3.10.16 \(\int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx\)

Optimal. Leaf size=94 \[ \frac {a^2 \sqrt {c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac {2 a b \sqrt {c x^2} (d x)^{m+3}}{d^3 (m+3) x}+\frac {b^2 \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x} \]

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Rubi [A]  time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 16, 43} \begin {gather*} \frac {a^2 \sqrt {c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac {2 a b \sqrt {c x^2} (d x)^{m+3}}{d^3 (m+3) x}+\frac {b^2 \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*Sqrt[c*x^2]*(a + b*x)^2,x]

[Out]

(a^2*(d*x)^(2 + m)*Sqrt[c*x^2])/(d^2*(2 + m)*x) + (2*a*b*(d*x)^(3 + m)*Sqrt[c*x^2])/(d^3*(3 + m)*x) + (b^2*(d*
x)^(4 + m)*Sqrt[c*x^2])/(d^4*(4 + m)*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 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, 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 (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx &=\frac {\sqrt {c x^2} \int x (d x)^m (a+b x)^2 \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int (d x)^{1+m} (a+b x)^2 \, dx}{d x}\\ &=\frac {\sqrt {c x^2} \int \left (a^2 (d x)^{1+m}+\frac {2 a b (d x)^{2+m}}{d}+\frac {b^2 (d x)^{3+m}}{d^2}\right ) \, dx}{d x}\\ &=\frac {a^2 (d x)^{2+m} \sqrt {c x^2}}{d^2 (2+m) x}+\frac {2 a b (d x)^{3+m} \sqrt {c x^2}}{d^3 (3+m) x}+\frac {b^2 (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 72, normalized size = 0.77 \begin {gather*} \frac {x \sqrt {c x^2} (d x)^m \left (a^2 \left (m^2+7 m+12\right )+2 a b \left (m^2+6 m+8\right ) x+b^2 \left (m^2+5 m+6\right ) x^2\right )}{(m+2) (m+3) (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*Sqrt[c*x^2]*(a + b*x)^2,x]

[Out]

(x*(d*x)^m*Sqrt[c*x^2]*(a^2*(12 + 7*m + m^2) + 2*a*b*(8 + 6*m + m^2)*x + b^2*(6 + 5*m + m^2)*x^2))/((2 + m)*(3
+ m)*(4 + m))

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IntegrateAlgebraic [F]  time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \sqrt {c x^2} (a+b x)^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d*x)^m*Sqrt[c*x^2]*(a + b*x)^2,x]

[Out]

Defer[IntegrateAlgebraic][(d*x)^m*Sqrt[c*x^2]*(a + b*x)^2, x]

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fricas [A]  time = 1.55, size = 94, normalized size = 1.00 \begin {gather*} \frac {{\left ({\left (b^{2} m^{2} + 5 \, b^{2} m + 6 \, b^{2}\right )} x^{3} + 2 \, {\left (a b m^{2} + 6 \, a b m + 8 \, a b\right )} x^{2} + {\left (a^{2} m^{2} + 7 \, a^{2} m + 12 \, a^{2}\right )} x\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 26 \, m + 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^2)^(1/2)*(b*x+a)^2,x, algorithm="fricas")

[Out]

((b^2*m^2 + 5*b^2*m + 6*b^2)*x^3 + 2*(a*b*m^2 + 6*a*b*m + 8*a*b)*x^2 + (a^2*m^2 + 7*a^2*m + 12*a^2)*x)*sqrt(c*
x^2)*(d*x)^m/(m^3 + 9*m^2 + 26*m + 24)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^2)^(1/2)*(b*x+a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Undef/Unsigned Inf encountered in limit

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maple [A]  time = 0.00, size = 95, normalized size = 1.01 \begin {gather*} \frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +5 b^{2} m \,x^{2}+a^{2} m^{2}+12 a b m x +6 b^{2} x^{2}+7 a^{2} m +16 a b x +12 a^{2}\right ) \sqrt {c \,x^{2}}\, x \left (d x \right )^{m}}{\left (m +4\right ) \left (m +3\right ) \left (m +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(c*x^2)^(1/2)*(b*x+a)^2,x)

[Out]

x*(b^2*m^2*x^2+2*a*b*m^2*x+5*b^2*m*x^2+a^2*m^2+12*a*b*m*x+6*b^2*x^2+7*a^2*m+16*a*b*x+12*a^2)*(d*x)^m*(c*x^2)^(
1/2)/(m+4)/(m+3)/(m+2)

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maxima [A]  time = 1.52, size = 64, normalized size = 0.68 \begin {gather*} \frac {b^{2} \sqrt {c} d^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, a b \sqrt {c} d^{m} x^{3} x^{m}}{m + 3} + \frac {a^{2} \sqrt {c} d^{m} x^{2} x^{m}}{m + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^2)^(1/2)*(b*x+a)^2,x, algorithm="maxima")

[Out]

b^2*sqrt(c)*d^m*x^4*x^m/(m + 4) + 2*a*b*sqrt(c)*d^m*x^3*x^m/(m + 3) + a^2*sqrt(c)*d^m*x^2*x^m/(m + 2)

