3.962 \(\int \frac{x^7 (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=99 \[ \frac{a^2 x (a+b x)^{n+1}}{b^3 c^2 (n+1) \sqrt{c x^2}}-\frac{2 a x (a+b x)^{n+2}}{b^3 c^2 (n+2) \sqrt{c x^2}}+\frac{x (a+b x)^{n+3}}{b^3 c^2 (n+3) \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0781562, antiderivative size = 99, 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 \[ \frac{a^2 x (a+b x)^{n+1}}{b^3 c^2 (n+1) \sqrt{c x^2}}-\frac{2 a x (a+b x)^{n+2}}{b^3 c^2 (n+2) \sqrt{c x^2}}+\frac{x (a+b x)^{n+3}}{b^3 c^2 (n+3) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(a + b*x)^n)/(c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 26.827, size = 92, normalized size = 0.93 \[ \frac{a^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b^{3} c^{3} x \left (n + 1\right )} - \frac{2 a \sqrt{c x^{2}} \left (a + b x\right )^{n + 2}}{b^{3} c^{3} x \left (n + 2\right )} + \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 3}}{b^{3} c^{3} x \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x+a)**n/(c*x**2)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0428646, size = 70, normalized size = 0.71 \[ \frac{x (a+b x)^{n+1} \left (2 a^2-2 a b (n+1) x+b^2 \left (n^2+3 n+2\right ) x^2\right )}{b^3 c^2 (n+1) (n+2) (n+3) \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(a + b*x)^n)/(c*x^2)^(5/2),x]

[Out]

(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*c
^2*(1 + n)*(2 + n)*(3 + n)*Sqrt[c*x^2])

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Maple [A]  time = 0.009, 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(x^7*(b*x+a)^n/(c*x^2)^(5/2),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)*x^5/(c
*x^2)^(5/2)/b^3/(n^3+6*n^2+11*n+6)

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Maxima [A]  time = 1.35746, size = 112, normalized size = 1.13 \[ \frac{{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} \sqrt{c} x^{3} +{\left (n^{2} + n\right )} a b^{2} \sqrt{c} x^{2} - 2 \, a^{2} b \sqrt{c} n x + 2 \, a^{3} \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^7/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.235441, size = 159, normalized size = 1.61 \[ -\frac{{\left (2 \, a^{2} b n x -{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3} - 2 \, a^{3} -{\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{3} c^{3} n^{3} + 6 \, b^{3} c^{3} n^{2} + 11 \, b^{3} c^{3} n + 6 \, b^{3} c^{3}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^7/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

-(2*a^2*b*n*x - (b^3*n^2 + 3*b^3*n + 2*b^3)*x^3 - 2*a^3 - (a*b^2*n^2 + a*b^2*n)*
x^2)*sqrt(c*x^2)*(b*x + a)^n/((b^3*c^3*n^3 + 6*b^3*c^3*n^2 + 11*b^3*c^3*n + 6*b^
3*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**7*(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. \[ \int \frac{{\left (b x + a\right )}^{n} x^{7}}{\left (c x^{2}\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^7/(c*x^2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x^7/(c*x^2)^(5/2), x)