3.10.38 \(\int (c x^2)^{5/2} (a+b x)^n \, dx\) [938]

Optimal. Leaf size=217 \[ -\frac {a^5 c^2 \sqrt {c x^2} (a+b x)^{1+n}}{b^6 (1+n) x}+\frac {5 a^4 c^2 \sqrt {c x^2} (a+b x)^{2+n}}{b^6 (2+n) x}-\frac {10 a^3 c^2 \sqrt {c x^2} (a+b x)^{3+n}}{b^6 (3+n) x}+\frac {10 a^2 c^2 \sqrt {c x^2} (a+b x)^{4+n}}{b^6 (4+n) x}-\frac {5 a c^2 \sqrt {c x^2} (a+b x)^{5+n}}{b^6 (5+n) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{6+n}}{b^6 (6+n) x} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 45} \begin {gather*} -\frac {a^5 c^2 \sqrt {c x^2} (a+b x)^{n+1}}{b^6 (n+1) x}+\frac {5 a^4 c^2 \sqrt {c x^2} (a+b x)^{n+2}}{b^6 (n+2) x}-\frac {10 a^3 c^2 \sqrt {c x^2} (a+b x)^{n+3}}{b^6 (n+3) x}+\frac {10 a^2 c^2 \sqrt {c x^2} (a+b x)^{n+4}}{b^6 (n+4) x}-\frac {5 a c^2 \sqrt {c x^2} (a+b x)^{n+5}}{b^6 (n+5) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{n+6}}{b^6 (n+6) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(5/2)*(a + b*x)^n,x]

[Out]

-((a^5*c^2*Sqrt[c*x^2]*(a + b*x)^(1 + n))/(b^6*(1 + n)*x)) + (5*a^4*c^2*Sqrt[c*x^2]*(a + b*x)^(2 + n))/(b^6*(2
+ n)*x) - (10*a^3*c^2*Sqrt[c*x^2]*(a + b*x)^(3 + n))/(b^6*(3 + n)*x) + (10*a^2*c^2*Sqrt[c*x^2]*(a + b*x)^(4 +
n))/(b^6*(4 + n)*x) - (5*a*c^2*Sqrt[c*x^2]*(a + b*x)^(5 + n))/(b^6*(5 + n)*x) + (c^2*Sqrt[c*x^2]*(a + b*x)^(6
+ n))/(b^6*(6 + n)*x)

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 \left (c x^2\right )^{5/2} (a+b x)^n \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^5 (a+b x)^n \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (-\frac {a^5 (a+b x)^n}{b^5}+\frac {5 a^4 (a+b x)^{1+n}}{b^5}-\frac {10 a^3 (a+b x)^{2+n}}{b^5}+\frac {10 a^2 (a+b x)^{3+n}}{b^5}-\frac {5 a (a+b x)^{4+n}}{b^5}+\frac {(a+b x)^{5+n}}{b^5}\right ) \, dx}{x}\\ &=-\frac {a^5 c^2 \sqrt {c x^2} (a+b x)^{1+n}}{b^6 (1+n) x}+\frac {5 a^4 c^2 \sqrt {c x^2} (a+b x)^{2+n}}{b^6 (2+n) x}-\frac {10 a^3 c^2 \sqrt {c x^2} (a+b x)^{3+n}}{b^6 (3+n) x}+\frac {10 a^2 c^2 \sqrt {c x^2} (a+b x)^{4+n}}{b^6 (4+n) x}-\frac {5 a c^2 \sqrt {c x^2} (a+b x)^{5+n}}{b^6 (5+n) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{6+n}}{b^6 (6+n) x}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 172, normalized size = 0.79 \begin {gather*} \frac {c^3 x (a+b x)^{1+n} \left (-120 a^5+120 a^4 b (1+n) x-60 a^3 b^2 \left (2+3 n+n^2\right ) x^2+20 a^2 b^3 \left (6+11 n+6 n^2+n^3\right ) x^3-5 a b^4 \left (24+50 n+35 n^2+10 n^3+n^4\right ) x^4+b^5 \left (120+274 n+225 n^2+85 n^3+15 n^4+n^5\right ) x^5\right )}{b^6 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n) \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*x*(a + b*x)^(1 + n)*(-120*a^5 + 120*a^4*b*(1 + n)*x - 60*a^3*b^2*(2 + 3*n + n^2)*x^2 + 20*a^2*b^3*(6 + 11
*n + 6*n^2 + n^3)*x^3 - 5*a*b^4*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)*x^4 + b^5*(120 + 274*n + 225*n^2 + 85*n^3
+ 15*n^4 + n^5)*x^5))/(b^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*Sqrt[c*x^2])