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mupad [B]  time = 0.26, size = 116, normalized size = 1.23 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {a^2\,x\,\sqrt {c\,x^2}\,\left (m^2+7\,m+12\right )}{m^3+9\,m^2+26\,m+24}+\frac {b^2\,x^3\,\sqrt {c\,x^2}\,\left (m^2+5\,m+6\right )}{m^3+9\,m^2+26\,m+24}+\frac {2\,a\,b\,x^2\,\sqrt {c\,x^2}\,\left (m^2+6\,m+8\right )}{m^3+9\,m^2+26\,m+24}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(c*x^2)^(1/2)*(a + b*x)^2,x)

[Out]

(d*x)^m*((a^2*x*(c*x^2)^(1/2)*(7*m + m^2 + 12))/(26*m + 9*m^2 + m^3 + 24) + (b^2*x^3*(c*x^2)^(1/2)*(5*m + m^2
+ 6))/(26*m + 9*m^2 + m^3 + 24) + (2*a*b*x^2*(c*x^2)^(1/2)*(6*m + m^2 + 8))/(26*m + 9*m^2 + m^3 + 24))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {\int \frac {a^{2} \sqrt {c x^{2}}}{x^{4}}\, dx + \int \frac {b^{2} \sqrt {c x^{2}}}{x^{2}}\, dx + \int \frac {2 a b \sqrt {c x^{2}}}{x^{3}}\, dx}{d^{4}} & \text {for}\: m = -4 \\\frac {\int \frac {a^{2} \sqrt {c x^{2}}}{x^{3}}\, dx + \int \frac {b^{2} \sqrt {c x^{2}}}{x}\, dx + \int \frac {2 a b \sqrt {c x^{2}}}{x^{2}}\, dx}{d^{3}} & \text {for}\: m = -3 \\\frac {\int b^{2} \sqrt {c x^{2}}\, dx + \int \frac {a^{2} \sqrt {c x^{2}}}{x^{2}}\, dx + \int \frac {2 a b \sqrt {c x^{2}}}{x}\, dx}{d^{2}} & \text {for}\: m = -2 \\\frac {a^{2} \sqrt {c} d^{m} m^{2} x x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {7 a^{2} \sqrt {c} d^{m} m x x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {12 a^{2} \sqrt {c} d^{m} x x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {2 a b \sqrt {c} d^{m} m^{2} x^{2} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {12 a b \sqrt {c} d^{m} m x^{2} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {16 a b \sqrt {c} d^{m} x^{2} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {b^{2} \sqrt {c} d^{m} m^{2} x^{3} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {5 b^{2} \sqrt {c} d^{m} m x^{3} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {6 b^{2} \sqrt {c} d^{m} x^{3} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(c*x**2)**(1/2)*(b*x+a)**2,x)

[Out]

Piecewise(((Integral(a**2*sqrt(c*x**2)/x**4, x) + Integral(b**2*sqrt(c*x**2)/x**2, x) + Integral(2*a*b*sqrt(c*
x**2)/x**3, x))/d**4, Eq(m, -4)), ((Integral(a**2*sqrt(c*x**2)/x**3, x) + Integral(b**2*sqrt(c*x**2)/x, x) + I
ntegral(2*a*b*sqrt(c*x**2)/x**2, x))/d**3, Eq(m, -3)), ((Integral(b**2*sqrt(c*x**2), x) + Integral(a**2*sqrt(c
*x**2)/x**2, x) + Integral(2*a*b*sqrt(c*x**2)/x, x))/d**2, Eq(m, -2)), (a**2*sqrt(c)*d**m*m**2*x*x**m*sqrt(x**
2)/(m**3 + 9*m**2 + 26*m + 24) + 7*a**2*sqrt(c)*d**m*m*x*x**m*sqrt(x**2)/(m**3 + 9*m**2 + 26*m + 24) + 12*a**2
*sqrt(c)*d**m*x*x**m*sqrt(x**2)/(m**3 + 9*m**2 + 26*m + 24) + 2*a*b*sqrt(c)*d**m*m**2*x**2*x**m*sqrt(x**2)/(m*
*3 + 9*m**2 + 26*m + 24) + 12*a*b*sqrt(c)*d**m*m*x**2*x**m*sqrt(x**2)/(m**3 + 9*m**2 + 26*m + 24) + 16*a*b*sqr
t(c)*d**m*x**2*x**m*sqrt(x**2)/(m**3 + 9*m**2 + 26*m + 24) + b**2*sqrt(c)*d**m*m**2*x**3*x**m*sqrt(x**2)/(m**3
+ 9*m**2 + 26*m + 24) + 5*b**2*sqrt(c)*d**m*m*x**3*x**m*sqrt(x**2)/(m**3 + 9*m**2 + 26*m + 24) + 6*b**2*sqrt(
c)*d**m*x**3*x**m*sqrt(x**2)/(m**3 + 9*m**2 + 26*m + 24), True))

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