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

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

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Rubi [A]  time = 0.04, antiderivative size = 135, 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, 43} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.05, size = 98, normalized size = 0.73 \begin {gather*} \frac {\left (c x^2\right )^{3/2} (a+b x)^{n+1} \left (-6 a^3+6 a^2 b (n+1) x-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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c x^2\right )^{3/2} (a+b x)^n \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.06, size = 164, normalized size = 1.21 \begin {gather*} \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 {gather*}

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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giac [B]  time = 1.14, size = 300, normalized size = 2.22 \begin {gather*} {\left (\frac {6 \, a^{4} a^{n} \mathrm {sgn}\relax (x)}{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}\relax (x) + {\left (b x + a\right )}^{n} a b^{3} n^{3} x^{3} \mathrm {sgn}\relax (x) + 6 \, {\left (b x + a\right )}^{n} b^{4} n^{2} x^{4} \mathrm {sgn}\relax (x) + 3 \, {\left (b x + a\right )}^{n} a b^{3} n^{2} x^{3} \mathrm {sgn}\relax (x) + 11 \, {\left (b x + a\right )}^{n} b^{4} n x^{4} \mathrm {sgn}\relax (x) - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} n^{2} x^{2} \mathrm {sgn}\relax (x) + 2 \, {\left (b x + a\right )}^{n} a b^{3} n x^{3} \mathrm {sgn}\relax (x) + 6 \, {\left (b x + a\right )}^{n} b^{4} x^{4} \mathrm {sgn}\relax (x) - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} n x^{2} \mathrm {sgn}\relax (x) + 6 \, {\left (b x + a\right )}^{n} a^{3} b n x \mathrm {sgn}\relax (x) - 6 \, {\left (b x + a\right )}^{n} a^{4} \mathrm {sgn}\relax (x)}{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 {gather*}

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)

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maple [A]  time = 0.01, size = 136, normalized size = 1.01 \begin {gather*} -\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \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 (b x +a \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) b^{4} x^{3}} \end {gather*}

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)^(n+1)*(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 = 1.44, size = 116, normalized size = 0.86 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.32, size = 219, normalized size = 1.62 \begin {gather*} \frac {{\left (a+b\,x\right )}^n\,\left (\frac {c\,x^4\,\sqrt {c\,x^2}\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {6\,a^4\,c\,\sqrt {c\,x^2}}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,a^3\,c\,n\,x\,\sqrt {c\,x^2}}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {3\,a^2\,c\,n\,x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,c\,n\,x^3\,\sqrt {c\,x^2}\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)*(a + b*x)^n,x)

[Out]

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

[Out]

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

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