 b^{3} x^{3}+8 a \,b^{2} x^{2}-6 a^{2} b x +5 a^{3}\right )}{35 \sqrt {x^{2} \left (b x +a \right )}\, x^{\frac {5}{2}} a^{4}}\) | \(55\) |
gosper | \(-\frac {2 \left (b x +a \right ) \left (-16 b^{3} x^{3}+8 a \,b^{2} x^{2}-6 a^{2} b x +5 a^{3}\right )}{35 x^{\frac {5}{2}} a^{4} \sqrt {b \,x^{3}+a \,x^{2}}}\) | \(57\) |
default | \(-\frac {2 \left (b x +a \right ) \left (-16 b^{3} x^{3}+8 a \,b^{2} x^{2}-6 a^{2} b x +5 a^{3}\right )}{35 x^{\frac {5}{2}} a^{4} \sqrt {b \,x^{3}+a \,x^{2}}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.84, size = 51, normalized size = 0.44 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} x^{3} - 8 \, a b^{2} x^{2} + 6 \, a^{2} b x - 5 \, a^{3}\right )} \sqrt {b x^{3} + a x^{2}}}{35 \, a^{4} x^{\frac {9}{2}}} \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 {1}{x^{\frac {7}{2}} \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 = 2.17, size = 107, normalized size = 0.92 \begin {gather*} \frac {64 \, {\left (35 \, {\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{6} - 21 \, a {\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{4} + 7 \, a^{2} {\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a^{3}\right )} b^{\frac {7}{2}}}{35 \, {\left ({\left (\sqrt {b} \sqrt {x} - \sqrt {b x + a}\right )}^{2} - a\right )}^{7} \mathrm {sgn}\left (x\right )} \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 {1}{x^{7/2}\,\sqrt {b\,x^3+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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