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

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

[Out]

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

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Rubi [A]  time = 0.0386992, antiderivative size = 135, 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} \[ -\frac{a^3 c \sqrt{c x^2} (a+b x)^{n+1}}{b^4 (n+1) x}+\frac{3 a^2 c \sqrt{c x^2} (a+b x)^{n+2}}{b^4 (n+2) x}-\frac{3 a c \sqrt{c x^2} (a+b x)^{n+3}}{b^4 (n+3) x}+\frac{c \sqrt{c x^2} (a+b x)^{n+4}}{b^4 (n+4) x} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(3/2)*(a + b*x)^n,x]

[Out]

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

Mathematica [A]  time = 0.0562247, size = 98, normalized size = 0.73 \[ \frac{\left (c x^2\right )^{3/2} (a+b x)^{n+1} \left (6 a^2 b (n+1) x-6 a^3-3 a b^2 \left (n^2+3 n+2\right ) x^2+b^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )}{b^4 (n+1) (n+2) (n+3) (n+4) x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 136, normalized size = 1. \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \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}bnx+6\,a{b}^{2}{x}^{2}-6\,{a}^{2}bx+6\,{a}^{3} \right ) }{{x}^{3}{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01576, size = 157, normalized size = 1.16 \begin{align*} \frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} c^{\frac{3}{2}} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} c^{\frac{3}{2}} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} c^{\frac{3}{2}} x^{2} + 6 \, a^{3} b c^{\frac{3}{2}} n x - 6 \, a^{4} c^{\frac{3}{2}}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.53218, size = 343, normalized size = 2.54 \begin{align*} \frac{{\left (6 \, a^{3} b c n x - 6 \, a^{4} c +{\left (b^{4} c n^{3} + 6 \, b^{4} c n^{2} + 11 \, b^{4} c n + 6 \, b^{4} c\right )} x^{4} +{\left (a b^{3} c n^{3} + 3 \, a b^{3} c n^{2} + 2 \, a b^{3} c n\right )} x^{3} - 3 \,{\left (a^{2} b^{2} c n^{2} + a^{2} b^{2} c n\right )} x^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [B]  time = 1.07286, size = 405, normalized size = 3. \begin{align*}{\left (\frac{6 \, a^{4} a^{n} \mathrm{sgn}\left (x\right )}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} + \frac{{\left (b x + a\right )}^{n} b^{4} n^{3} x^{4} \mathrm{sgn}\left (x\right ) +{\left (b x + a\right )}^{n} a b^{3} n^{3} x^{3} \mathrm{sgn}\left (x\right ) + 6 \,{\left (b x + a\right )}^{n} b^{4} n^{2} x^{4} \mathrm{sgn}\left (x\right ) + 3 \,{\left (b x + a\right )}^{n} a b^{3} n^{2} x^{3} \mathrm{sgn}\left (x\right ) + 11 \,{\left (b x + a\right )}^{n} b^{4} n x^{4} \mathrm{sgn}\left (x\right ) - 3 \,{\left (b x + a\right )}^{n} a^{2} b^{2} n^{2} x^{2} \mathrm{sgn}\left (x\right ) + 2 \,{\left (b x + a\right )}^{n} a b^{3} n x^{3} \mathrm{sgn}\left (x\right ) + 6 \,{\left (b x + a\right )}^{n} b^{4} x^{4} \mathrm{sgn}\left (x\right ) - 3 \,{\left (b x + a\right )}^{n} a^{2} b^{2} n x^{2} \mathrm{sgn}\left (x\right ) + 6 \,{\left (b x + a\right )}^{n} a^{3} b n x \mathrm{sgn}\left (x\right ) - 6 \,{\left (b x + a\right )}^{n} a^{4} \mathrm{sgn}\left (x\right )}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}}\right )} c^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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