3.942 \(\int \frac{\left (c x^2\right )^{5/2} (a+b x)^n}{x^4} \, dx\)

Optimal. Leaf size=69 \[ \frac{c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac{a c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]

[Out]

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

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Rubi [A]  time = 0.0500284, antiderivative size = 69, 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 \[ \frac{c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac{a c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]

Antiderivative was successfully verified.

[In]  Int[((c*x^2)^(5/2)*(a + b*x)^n)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 22.2113, size = 58, normalized size = 0.84 \[ - \frac{a c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b^{2} x \left (n + 1\right )} + \frac{c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 2}}{b^{2} x \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(5/2)*(b*x+a)**n/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0394769, size = 46, normalized size = 0.67 \[ \frac{c^3 x (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((c*x^2)^(5/2)*(a + b*x)^n)/x^4,x]

[Out]

(c^3*x*(a + b*x)^(1 + n)*(-a + b*(1 + n)*x))/(b^2*(1 + n)*(2 + n)*Sqrt[c*x^2])

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Maple [A]  time = 0.003, size = 46, normalized size = 0.7 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -bxn-bx+a \right ) }{{x}^{5}{b}^{2} \left ({n}^{2}+3\,n+2 \right ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(5/2)*(b*x+a)^n/x^4,x)

[Out]

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

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Maxima [A]  time = 1.34875, size = 69, normalized size = 1. \[ \frac{{\left (b^{2} c^{\frac{5}{2}}{\left (n + 1\right )} x^{2} + a b c^{\frac{5}{2}} n x - a^{2} c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)^n/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.229163, size = 103, normalized size = 1.49 \[ \frac{{\left (a b c^{2} n x - a^{2} c^{2} +{\left (b^{2} c^{2} n + b^{2} c^{2}\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)^n/x^4,x, algorithm="fricas")

[Out]

(a*b*c^2*n*x - a^2*c^2 + (b^2*c^2*n + b^2*c^2)*x^2)*sqrt(c*x^2)*(b*x + a)^n/((b^
2*n^2 + 3*b^2*n + 2*b^2)*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((c*x**2)**(5/2)*(b*x+a)**n/x**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{5}{2}}{\left (b x + a\right )}^{n}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)^n/x^4,x, algorithm="giac")

[Out]

integrate((c*x^2)^(5/2)*(b*x + a)^n/x^4, x)