Optimal. Leaf size=120 \[ \frac {2 a A x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 (A b+a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b B x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {784, 77}
\begin {gather*} \frac {2 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac {2 a A x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 b B x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 784
Rubi steps
\begin {align*} \int x^{5/2} (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^{5/2} \left (a b+b^2 x\right ) (A+B x) \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a A b x^{5/2}+b (A b+a B) x^{7/2}+b^2 B x^{9/2}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 a A x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 (A b+a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b B x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 0.41 \begin {gather*} \frac {2 x^{7/2} \sqrt {(a+b x)^2} \left (99 a A+77 A b x+77 a B x+63 b B x^2\right )}{693 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
2.
time = 0.31, size = 34, normalized size = 0.28
method | result | size |
default | \(\frac {2 \,\mathrm {csgn}\left (b x +a \right ) x^{\frac {7}{2}} \left (63 b B \,x^{2}+77 A b x +77 B a x +99 A a \right )}{693}\) | \(34\) |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (63 b B \,x^{2}+77 A b x +77 B a x +99 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{693 \left (b x +a \right )}\) | \(44\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (63 b B \,x^{2}+77 A b x +77 B a x +99 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{693 \left (b x +a \right )}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 35, normalized size = 0.29 \begin {gather*} \frac {2}{99} \, {\left (9 \, b x^{2} + 11 \, a x\right )} B x^{\frac {7}{2}} + \frac {2}{63} \, {\left (7 \, b x^{2} + 9 \, a x\right )} A x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.06, size = 32, normalized size = 0.27 \begin {gather*} \frac {2}{693} \, {\left (63 \, B b x^{5} + 99 \, A a x^{3} + 77 \, {\left (B a + A b\right )} x^{4}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.65, size = 53, normalized size = 0.44 \begin {gather*} \frac {2}{11} \, B b x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, B a x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, A b x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A a x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\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 x^{5/2}\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (A+B\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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