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Maple [A]
time = 0.14, size = 280, normalized size = 1.29

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{1+n} \left (c \,x^{2}\right )^{\frac {5}{2}} \left (-b^{5} n^{5} x^{5}-15 b^{5} n^{4} x^{5}+5 a \,b^{4} n^{4} x^{4}-85 b^{5} n^{3} x^{5}+50 a \,b^{4} n^{3} x^{4}-225 b^{5} n^{2} x^{5}-20 a^{2} b^{3} n^{3} x^{3}+175 a \,b^{4} n^{2} x^{4}-274 b^{5} n \,x^{5}-120 a^{2} b^{3} n^{2} x^{3}+250 x^{4} a n \,b^{4}-120 b^{5} x^{5}+60 a^{3} b^{2} n^{2} x^{2}-220 a^{2} n \,x^{3} b^{3}+120 a \,b^{4} x^{4}+180 a^{3} n \,x^{2} b^{2}-120 a^{2} b^{3} x^{3}-120 a^{4} b n x +120 a^{3} b^{2} x^{2}-120 a^{4} b x +120 a^{5}\right )}{x^{5} b^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}\)	\(280\)
risch	\(-\frac {c^{2} \sqrt {c \,x^{2}}\, \left (-b^{6} n^{5} x^{6}-a \,b^{5} n^{5} x^{5}-15 b^{6} n^{4} x^{6}-10 a \,b^{5} n^{4} x^{5}-85 b^{6} n^{3} x^{6}+5 a^{2} b^{4} n^{4} x^{4}-35 a \,b^{5} n^{3} x^{5}-225 b^{6} n^{2} x^{6}+30 a^{2} b^{4} n^{3} x^{4}-50 a \,b^{5} n^{2} x^{5}-274 b^{6} n \,x^{6}-20 a^{3} b^{3} n^{3} x^{3}+55 a^{2} b^{4} n^{2} x^{4}-24 x^{5} a n \,b^{5}-120 x^{6} b^{6}-60 a^{3} b^{3} n^{2} x^{3}+30 a^{2} n \,x^{4} b^{4}+60 a^{4} b^{2} n^{2} x^{2}-40 a^{3} n \,x^{3} b^{3}+60 a^{4} b^{2} n \,x^{2}-120 a^{5} b n x +120 a^{6}\right ) \left (b x +a \right )^{n}}{x \left (5+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{6}}\)	\(309\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(5/2)*(b*x+a)^n,x,method=_RETURNVERBOSE)

[Out]

-(b*x+a)^(1+n)*(c*x^2)^(5/2)*(-b^5*n^5*x^5-15*b^5*n^4*x^5+5*a*b^4*n^4*x^4-85*b^5*n^3*x^5+50*a*b^4*n^3*x^4-225*
b^5*n^2*x^5-20*a^2*b^3*n^3*x^3+175*a*b^4*n^2*x^4-274*b^5*n*x^5-120*a^2*b^3*n^2*x^3+250*a*b^4*n*x^4-120*b^5*x^5
+60*a^3*b^2*n^2*x^2-220*a^2*b^3*n*x^3+120*a*b^4*x^4+180*a^3*b^2*n*x^2-120*a^2*b^3*x^3-120*a^4*b*n*x+120*a^3*b^
2*x^2-120*a^4*b*x+120*a^5)/x^5/b^6/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+720)

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Maxima [A]
time = 0.30, size = 203, normalized size = 0.94 \begin {gather*} \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} c^{\frac {5}{2}} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} c^{\frac {5}{2}} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} c^{\frac {5}{2}} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} c^{\frac {5}{2}} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} c^{\frac {5}{2}} x^{2} + 120 \, a^{5} b c^{\frac {5}{2}} n x - 120 \, a^{6} c^{\frac {5}{2}}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^6*c^(5/2)*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a*b
^5*c^(5/2)*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*c^(5/2)*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^3*c^(5/2)*x
^3 - 60*(n^2 + n)*a^4*b^2*c^(5/2)*x^2 + 120*a^5*b*c^(5/2)*n*x - 120*a^6*c^(5/2))*(b*x + a)^n/((n^6 + 21*n^5 +
175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^6)

