3.963 \(\int \frac{x^6 (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=65 \[ \frac{x (a+b x)^{n+2}}{b^2 c^2 (n+2) \sqrt{c x^2}}-\frac{a x (a+b x)^{n+1}}{b^2 c^2 (n+1) \sqrt{c x^2}} \]

[Out]

-((a*x*(a + b*x)^(1 + n))/(b^2*c^2*(1 + n)*Sqrt[c*x^2])) + (x*(a + b*x)^(2 + n))
/(b^2*c^2*(2 + n)*Sqrt[c*x^2])

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Rubi [A]  time = 0.0505711, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x (a+b x)^{n+2}}{b^2 c^2 (n+2) \sqrt{c x^2}}-\frac{a x (a+b x)^{n+1}}{b^2 c^2 (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(a + b*x)^n)/(c*x^2)^(5/2),x]

[Out]

-((a*x*(a + b*x)^(1 + n))/(b^2*c^2*(1 + n)*Sqrt[c*x^2])) + (x*(a + b*x)^(2 + n))
/(b^2*c^2*(2 + n)*Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 20.2408, size = 58, normalized size = 0.89 \[ - \frac{a \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b^{2} c^{3} x \left (n + 1\right )} + \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 2}}{b^{2} c^{3} x \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(b*x+a)**n/(c*x**2)**(5/2),x)

[Out]

-a*sqrt(c*x**2)*(a + b*x)**(n + 1)/(b**2*c**3*x*(n + 1)) + sqrt(c*x**2)*(a + b*x
)**(n + 2)/(b**2*c**3*x*(n + 2))

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Mathematica [A]  time = 0.0348481, size = 46, normalized size = 0.71 \[ \frac{x (a+b x)^{n+1} (b (n+1) x-a)}{b^2 c^2 (n+1) (n+2) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(a + b*x)^n)/(c*x^2)^(5/2),x]

[Out]

(x*(a + b*x)^(1 + n)*(-a + b*(1 + n)*x))/(b^2*c^2*(1 + n)*(2 + n)*Sqrt[c*x^2])

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Maple [A]  time = 0.004, size = 46, normalized size = 0.7 \[ -{\frac{ \left ( bx+a \right ) ^{1+n}{x}^{5} \left ( -bxn-bx+a \right ) }{{b}^{2} \left ({n}^{2}+3\,n+2 \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x+a)^n/(c*x^2)^(5/2),x)

[Out]

-(b*x+a)^(1+n)*x^5*(-b*n*x-b*x+a)/(c*x^2)^(5/2)/b^2/(n^2+3*n+2)

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Maxima [A]  time = 1.3585, size = 61, normalized size = 0.94 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^6/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n/((n^2 + 3*n + 2)*b^2*c^(5/2))

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Fricas [A]  time = 0.245831, size = 97, normalized size = 1.49 \[ \frac{{\left (a b n x +{\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{2} c^{3} n^{2} + 3 \, b^{2} c^{3} n + 2 \, b^{2} c^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^6/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

(a*b*n*x + (b^2*n + b^2)*x^2 - a^2)*sqrt(c*x^2)*(b*x + a)^n/((b^2*c^3*n^2 + 3*b^
2*c^3*n + 2*b^2*c^3)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(b*x+a)**n/(c*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n} x^{6}}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^6/(c*x^2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x^6/(c*x^2)^(5/2), x)