3.941 \(\int \frac{\left (c x^2\right )^{5/2} (a+b x)^n}{x^3} \, dx\)

Optimal. Leaf size=105 \[ \frac{a^2 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^3 (n+1) x}-\frac{2 a c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^3 (n+2) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^3 (n+3) x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0785405, antiderivative size = 105, 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 \[ \frac{a^2 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^3 (n+1) x}-\frac{2 a c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^3 (n+2) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^3 (n+3) x} \]

Antiderivative was successfully verified.

[In]  Int[((c*x^2)^(5/2)*(a + b*x)^n)/x^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 30.3439, size = 92, normalized size = 0.88 \[ \frac{a^{2} c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b^{3} x \left (n + 1\right )} - \frac{2 a c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 2}}{b^{3} x \left (n + 2\right )} + \frac{c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 3}}{b^{3} x \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(5/2)*(b*x+a)**n/x**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0719232, size = 70, normalized size = 0.67 \[ \frac{c^3 x (a+b x)^{n+1} \left (2 a^2-2 a b (n+1) x+b^2 \left (n^2+3 n+2\right ) x^2\right )}{b^3 (n+1) (n+2) (n+3) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((c*x^2)^(5/2)*(a + b*x)^n)/x^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 83, normalized size = 0.8 \[{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{2}{n}^{2}{x}^{2}+3\,{b}^{2}n{x}^{2}-2\,abnx+2\,{b}^{2}{x}^{2}-2\,abx+2\,{a}^{2} \right ) }{{x}^{5}{b}^{3} \left ({n}^{3}+6\,{n}^{2}+11\,n+6 \right ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(5/2)*(b*x+a)^n/x^3,x)

[Out]

(b*x+a)^(1+n)*(b^2*n^2*x^2+3*b^2*n*x^2-2*a*b*n*x+2*b^2*x^2-2*a*b*x+2*a^2)*(c*x^2
)^(5/2)/x^5/b^3/(n^3+6*n^2+11*n+6)

_______________________________________________________________________________________

Maxima [A]  time = 1.35028, size = 108, normalized size = 1.03 \[ \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} c^{\frac{5}{2}} x^{3} +{\left (n^{2} + n\right )} a b^{2} c^{\frac{5}{2}} x^{2} - 2 \, a^{2} b c^{\frac{5}{2}} n x + 2 \, a^{3} c^{\frac{5}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)^n/x^3,x, algorithm="maxima")

[Out]

((n^2 + 3*n + 2)*b^3*c^(5/2)*x^3 + (n^2 + n)*a*b^2*c^(5/2)*x^2 - 2*a^2*b*c^(5/2)
*n*x + 2*a^3*c^(5/2))*(b*x + a)^n/((n^3 + 6*n^2 + 11*n + 6)*b^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.228433, size = 171, normalized size = 1.63 \[ -\frac{{\left (2 \, a^{2} b c^{2} n x - 2 \, a^{3} c^{2} -{\left (b^{3} c^{2} n^{2} + 3 \, b^{3} c^{2} n + 2 \, b^{3} c^{2}\right )} x^{3} -{\left (a b^{2} c^{2} n^{2} + a b^{2} c^{2} n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{3} n^{3} + 6 \, b^{3} n^{2} + 11 \, b^{3} n + 6 \, b^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)^n/x^3,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{5}{2}}{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)^n/x^3,x, algorithm="giac")

[Out]

integrate((c*x^2)^(5/2)*(b*x + a)^n/x^3, x)