Optimal. Leaf size=19 \[ \frac {2}{7} d \left (a+b x+c x^2\right )^{7/2} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {643}
\begin {gather*} \frac {2}{7} d \left (a+b x+c x^2\right )^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rubi steps
\begin {align*} \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {2}{7} d \left (a+b x+c x^2\right )^{7/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 0.95 \begin {gather*} \frac {2}{7} d (a+x (b+c x))^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.65, size = 16, normalized size = 0.84
method | result | size |
gosper | \(\frac {2 d \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7}\) | \(16\) |
default | \(\frac {2 d \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7}\) | \(16\) |
risch | \(\frac {2 d \left (c^{3} x^{6}+3 b \,c^{2} x^{5}+3 c^{2} a \,x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{7}\) | \(94\) |
trager | \(d \left (\frac {2}{7} c^{3} x^{6}+\frac {6}{7} b \,c^{2} x^{5}+\frac {6}{7} c^{2} a \,x^{4}+\frac {6}{7} b^{2} c \,x^{4}+\frac {12}{7} a b c \,x^{3}+\frac {2}{7} b^{3} x^{3}+\frac {6}{7} a^{2} c \,x^{2}+\frac {6}{7} a \,b^{2} x^{2}+\frac {6}{7} a^{2} b x +\frac {2}{7} a^{3}\right ) \sqrt {c \,x^{2}+b x +a}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 15, normalized size = 0.79 \begin {gather*} \frac {2}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (15) = 30\).
time = 2.17, size = 94, normalized size = 4.95 \begin {gather*} \frac {2}{7} \, {\left (c^{3} d x^{6} + 3 \, b c^{2} d x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} d x^{4} + 3 \, a^{2} b d x + {\left (b^{3} + 6 \, a b c\right )} d x^{3} + a^{3} d + 3 \, {\left (a b^{2} + a^{2} c\right )} d x^{2}\right )} \sqrt {c x^{2} + b x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 260 vs.
\(2 (17) = 34\).
time = 0.24, size = 260, normalized size = 13.68 \begin {gather*} \frac {2 a^{3} d \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a^{2} b d x \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a^{2} c d x^{2} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a b^{2} d x^{2} \sqrt {a + b x + c x^{2}}}{7} + \frac {12 a b c d x^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a c^{2} d x^{4} \sqrt {a + b x + c x^{2}}}{7} + \frac {2 b^{3} d x^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 b^{2} c d x^{4} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 b c^{2} d x^{5} \sqrt {a + b x + c x^{2}}}{7} + \frac {2 c^{3} d x^{6} \sqrt {a + b x + c x^{2}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.59, size = 15, normalized size = 0.79 \begin {gather*} \frac {2}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.67, size = 15, normalized size = 0.79 \begin {gather*} \frac {2\,d\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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