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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ \frac {a^2 x (a+b x)^{n+1}}{b^3 c (n+1) \sqrt {c x^2}}-\frac {2 a x (a+b x)^{n+2}}{b^3 c (n+2) \sqrt {c x^2}}+\frac {x (a+b x)^{n+3}}{b^3 c (n+3) \sqrt {c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {x^5 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int x^2 (a+b x)^n \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2 (a+b x)^n}{b^2}-\frac {2 a (a+b x)^{1+n}}{b^2}+\frac {(a+b x)^{2+n}}{b^2}\right ) \, dx}{c \sqrt {c x^2}}\\ &=\frac {a^2 x (a+b x)^{1+n}}{b^3 c (1+n) \sqrt {c x^2}}-\frac {2 a x (a+b x)^{2+n}}{b^3 c (2+n) \sqrt {c x^2}}+\frac {x (a+b x)^{3+n}}{b^3 c (3+n) \sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 69, normalized size = 0.70 \[ \frac {x^3 (a+b x)^{n+1} \left (2 a^2-2 a b (n+1) x+b^2 \left (n^2+3 n+2\right ) x^2\right )}{b^3 (n+1) (n+2) (n+3) \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(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)*(c*x^
2)^(3/2))

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fricas [A]  time = 0.44, size = 118, normalized size = 1.19 \[ -\frac {{\left (2 \, a^{2} b n x - {\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{3} - 2 \, a^{3} - {\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{3} c^{2} n^{3} + 6 \, b^{3} c^{2} n^{2} + 11 \, b^{3} c^{2} n + 6 \, b^{3} c^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n} x^{5}}{\left (c x^{2}\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 83, normalized size = 0.84 \[ \frac {\left (b^{2} n^{2} x^{2}+3 b^{2} n \,x^{2}-2 a b n x +2 b^{2} x^{2}-2 a b x +2 a^{2}\right ) x^{3} \left (b x +a \right )^{n +1}}{\left (c \,x^{2}\right )^{\frac {3}{2}} \left (n^{3}+6 n^{2}+11 n +6\right ) b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.46, size = 83, normalized size = 0.84 \[ \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} \sqrt {c} x^{3} + {\left (n^{2} + n\right )} a b^{2} \sqrt {c} x^{2} - 2 \, a^{2} b \sqrt {c} n x + 2 \, a^{3} \sqrt {c}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 0.31, size = 133, normalized size = 1.34 \[ \frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^4\,\left (n^2+3\,n+2\right )}{c\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {2\,a^3\,x}{b^3\,c\,\left (n^3+6\,n^2+11\,n+6\right )}-\frac {2\,a^2\,n\,x^2}{b^2\,c\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {a\,n\,x^3\,\left (n+1\right )}{b\,c\,\left (n^3+6\,n^2+11\,n+6\right )}\right )}{\sqrt {c\,x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {a^{n} x^{6}}{3 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\\int \frac {x^{5}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{3}}\, dx & \text {for}\: n = -3 \\\int \frac {x^{5}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {x^{5}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\\frac {2 a^{3} x^{3} \left (a + b x\right )^{n}}{b^{3} c^{\frac {3}{2}} n^{3} \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 11 b^{3} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} - \frac {2 a^{2} b n x^{4} \left (a + b x\right )^{n}}{b^{3} c^{\frac {3}{2}} n^{3} \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 11 b^{3} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {a b^{2} n^{2} x^{5} \left (a + b x\right )^{n}}{b^{3} c^{\frac {3}{2}} n^{3} \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 11 b^{3} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {a b^{2} n x^{5} \left (a + b x\right )^{n}}{b^{3} c^{\frac {3}{2}} n^{3} \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 11 b^{3} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {b^{3} n^{2} x^{6} \left (a + b x\right )^{n}}{b^{3} c^{\frac {3}{2}} n^{3} \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 11 b^{3} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {3 b^{3} n x^{6} \left (a + b x\right )^{n}}{b^{3} c^{\frac {3}{2}} n^{3} \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 11 b^{3} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} + \frac {2 b^{3} x^{6} \left (a + b x\right )^{n}}{b^{3} c^{\frac {3}{2}} n^{3} \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} n^{2} \left (x^{2}\right )^{\frac {3}{2}} + 11 b^{3} c^{\frac {3}{2}} n \left (x^{2}\right )^{\frac {3}{2}} + 6 b^{3} c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**n*x**6/(3*c**(3/2)*(x**2)**(3/2)), Eq(b, 0)), (Integral(x**5/((c*x**2)**(3/2)*(a + b*x)**3), x),
Eq(n, -3)), (Integral(x**5/((c*x**2)**(3/2)*(a + b*x)**2), x), Eq(n, -2)), (Integral(x**5/((c*x**2)**(3/2)*(a
+ b*x)), x), Eq(n, -1)), (2*a**3*x**3*(a + b*x)**n/(b**3*c**(3/2)*n**3*(x**2)**(3/2) + 6*b**3*c**(3/2)*n**2*(
x**2)**(3/2) + 11*b**3*c**(3/2)*n*(x**2)**(3/2) + 6*b**3*c**(3/2)*(x**2)**(3/2)) - 2*a**2*b*n*x**4*(a + b*x)**
n/(b**3*c**(3/2)*n**3*(x**2)**(3/2) + 6*b**3*c**(3/2)*n**2*(x**2)**(3/2) + 11*b**3*c**(3/2)*n*(x**2)**(3/2) +
6*b**3*c**(3/2)*(x**2)**(3/2)) + a*b**2*n**2*x**5*(a + b*x)**n/(b**3*c**(3/2)*n**3*(x**2)**(3/2) + 6*b**3*c**(
3/2)*n**2*(x**2)**(3/2) + 11*b**3*c**(3/2)*n*(x**2)**(3/2) + 6*b**3*c**(3/2)*(x**2)**(3/2)) + a*b**2*n*x**5*(a
+ b*x)**n/(b**3*c**(3/2)*n**3*(x**2)**(3/2) + 6*b**3*c**(3/2)*n**2*(x**2)**(3/2) + 11*b**3*c**(3/2)*n*(x**2)*
*(3/2) + 6*b**3*c**(3/2)*(x**2)**(3/2)) + b**3*n**2*x**6*(a + b*x)**n/(b**3*c**(3/2)*n**3*(x**2)**(3/2) + 6*b*
*3*c**(3/2)*n**2*(x**2)**(3/2) + 11*b**3*c**(3/2)*n*(x**2)**(3/2) + 6*b**3*c**(3/2)*(x**2)**(3/2)) + 3*b**3*n*
x**6*(a + b*x)**n/(b**3*c**(3/2)*n**3*(x**2)**(3/2) + 6*b**3*c**(3/2)*n**2*(x**2)**(3/2) + 11*b**3*c**(3/2)*n*
(x**2)**(3/2) + 6*b**3*c**(3/2)*(x**2)**(3/2)) + 2*b**3*x**6*(a + b*x)**n/(b**3*c**(3/2)*n**3*(x**2)**(3/2) +
6*b**3*c**(3/2)*n**2*(x**2)**(3/2) + 11*b**3*c**(3/2)*n*(x**2)**(3/2) + 6*b**3*c**(3/2)*(x**2)**(3/2)), True))

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