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Fricas [A]
time = 0.73, size = 352, normalized size = 1.62 \begin {gather*} \frac {{\left (120 \, a^{5} b c^{2} n x - 120 \, a^{6} c^{2} + {\left (b^{6} c^{2} n^{5} + 15 \, b^{6} c^{2} n^{4} + 85 \, b^{6} c^{2} n^{3} + 225 \, b^{6} c^{2} n^{2} + 274 \, b^{6} c^{2} n + 120 \, b^{6} c^{2}\right )} x^{6} + {\left (a b^{5} c^{2} n^{5} + 10 \, a b^{5} c^{2} n^{4} + 35 \, a b^{5} c^{2} n^{3} + 50 \, a b^{5} c^{2} n^{2} + 24 \, a b^{5} c^{2} n\right )} x^{5} - 5 \, {\left (a^{2} b^{4} c^{2} n^{4} + 6 \, a^{2} b^{4} c^{2} n^{3} + 11 \, a^{2} b^{4} c^{2} n^{2} + 6 \, a^{2} b^{4} c^{2} n\right )} x^{4} + 20 \, {\left (a^{3} b^{3} c^{2} n^{3} + 3 \, a^{3} b^{3} c^{2} n^{2} + 2 \, a^{3} b^{3} c^{2} n\right )} x^{3} - 60 \, {\left (a^{4} b^{2} c^{2} n^{2} + a^{4} b^{2} c^{2} n\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(120*a^5*b*c^2*n*x - 120*a^6*c^2 + (b^6*c^2*n^5 + 15*b^6*c^2*n^4 + 85*b^6*c^2*n^3 + 225*b^6*c^2*n^2 + 274*b^6*
c^2*n + 120*b^6*c^2)*x^6 + (a*b^5*c^2*n^5 + 10*a*b^5*c^2*n^4 + 35*a*b^5*c^2*n^3 + 50*a*b^5*c^2*n^2 + 24*a*b^5*
c^2*n)*x^5 - 5*(a^2*b^4*c^2*n^4 + 6*a^2*b^4*c^2*n^3 + 11*a^2*b^4*c^2*n^2 + 6*a^2*b^4*c^2*n)*x^4 + 20*(a^3*b^3*
c^2*n^3 + 3*a^3*b^3*c^2*n^2 + 2*a^3*b^3*c^2*n)*x^3 - 60*(a^4*b^2*c^2*n^2 + a^4*b^2*c^2*n)*x^2)*sqrt(c*x^2)*(b*
x + a)^n/((b^6*n^6 + 21*b^6*n^5 + 175*b^6*n^4 + 735*b^6*n^3 + 1624*b^6*n^2 + 1764*b^6*n + 720*b^6)*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c x^{2}\right )^{\frac {5}{2}} \left (a + b x\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (205) = 410\).
time = 0.92, size = 640, normalized size = 2.95 \begin {gather*} {\left (\frac {120 \, a^{6} a^{n} c^{2} \mathrm {sgn}\left (x\right )}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}} + \frac {{\left (b x + a\right )}^{n} b^{6} c^{2} n^{5} x^{6} \mathrm {sgn}\left (x\right ) + {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{5} x^{5} \mathrm {sgn}\left (x\right ) + 15 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n^{4} x^{6} \mathrm {sgn}\left (x\right ) + 10 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{4} x^{5} \mathrm {sgn}\left (x\right ) + 85 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n^{3} x^{6} \mathrm {sgn}\left (x\right ) - 5 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n^{4} x^{4} \mathrm {sgn}\left (x\right ) + 35 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{3} x^{5} \mathrm {sgn}\left (x\right ) + 225 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n^{2} x^{6} \mathrm {sgn}\left (x\right ) - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n^{3} x^{4} \mathrm {sgn}\left (x\right ) + 50 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{2} x^{5} \mathrm {sgn}\left (x\right ) + 274 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n x^{6} \mathrm {sgn}\left (x\right ) + 20 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c^{2} n^{3} x^{3} \mathrm {sgn}\left (x\right ) - 55 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n^{2} x^{4} \mathrm {sgn}\left (x\right ) + 24 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n x^{5} \mathrm {sgn}\left (x\right ) + 120 \, {\left (b x + a\right )}^{n} b^{6} c^{2} x^{6} \mathrm {sgn}\left (x\right ) + 60 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c^{2} n^{2} x^{3} \mathrm {sgn}\left (x\right ) - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n x^{4} \mathrm {sgn}\left (x\right ) - 60 \, {\left (b x + a\right )}^{n} a^{4} b^{2} c^{2} n^{2} x^{2} \mathrm {sgn}\left (x\right ) + 40 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c^{2} n x^{3} \mathrm {sgn}\left (x\right ) - 60 \, {\left (b x + a\right )}^{n} a^{4} b^{2} c^{2} n x^{2} \mathrm {sgn}\left (x\right ) + 120 \, {\left (b x + a\right )}^{n} a^{5} b c^{2} n x \mathrm {sgn}\left (x\right ) - 120 \, {\left (b x + a\right )}^{n} a^{6} c^{2} \mathrm {sgn}\left (x\right )}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}}\right )} \sqrt {c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(120*a^6*a^n*c^2*sgn(x)/(b^6*n^6 + 21*b^6*n^5 + 175*b^6*n^4 + 735*b^6*n^3 + 1624*b^6*n^2 + 1764*b^6*n + 720*b^
6) + ((b*x + a)^n*b^6*c^2*n^5*x^6*sgn(x) + (b*x + a)^n*a*b^5*c^2*n^5*x^5*sgn(x) + 15*(b*x + a)^n*b^6*c^2*n^4*x
^6*sgn(x) + 10*(b*x + a)^n*a*b^5*c^2*n^4*x^5*sgn(x) + 85*(b*x + a)^n*b^6*c^2*n^3*x^6*sgn(x) - 5*(b*x + a)^n*a^
2*b^4*c^2*n^4*x^4*sgn(x) + 35*(b*x + a)^n*a*b^5*c^2*n^3*x^5*sgn(x) + 225*(b*x + a)^n*b^6*c^2*n^2*x^6*sgn(x) -
30*(b*x + a)^n*a^2*b^4*c^2*n^3*x^4*sgn(x) + 50*(b*x + a)^n*a*b^5*c^2*n^2*x^5*sgn(x) + 274*(b*x + a)^n*b^6*c^2*
n*x^6*sgn(x) + 20*(b*x + a)^n*a^3*b^3*c^2*n^3*x^3*sgn(x) - 55*(b*x + a)^n*a^2*b^4*c^2*n^2*x^4*sgn(x) + 24*(b*x
+ a)^n*a*b^5*c^2*n*x^5*sgn(x) + 120*(b*x + a)^n*b^6*c^2*x^6*sgn(x) + 60*(b*x + a)^n*a^3*b^3*c^2*n^2*x^3*sgn(x
) - 30*(b*x + a)^n*a^2*b^4*c^2*n*x^4*sgn(x) - 60*(b*x + a)^n*a^4*b^2*c^2*n^2*x^2*sgn(x) + 40*(b*x + a)^n*a^3*b
^3*c^2*n*x^3*sgn(x) - 60*(b*x + a)^n*a^4*b^2*c^2*n*x^2*sgn(x) + 120*(b*x + a)^n*a^5*b*c^2*n*x*sgn(x) - 120*(b*
x + a)^n*a^6*c^2*sgn(x))/(b^6*n^6 + 21*b^6*n^5 + 175*b^6*n^4 + 735*b^6*n^3 + 1624*b^6*n^2 + 1764*b^6*n + 720*b
^6))*sqrt(c)

