3.930 \(\int x \left (c x^2\right )^{3/2} (a+b x)^n \, dx\)

Optimal. Leaf size=169 \[ \frac{a^4 c \sqrt{c x^2} (a+b x)^{n+1}}{b^5 (n+1) x}-\frac{4 a^3 c \sqrt{c x^2} (a+b x)^{n+2}}{b^5 (n+2) x}+\frac{6 a^2 c \sqrt{c x^2} (a+b x)^{n+3}}{b^5 (n+3) x}-\frac{4 a c \sqrt{c x^2} (a+b x)^{n+4}}{b^5 (n+4) x}+\frac{c \sqrt{c x^2} (a+b x)^{n+5}}{b^5 (n+5) x} \]

[Out]

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

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Rubi [A]  time = 0.134337, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{a^4 c \sqrt{c x^2} (a+b x)^{n+1}}{b^5 (n+1) x}-\frac{4 a^3 c \sqrt{c x^2} (a+b x)^{n+2}}{b^5 (n+2) x}+\frac{6 a^2 c \sqrt{c x^2} (a+b x)^{n+3}}{b^5 (n+3) x}-\frac{4 a c \sqrt{c x^2} (a+b x)^{n+4}}{b^5 (n+4) x}+\frac{c \sqrt{c x^2} (a+b x)^{n+5}}{b^5 (n+5) x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(c*x**2)**(3/2)*(b*x+a)**n,x)

[Out]

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

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Mathematica [A]  time = 0.117044, size = 132, normalized size = 0.78 \[ \frac{\left (c x^2\right )^{3/2} (a+b x)^{n+1} \left (24 a^4-24 a^3 b (n+1) x+12 a^2 b^2 \left (n^2+3 n+2\right ) x^2-4 a b^3 \left (n^3+6 n^2+11 n+6\right ) x^3+b^4 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4\right )}{b^5 (n+1) (n+2) (n+3) (n+4) (n+5) x^3} \]

Antiderivative was successfully verified.

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

[Out]

((c*x^2)^(3/2)*(a + b*x)^(1 + n)*(24*a^4 - 24*a^3*b*(1 + n)*x + 12*a^2*b^2*(2 +
3*n + n^2)*x^2 - 4*a*b^3*(6 + 11*n + 6*n^2 + n^3)*x^3 + b^4*(24 + 50*n + 35*n^2
+ 10*n^3 + n^4)*x^4))/(b^5*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*x^3)

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Maple [A]  time = 0.01, size = 199, normalized size = 1.2 \[{\frac{ \left ( bx+a \right ) ^{1+n} \left ({b}^{4}{n}^{4}{x}^{4}+10\,{b}^{4}{n}^{3}{x}^{4}-4\,a{b}^{3}{n}^{3}{x}^{3}+35\,{b}^{4}{n}^{2}{x}^{4}-24\,a{b}^{3}{n}^{2}{x}^{3}+50\,{b}^{4}n{x}^{4}+12\,{a}^{2}{b}^{2}{n}^{2}{x}^{2}-44\,a{b}^{3}n{x}^{3}+24\,{x}^{4}{b}^{4}+36\,{a}^{2}{b}^{2}n{x}^{2}-24\,{x}^{3}a{b}^{3}-24\,{a}^{3}bnx+24\,{x}^{2}{a}^{2}{b}^{2}-24\,x{a}^{3}b+24\,{a}^{4} \right ) }{{x}^{3}{b}^{5} \left ({n}^{5}+15\,{n}^{4}+85\,{n}^{3}+225\,{n}^{2}+274\,n+120 \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x+a)^(1+n)*(b^4*n^4*x^4+10*b^4*n^3*x^4-4*a*b^3*n^3*x^3+35*b^4*n^2*x^4-24*a*b^
3*n^2*x^3+50*b^4*n*x^4+12*a^2*b^2*n^2*x^2-44*a*b^3*n*x^3+24*b^4*x^4+36*a^2*b^2*n
*x^2-24*a*b^3*x^3-24*a^3*b*n*x+24*a^2*b^2*x^2-24*a^3*b*x+24*a^4)*(c*x^2)^(3/2)/x
^3/b^5/(n^5+15*n^4+85*n^3+225*n^2+274*n+120)

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Maxima [A]  time = 1.37653, size = 212, normalized size = 1.25 \[ \frac{{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} c^{\frac{3}{2}} x^{5} +{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} c^{\frac{3}{2}} x^{4} - 4 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} c^{\frac{3}{2}} x^{3} + 12 \,{\left (n^{2} + n\right )} a^{3} b^{2} c^{\frac{3}{2}} x^{2} - 24 \, a^{4} b c^{\frac{3}{2}} n x + 24 \, a^{5} c^{\frac{3}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*c^(3/2)*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6
*n)*a*b^4*c^(3/2)*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*c^(3/2)*x^3 + 12*(n^2 + n)
*a^3*b^2*c^(3/2)*x^2 - 24*a^4*b*c^(3/2)*n*x + 24*a^5*c^(3/2))*(b*x + a)^n/((n^5
+ 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5)

