3.955 \(\int \frac{x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0495366, 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 \[ \frac{x (a+b x)^{n+2}}{b^2 c (n+2) \sqrt{c x^2}}-\frac{a x (a+b x)^{n+1}}{b^2 c (n+1) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(a + b*x)^n)/(c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.4463, size = 58, normalized size = 0.89 \[ - \frac{a \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b^{2} c^{2} x \left (n + 1\right )} + \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 2}}{b^{2} c^{2} x \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x+a)**n/(c*x**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0373625, size = 45, normalized size = 0.69 \[ \frac{x^3 (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(a + b*x)^n)/(c*x^2)^(3/2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x+a)^n/(c*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.36547, size = 61, normalized size = 0.94 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^4/(c*x^2)^(3/2),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^(3/2))

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Fricas [A]  time = 0.241518, size = 97, normalized size = 1.49 \[ \frac{{\left (a b n x +{\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{2} c^{2} n^{2} + 3 \, b^{2} c^{2} n + 2 \, b^{2} c^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^4/(c*x^2)^(3/2),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^2*n^2 + 3*b^
2*c^2*n + 2*b^2*c^2)*x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x+a)**n/(c*x**2)**(3/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n} x^{4}}{\left (c x^{2}\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^4/(c*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x^4/(c*x^2)^(3/2), x)