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

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

[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)

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Rubi [A]  time = 0.073794, antiderivative size = 217, normalized size of antiderivative = 1., 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, 43} \[ -\frac{a^5 c^2 \sqrt{c x^2} (a+b x)^{n+1}}{b^6 (n+1) x}+\frac{5 a^4 c^2 \sqrt{c x^2} (a+b x)^{n+2}}{b^6 (n+2) x}-\frac{10 a^3 c^2 \sqrt{c x^2} (a+b x)^{n+3}}{b^6 (n+3) x}+\frac{10 a^2 c^2 \sqrt{c x^2} (a+b x)^{n+4}}{b^6 (n+4) x}-\frac{5 a c^2 \sqrt{c x^2} (a+b x)^{n+5}}{b^6 (n+5) x}+\frac{c^2 \sqrt{c x^2} (a+b x)^{n+6}}{b^6 (n+6) x} \]

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 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 \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*}

Mathematica [A]  time = 0.1091, size = 172, normalized size = 0.79 \[ \frac{c^3 x (a+b x)^{n+1} \left (-60 a^3 b^2 \left (n^2+3 n+2\right ) x^2+20 a^2 b^3 \left (n^3+6 n^2+11 n+6\right ) x^3+120 a^4 b (n+1) x-120 a^5-5 a b^4 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4+b^5 \left (n^5+15 n^4+85 n^3+225 n^2+274 n+120\right ) x^5\right )}{b^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) \sqrt{c x^2}} \]

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 = 0.006, size = 280, normalized size = 1.3 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \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\,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}bnx+120\,{a}^{3}{b}^{2}{x}^{2}-120\,{a}^{4}bx+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 ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.12252, size = 274, normalized size = 1.26 \begin{align*} \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{align*}

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 = 1.71717, size = 733, normalized size = 3.38 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2}\right )^{\frac{5}{2}} \left (a + b x\right )^{n}\, dx \end{align*}

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]  time = 1.08614, size = 864, normalized size = 3.98 \begin{align*} \text{result too large to display} \end{align*}

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)