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Fricas [A]  time = 0.226257, size = 315, normalized size = 1.86 \[ -\frac{{\left (24 \, a^{4} b c n x - 24 \, a^{5} c -{\left (b^{5} c n^{4} + 10 \, b^{5} c n^{3} + 35 \, b^{5} c n^{2} + 50 \, b^{5} c n + 24 \, b^{5} c\right )} x^{5} -{\left (a b^{4} c n^{4} + 6 \, a b^{4} c n^{3} + 11 \, a b^{4} c n^{2} + 6 \, a b^{4} c n\right )} x^{4} + 4 \,{\left (a^{2} b^{3} c n^{3} + 3 \, a^{2} b^{3} c n^{2} + 2 \, a^{2} b^{3} c n\right )} x^{3} - 12 \,{\left (a^{3} b^{2} c n^{2} + a^{3} b^{2} c n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(24*a^4*b*c*n*x - 24*a^5*c - (b^5*c*n^4 + 10*b^5*c*n^3 + 35*b^5*c*n^2 + 50*b^5*
c*n + 24*b^5*c)*x^5 - (a*b^4*c*n^4 + 6*a*b^4*c*n^3 + 11*a*b^4*c*n^2 + 6*a*b^4*c*
n)*x^4 + 4*(a^2*b^3*c*n^3 + 3*a^2*b^3*c*n^2 + 2*a^2*b^3*c*n)*x^3 - 12*(a^3*b^2*c
*n^2 + a^3*b^2*c*n)*x^2)*sqrt(c*x^2)*(b*x + a)^n/((b^5*n^5 + 15*b^5*n^4 + 85*b^5
*n^3 + 225*b^5*n^2 + 274*b^5*n + 120*b^5)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(c*x**2)**(3/2)*(b*x+a)**n,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212697, size = 621, normalized size = 3.67 \[ -{\left (\frac{24 \, a^{5} e^{\left (n{\rm ln}\left (a\right )\right )}{\rm sign}\left (x\right )}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} - \frac{b^{5} n^{4} x^{5} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + a b^{4} n^{4} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 10 \, b^{5} n^{3} x^{5} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 6 \, a b^{4} n^{3} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 35 \, b^{5} n^{2} x^{5} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) - 4 \, a^{2} b^{3} n^{3} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 11 \, a b^{4} n^{2} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 50 \, b^{5} n x^{5} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) - 12 \, a^{2} b^{3} n^{2} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 6 \, a b^{4} n x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 24 \, b^{5} x^{5} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 12 \, a^{3} b^{2} n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) - 8 \, a^{2} b^{3} n x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 12 \, a^{3} b^{2} n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) - 24 \, a^{4} b n x e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right ) + 24 \, a^{5} e^{\left (n{\rm ln}\left (b x + a\right )\right )}{\rm sign}\left (x\right )}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}}\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(24*a^5*e^(n*ln(a))*sign(x)/(b^5*n^5 + 15*b^5*n^4 + 85*b^5*n^3 + 225*b^5*n^2 +
274*b^5*n + 120*b^5) - (b^5*n^4*x^5*e^(n*ln(b*x + a))*sign(x) + a*b^4*n^4*x^4*e^
(n*ln(b*x + a))*sign(x) + 10*b^5*n^3*x^5*e^(n*ln(b*x + a))*sign(x) + 6*a*b^4*n^3
*x^4*e^(n*ln(b*x + a))*sign(x) + 35*b^5*n^2*x^5*e^(n*ln(b*x + a))*sign(x) - 4*a^
2*b^3*n^3*x^3*e^(n*ln(b*x + a))*sign(x) + 11*a*b^4*n^2*x^4*e^(n*ln(b*x + a))*sig
n(x) + 50*b^5*n*x^5*e^(n*ln(b*x + a))*sign(x) - 12*a^2*b^3*n^2*x^3*e^(n*ln(b*x +
a))*sign(x) + 6*a*b^4*n*x^4*e^(n*ln(b*x + a))*sign(x) + 24*b^5*x^5*e^(n*ln(b*x
+ a))*sign(x) + 12*a^3*b^2*n^2*x^2*e^(n*ln(b*x + a))*sign(x) - 8*a^2*b^3*n*x^3*e
^(n*ln(b*x + a))*sign(x) + 12*a^3*b^2*n*x^2*e^(n*ln(b*x + a))*sign(x) - 24*a^4*b
*n*x*e^(n*ln(b*x + a))*sign(x) + 24*a^5*e^(n*ln(b*x + a))*sign(x))/(b^5*n^5 + 15
*b^5*n^4 + 85*b^5*n^3 + 225*b^5*n^2 + 274*b^5*n + 120*b^5))*c^(3/2)