3.1449 \(\int \frac{(a+b x)^5}{(a c+b c x)^{7/2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{5/2}}{5 b c^6} \]

[Out]

(2*(a*c + b*c*x)^(5/2))/(5*b*c^6)

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Rubi [A]  time = 0.0133471, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 (a c+b c x)^{5/2}}{5 b c^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(a*c + b*c*x)^(7/2),x]

[Out]

(2*(a*c + b*c*x)^(5/2))/(5*b*c^6)

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Rubi in Sympy [A]  time = 4.37663, size = 19, normalized size = 0.86 \[ \frac{2 \left (a c + b c x\right )^{\frac{5}{2}}}{5 b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(b*c*x+a*c)**(7/2),x)

[Out]

2*(a*c + b*c*x)**(5/2)/(5*b*c**6)

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Mathematica [A]  time = 0.0194937, size = 25, normalized size = 1.14 \[ \frac{2 (a+b x)^6}{5 b (c (a+b x))^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(a*c + b*c*x)^(7/2),x]

[Out]

(2*(a + b*x)^6)/(5*b*(c*(a + b*x))^(7/2))

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Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \[{\frac{2\, \left ( bx+a \right ) ^{6}}{5\,b} \left ( bcx+ac \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(b*c*x+a*c)^(7/2),x)

[Out]

2/5*(b*x+a)^6/b/(b*c*x+a*c)^(7/2)

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Maxima [A]  time = 1.32904, size = 24, normalized size = 1.09 \[ \frac{2 \,{\left (b c x + a c\right )}^{\frac{5}{2}}}{5 \, b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^(7/2),x, algorithm="maxima")

[Out]

2/5*(b*c*x + a*c)^(5/2)/(b*c^6)

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Fricas [A]  time = 0.210368, size = 46, normalized size = 2.09 \[ \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b c x + a c}}{5 \, b c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^(7/2),x, algorithm="fricas")

[Out]

2/5*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*c*x + a*c)/(b*c^4)

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Sympy [A]  time = 5.38577, size = 80, normalized size = 3.64 \[ \begin{cases} \frac{2 a^{2} \sqrt{a c + b c x}}{5 b c^{4}} + \frac{4 a x \sqrt{a c + b c x}}{5 c^{4}} + \frac{2 b x^{2} \sqrt{a c + b c x}}{5 c^{4}} & \text{for}\: b \neq 0 \\\frac{a^{5} x}{\left (a c\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(b*c*x+a*c)**(7/2),x)

[Out]

Piecewise((2*a**2*sqrt(a*c + b*c*x)/(5*b*c**4) + 4*a*x*sqrt(a*c + b*c*x)/(5*c**4
) + 2*b*x**2*sqrt(a*c + b*c*x)/(5*c**4), Ne(b, 0)), (a**5*x/(a*c)**(7/2), True))

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GIAC/XCAS [A]  time = 0.215109, size = 166, normalized size = 7.55 \[ \frac{2 \,{\left (15 \, \sqrt{b c x + a c} a^{2} - \frac{10 \,{\left (3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}\right )} a}{c} + \frac{15 \, \sqrt{b c x + a c} a^{2} b^{8} c^{10} - 10 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a b^{8} c^{9} + 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}} b^{8} c^{8}}{b^{8} c^{10}}\right )}}{15 \, b c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^(7/2),x, algorithm="giac")

[Out]

2/15*(15*sqrt(b*c*x + a*c)*a^2 - 10*(3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/
2))*a/c + (15*sqrt(b*c*x + a*c)*a^2*b^8*c^10 - 10*(b*c*x + a*c)^(3/2)*a*b^8*c^9
+ 3*(b*c*x + a*c)^(5/2)*b^8*c^8)/(b^8*c^10))/(b*c^4)