3.10.53 \(\int \frac {x^6 (a+b x)^n}{(c x^2)^{3/2}} \, dx\) [953]

Optimal. Leaf size=135 \[ -\frac {a^3 x (a+b x)^{1+n}}{b^4 c (1+n) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{2+n}}{b^4 c (2+n) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{3+n}}{b^4 c (3+n) \sqrt {c x^2}}+\frac {x (a+b x)^{4+n}}{b^4 c (4+n) \sqrt {c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 135, 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, 45} \begin {gather*} -\frac {a^3 x (a+b x)^{n+1}}{b^4 c (n+1) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{n+2}}{b^4 c (n+2) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{n+3}}{b^4 c (n+3) \sqrt {c x^2}}+\frac {x (a+b x)^{n+4}}{b^4 c (n+4) \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^6*(a + b*x)^n)/(c*x^2)^(3/2),x]

[Out]

-((a^3*x*(a + b*x)^(1 + n))/(b^4*c*(1 + n)*Sqrt[c*x^2])) + (3*a^2*x*(a + b*x)^(2 + n))/(b^4*c*(2 + n)*Sqrt[c*x
^2]) - (3*a*x*(a + b*x)^(3 + n))/(b^4*c*(3 + n)*Sqrt[c*x^2]) + (x*(a + b*x)^(4 + n))/(b^4*c*(4 + 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 45

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^6 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int x^3 (a+b x)^n \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (-\frac {a^3 (a+b x)^n}{b^3}+\frac {3 a^2 (a+b x)^{1+n}}{b^3}-\frac {3 a (a+b x)^{2+n}}{b^3}+\frac {(a+b x)^{3+n}}{b^3}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a^3 x (a+b x)^{1+n}}{b^4 c (1+n) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{2+n}}{b^4 c (2+n) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{3+n}}{b^4 c (3+n) \sqrt {c x^2}}+\frac {x (a+b x)^{4+n}}{b^4 c (4+n) \sqrt {c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 98, normalized size = 0.73 \begin {gather*} \frac {x^3 (a+b x)^{1+n} \left (-6 a^3+6 a^2 b (1+n) x-3 a b^2 \left (2+3 n+n^2\right ) x^2+b^3 \left (6+11 n+6 n^2+n^3\right ) x^3\right )}{b^4 (1+n) (2+n) (3+n) (4+n) \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(a + b*x)^n)/(c*x^2)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 136, normalized size = 1.01

