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

Optimal. Leaf size=143 \[ -\frac{a^3 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^4 (n+1) x}+\frac{3 a^2 c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^4 (n+2) x}-\frac{3 a c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^4 (n+3) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+4}}{b^4 (n+4) x} \]

[Out]

-((a^3*c^2*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b^4*(1 + n)*x)) + (3*a^2*c^2*Sqrt[c*x
^2]*(a + b*x)^(2 + n))/(b^4*(2 + n)*x) - (3*a*c^2*Sqrt[c*x^2]*(a + b*x)^(3 + n))
/(b^4*(3 + n)*x) + (c^2*Sqrt[c*x^2]*(a + b*x)^(4 + n))/(b^4*(4 + n)*x)

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Rubi [A]  time = 0.107941, antiderivative size = 143, 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 \[ -\frac{a^3 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^4 (n+1) x}+\frac{3 a^2 c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^4 (n+2) x}-\frac{3 a c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^4 (n+3) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+4}}{b^4 (n+4) x} \]

Antiderivative was successfully verified.

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

[Out]

-((a^3*c^2*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b^4*(1 + n)*x)) + (3*a^2*c^2*Sqrt[c*x
^2]*(a + b*x)^(2 + n))/(b^4*(2 + n)*x) - (3*a*c^2*Sqrt[c*x^2]*(a + b*x)^(3 + n))
/(b^4*(3 + n)*x) + (c^2*Sqrt[c*x^2]*(a + b*x)^(4 + n))/(b^4*(4 + n)*x)

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Rubi in Sympy [A]  time = 39.0175, size = 126, normalized size = 0.88 \[ - \frac{a^{3} c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b^{4} x \left (n + 1\right )} + \frac{3 a^{2} c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 2}}{b^{4} x \left (n + 2\right )} - \frac{3 a c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 3}}{b^{4} x \left (n + 3\right )} + \frac{c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 4}}{b^{4} x \left (n + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3*c**2*sqrt(c*x**2)*(a + b*x)**(n + 1)/(b**4*x*(n + 1)) + 3*a**2*c**2*sqrt(c
*x**2)*(a + b*x)**(n + 2)/(b**4*x*(n + 2)) - 3*a*c**2*sqrt(c*x**2)*(a + b*x)**(n
+ 3)/(b**4*x*(n + 3)) + c**2*sqrt(c*x**2)*(a + b*x)**(n + 4)/(b**4*x*(n + 4))

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Mathematica [A]  time = 0.0135622, size = 99, normalized size = 0.69 \[ \frac{c \left (c x^2\right )^{3/2} (a+b x)^{n+1} \left (-6 a^3+6 a^2 b (n+1) x-3 a b^2 \left (n^2+3 n+2\right ) x^2+b^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )}{b^4 (n+1) (n+2) (n+3) (n+4) x^3} \]

Antiderivative was successfully verified.

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

[Out]

(c*(c*x^2)^(3/2)*(a + b*x)^(1 + n)*(-6*a^3 + 6*a^2*b*(1 + n)*x - 3*a*b^2*(2 + 3*
n + n^2)*x^2 + b^3*(6 + 11*n + 6*n^2 + n^3)*x^3))/(b^4*(1 + n)*(2 + n)*(3 + n)*(
4 + n)*x^3)

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Maple [A]  time = 0.008, size = 136, normalized size = 1. \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}{n}^{3}{x}^{3}-6\,{b}^{3}{n}^{2}{x}^{3}+3\,a{b}^{2}{n}^{2}{x}^{2}-11\,{b}^{3}n{x}^{3}+9\,a{b}^{2}n{x}^{2}-6\,{b}^{3}{x}^{3}-6\,{a}^{2}bnx+6\,a{b}^{2}{x}^{2}-6\,{a}^{2}bx+6\,{a}^{3} \right ) }{{x}^{5}{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*x+a)^(1+n)*(c*x^2)^(5/2)*(-b^3*n^3*x^3-6*b^3*n^2*x^3+3*a*b^2*n^2*x^2-11*b^3*
n*x^3+9*a*b^2*n*x^2-6*b^3*x^3-6*a^2*b*n*x+6*a*b^2*x^2-6*a^2*b*x+6*a^3)/x^5/b^4/(
n^4+10*n^3+35*n^2+50*n+24)

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Maxima [A]  time = 1.37417, size = 157, normalized size = 1.1 \[ \frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} c^{\frac{5}{2}} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} c^{\frac{5}{2}} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} c^{\frac{5}{2}} x^{2} + 6 \, a^{3} b c^{\frac{5}{2}} n x - 6 \, a^{4} c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((n^3 + 6*n^2 + 11*n + 6)*b^4*c^(5/2)*x^4 + (n^3 + 3*n^2 + 2*n)*a*b^3*c^(5/2)*x^
3 - 3*(n^2 + n)*a^2*b^2*c^(5/2)*x^2 + 6*a^3*b*c^(5/2)*n*x - 6*a^4*c^(5/2))*(b*x
+ a)^n/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4)

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Fricas [A]  time = 0.22562, size = 251, normalized size = 1.76 \[ \frac{{\left (6 \, a^{3} b c^{2} n x - 6 \, a^{4} c^{2} +{\left (b^{4} c^{2} n^{3} + 6 \, b^{4} c^{2} n^{2} + 11 \, b^{4} c^{2} n + 6 \, b^{4} c^{2}\right )} x^{4} +{\left (a b^{3} c^{2} n^{3} + 3 \, a b^{3} c^{2} n^{2} + 2 \, a b^{3} c^{2} n\right )} x^{3} - 3 \,{\left (a^{2} b^{2} c^{2} n^{2} + a^{2} b^{2} c^{2} n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(6*a^3*b*c^2*n*x - 6*a^4*c^2 + (b^4*c^2*n^3 + 6*b^4*c^2*n^2 + 11*b^4*c^2*n + 6*b
^4*c^2)*x^4 + (a*b^3*c^2*n^3 + 3*a*b^3*c^2*n^2 + 2*a*b^3*c^2*n)*x^3 - 3*(a^2*b^2
*c^2*n^2 + a^2*b^2*c^2*n)*x^2)*sqrt(c*x^2)*(b*x + a)^n/((b^4*n^4 + 10*b^4*n^3 +
35*b^4*n^2 + 50*b^4*n + 24*b^4)*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**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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