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Mupad [B]
time = 0.50, size = 424, normalized size = 1.95 \begin {gather*} \frac {{\left (a+b\,x\right )}^n\,\left (\frac {c^2\,x^6\,\sqrt {c\,x^2}\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}-\frac {120\,a^6\,c^2\,\sqrt {c\,x^2}}{b^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {120\,a^5\,c^2\,n\,x\,\sqrt {c\,x^2}}{b^5\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {5\,a^2\,c^2\,n\,x^4\,\sqrt {c\,x^2}\,\left (n^3+6\,n^2+11\,n+6\right )}{b^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {60\,a^4\,c^2\,n\,x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{b^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,c^2\,n\,x^5\,\sqrt {c\,x^2}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {20\,a^3\,c^2\,n\,x^3\,\sqrt {c\,x^2}\,\left (n^2+3\,n+2\right )}{b^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(5/2)*(a + b*x)^n,x)

[Out]

((a + b*x)^n*((c^2*x^6*(c*x^2)^(1/2)*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(1764*n + 1624*n^2 + 735
*n^3 + 175*n^4 + 21*n^5 + n^6 + 720) - (120*a^6*c^2*(c*x^2)^(1/2))/(b^6*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4
+ 21*n^5 + n^6 + 720)) + (120*a^5*c^2*n*x*(c*x^2)^(1/2))/(b^5*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5
+ n^6 + 720)) - (5*a^2*c^2*n*x^4*(c*x^2)^(1/2)*(11*n + 6*n^2 + n^3 + 6))/(b^2*(1764*n + 1624*n^2 + 735*n^3 +
175*n^4 + 21*n^5 + n^6 + 720)) - (60*a^4*c^2*n*x^2*(c*x^2)^(1/2)*(n + 1))/(b^4*(1764*n + 1624*n^2 + 735*n^3 +
175*n^4 + 21*n^5 + n^6 + 720)) + (a*c^2*n*x^5*(c*x^2)^(1/2)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(b*(1764*n +
1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (20*a^3*c^2*n*x^3*(c*x^2)^(1/2)*(3*n + n^2 + 2))/(b^3*(1
764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720))))/x

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