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{1+n} x^{3} \left (-b^{3} n^{3} x^{3}-6 b^{3} n^{2} x^{3}+3 a \,b^{2} n^{2} x^{2}-11 b^{3} n \,x^{3}+9 a \,b^{2} n \,x^{2}-6 b^{3} x^{3}-6 a^{2} b n x +6 a \,b^{2} x^{2}-6 a^{2} b x +6 a^{3}\right )}{\left (c \,x^{2}\right )^{\frac {3}{2}} b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\)	\(136\)
risch	\(-\frac {x \left (-b^{4} n^{3} x^{4}-a \,b^{3} n^{3} x^{3}-6 b^{4} n^{2} x^{4}-3 a \,b^{3} n^{2} x^{3}-11 b^{4} n \,x^{4}+3 a^{2} b^{2} n^{2} x^{2}-2 x^{3} a n \,b^{3}-6 b^{4} x^{4}+3 a^{2} n \,x^{2} b^{2}-6 a^{3} b n x +6 a^{4}\right ) \left (b x +a \right )^{n}}{c \sqrt {c \,x^{2}}\, \left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x+a)^n/(c*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 104, normalized size = 0.77 \begin {gather*} \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4} c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^n/(c*x^2)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.73, size = 168, normalized size = 1.24 \begin {gather*} \frac {{\left (6 \, a^{3} b n x + {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 6 \, a^{4} + {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 3 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{4} c^{2} n^{4} + 10 \, b^{4} c^{2} n^{3} + 35 \, b^{4} c^{2} n^{2} + 50 \, b^{4} c^{2} n + 24 \, b^{4} c^{2}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^n/(c*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {a^{n} x^{7}}{4 \left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\\int \frac {x^{6}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{4}}\, dx & \text {for}\: n = -4 \\\int \frac {x^{6}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{3}}\, dx & \text {for}\: n = -3 \\\int \frac {x^{6}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {x^{6}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\- \frac {6 a^{4} x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {6 a^{3} b n x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {3 a^{2} b^{2} n^{2} x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {3 a^{2} b^{2} n x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {a b^{3} n^{3} x^{6} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {3 a b^{3} n^{2} x^{6} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {2 a b^{3} n x^{6} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b^{4} n^{3} x^{7} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {6 b^{4} n^{2} x^{7} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {11 b^{4} n x^{7} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {6 b^{4} x^{7} \left (a + b x\right )^{n}}{b^{4} n^{4} \left (c x^{2}\right )^{\frac {3}{2}} + 10 b^{4} n^{3} \left (c x^{2}\right )^{\frac {3}{2}} + 35 b^{4} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 50 b^{4} n \left (c x^{2}\right )^{\frac {3}{2}} + 24 b^{4} \left (c x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**n*x**7/(4*(c*x**2)**(3/2)), Eq(b, 0)), (Integral(x**6/((c*x**2)**(3/2)*(a + b*x)**4), x), Eq(n,
-4)), (Integral(x**6/((c*x**2)**(3/2)*(a + b*x)**3), x), Eq(n, -3)), (Integral(x**6/((c*x**2)**(3/2)*(a + b*x)
**2), x), Eq(n, -2)), (Integral(x**6/((c*x**2)**(3/2)*(a + b*x)), x), Eq(n, -1)), (-6*a**4*x**3*(a + b*x)**n/(
b**4*n**4*(c*x**2)**(3/2) + 10*b**4*n**3*(c*x**2)**(3/2) + 35*b**4*n**2*(c*x**2)**(3/2) + 50*b**4*n*(c*x**2)**
(3/2) + 24*b**4*(c*x**2)**(3/2)) + 6*a**3*b*n*x**4*(a + b*x)**n/(b**4*n**4*(c*x**2)**(3/2) + 10*b**4*n**3*(c*x
**2)**(3/2) + 35*b**4*n**2*(c*x**2)**(3/2) + 50*b**4*n*(c*x**2)**(3/2) + 24*b**4*(c*x**2)**(3/2)) - 3*a**2*b**
2*n**2*x**5*(a + b*x)**n/(b**4*n**4*(c*x**2)**(3/2) + 10*b**4*n**3*(c*x**2)**(3/2) + 35*b**4*n**2*(c*x**2)**(3
/2) + 50*b**4*n*(c*x**2)**(3/2) + 24*b**4*(c*x**2)**(3/2)) - 3*a**2*b**2*n*x**5*(a + b*x)**n/(b**4*n**4*(c*x**
2)**(3/2) + 10*b**4*n**3*(c*x**2)**(3/2) + 35*b**4*n**2*(c*x**2)**(3/2) + 50*b**4*n*(c*x**2)**(3/2) + 24*b**4*
(c*x**2)**(3/2)) + a*b**3*n**3*x**6*(a + b*x)**n/(b**4*n**4*(c*x**2)**(3/2) + 10*b**4*n**3*(c*x**2)**(3/2) + 3
5*b**4*n**2*(c*x**2)**(3/2) + 50*b**4*n*(c*x**2)**(3/2) + 24*b**4*(c*x**2)**(3/2)) + 3*a*b**3*n**2*x**6*(a + b
*x)**n/(b**4*n**4*(c*x**2)**(3/2) + 10*b**4*n**3*(c*x**2)**(3/2) + 35*b**4*n**2*(c*x**2)**(3/2) + 50*b**4*n*(c
*x**2)**(3/2) + 24*b**4*(c*x**2)**(3/2)) + 2*a*b**3*n*x**6*(a + b*x)**n/(b**4*n**4*(c*x**2)**(3/2) + 10*b**4*n
**3*(c*x**2)**(3/2) + 35*b**4*n**2*(c*x**2)**(3/2) + 50*b**4*n*(c*x**2)**(3/2) + 24*b**4*(c*x**2)**(3/2)) + b*
*4*n**3*x**7*(a + b*x)**n/(b**4*n**4*(c*x**2)**(3/2) + 10*b**4*n**3*(c*x**2)**(3/2) + 35*b**4*n**2*(c*x**2)**(
3/2) + 50*b**4*n*(c*x**2)**(3/2) + 24*b**4*(c*x**2)**(3/2)) + 6*b**4*n**2*x**7*(a + b*x)**n/(b**4*n**4*(c*x**2
)**(3/2) + 10*b**4*n**3*(c*x**2)**(3/2) + 35*b**4*n**2*(c*x**2)**(3/2) + 50*b**4*n*(c*x**2)**(3/2) + 24*b**4*(
c*x**2)**(3/2)) + 11*b**4*n*x**7*(a + b*x)**n/(b**4*n**4*(c*x**2)**(3/2) + 10*b**4*n**3*(c*x**2)**(3/2) + 35*b
**4*n**2*(c*x**2)**(3/2) + 50*b**4*n*(c*x**2)**(3/2) + 24*b**4*(c*x**2)**(3/2)) + 6*b**4*x**7*(a + b*x)**n/(b*
*4*n**4*(c*x**2)**(3/2) + 10*b**4*n**3*(c*x**2)**(3/2) + 35*b**4*n**2*(c*x**2)**(3/2) + 50*b**4*n*(c*x**2)**(3
/2) + 24*b**4*(c*x**2)**(3/2)), True))

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^n/(c*x^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

________________________________________________________________________________________

Mupad [B]
time = 0.40, size = 201, normalized size = 1.49 \begin {gather*} \frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^5\,\left (n^3+6\,n^2+11\,n+6\right )}{c\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {6\,a^4\,x}{b^4\,c\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,a^3\,n\,x^2}{b^3\,c\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x^4\,\left (n^2+3\,n+2\right )}{b\,c\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {3\,a^2\,n\,x^3\,\left (n+1\right )}{b^2\,c\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right )}{\sqrt {c\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6*(a + b*x)^n)/(c*x^2)^(3/2),x)

[Out]

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

________________________________________________________________________________________