3.303 \(\int (a+b x)^{5/2} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 (a+b x)^{7/2}}{7 b} \]

[Out]

(2*(a + b*x)^(7/2))/(7*b)

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Rubi [A]  time = 0.00683132, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2 (a+b x)^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2),x]

[Out]

(2*(a + b*x)^(7/2))/(7*b)

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Rubi in Sympy [A]  time = 1.26961, size = 12, normalized size = 0.75 \[ \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2),x)

[Out]

2*(a + b*x)**(7/2)/(7*b)

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Mathematica [A]  time = 0.00735577, size = 16, normalized size = 1. \[ \frac{2 (a+b x)^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2),x]

[Out]

(2*(a + b*x)^(7/2))/(7*b)

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Maple [A]  time = 0.006, size = 13, normalized size = 0.8 \[{\frac{2}{7\,b} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2),x)

[Out]

2/7*(b*x+a)^(7/2)/b

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Maxima [A]  time = 1.34122, size = 16, normalized size = 1. \[ \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}}}{7 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2),x, algorithm="maxima")

[Out]

2/7*(b*x + a)^(7/2)/b

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Fricas [A]  time = 0.21242, size = 53, normalized size = 3.31 \[ \frac{2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x + a}}{7 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2),x, algorithm="fricas")

[Out]

2/7*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(b*x + a)/b

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Sympy [A]  time = 0.092839, size = 12, normalized size = 0.75 \[ \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2),x)

[Out]

2*(a + b*x)**(7/2)/(7*b)

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GIAC/XCAS [A]  time = 0.217145, size = 116, normalized size = 7.25 \[ \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} + 14 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} a + \frac{15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}}{b^{12}}\right )}}{105 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2),x, algorithm="giac")

[Out]

2/105*(35*(b*x + a)^(3/2)*a^2 + 14*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*a +
(15*(b*x + a)^(7/2)*b^12 - 42*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)^(3/2)*a^2*b
^12)/b^12)/b