3.10.15 \(\int (d x)^m (c x^2)^{3/2} (a+b x)^2 \, dx\)

Optimal. Leaf size=97 \[ \frac {a^2 c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {2 a b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x}+\frac {b^2 c \sqrt {c x^2} (d x)^{m+6}}{d^6 (m+6) x} \]

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Rubi [A]  time = 0.04, antiderivative size = 97, 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 c \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac {2 a b c \sqrt {c x^2} (d x)^{m+5}}{d^5 (m+5) x}+\frac {b^2 c \sqrt {c x^2} (d x)^{m+6}}{d^6 (m+6) x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*c*(d*x)^(4 + m)*Sqrt[c*x^2])/(d^4*(4 + m)*x) + (2*a*b*c*(d*x)^(5 + m)*Sqrt[c*x^2])/(d^5*(5 + m)*x) + (b^2
*c*(d*x)^(6 + m)*Sqrt[c*x^2])/(d^6*(6 + 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 \left (c x^2\right )^{3/2} (a+b x)^2 \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^3 (d x)^m (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int (d x)^{3+m} (a+b x)^2 \, dx}{d^3 x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a^2 (d x)^{3+m}+\frac {2 a b (d x)^{4+m}}{d}+\frac {b^2 (d x)^{5+m}}{d^2}\right ) \, dx}{d^3 x}\\ &=\frac {a^2 c (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x}+\frac {2 a b c (d x)^{5+m} \sqrt {c x^2}}{d^5 (5+m) x}+\frac {b^2 c (d x)^{6+m} \sqrt {c x^2}}{d^6 (6+m) x}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 48, normalized size = 0.49 \begin {gather*} x \left (c x^2\right )^{3/2} (d x)^m \left (\frac {a^2}{m+4}+\frac {2 a b x}{m+5}+\frac {b^2 x^2}{m+6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (c x^2\right )^{3/2} (a+b x)^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.42, size = 105, normalized size = 1.08 \begin {gather*} \frac {{\left ({\left (b^{2} c m^{2} + 9 \, b^{2} c m + 20 \, b^{2} c\right )} x^{5} + 2 \, {\left (a b c m^{2} + 10 \, a b c m + 24 \, a b c\right )} x^{4} + {\left (a^{2} c m^{2} + 11 \, a^{2} c m + 30 \, a^{2} c\right )} x^{3}\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 15 \, m^{2} + 74 \, m + 120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*c*m^2 + 9*b^2*c*m + 20*b^2*c)*x^5 + 2*(a*b*c*m^2 + 10*a*b*c*m + 24*a*b*c)*x^4 + (a^2*c*m^2 + 11*a^2*c*m
+ 30*a^2*c)*x^3)*sqrt(c*x^2)*(d*x)^m/(m^3 + 15*m^2 + 74*m + 120)

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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)^(3/2)*(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 = 0.98 \begin {gather*} \frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +9 b^{2} m \,x^{2}+a^{2} m^{2}+20 a b m x +20 b^{2} x^{2}+11 a^{2} m +48 a b x +30 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {3}{2}} x \left (d x \right )^{m}}{\left (m +6\right ) \left (m +5\right ) \left (m +4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(b^2*m^2*x^2+2*a*b*m^2*x+9*b^2*m*x^2+a^2*m^2+20*a*b*m*x+20*b^2*x^2+11*a^2*m+48*a*b*x+30*a^2)*(d*x)^m*(c*x^2)
^(3/2)/(m+6)/(m+5)/(m+4)

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maxima [A]  time = 1.55, size = 64, normalized size = 0.66 \begin {gather*} \frac {b^{2} c^{\frac {3}{2}} d^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, a b c^{\frac {3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac {a^{2} c^{\frac {3}{2}} d^{m} x^{4} x^{m}}{m + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.28, size = 121, normalized size = 1.25 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {a^2\,c\,x^3\,\sqrt {c\,x^2}\,\left (m^2+11\,m+30\right )}{m^3+15\,m^2+74\,m+120}+\frac {b^2\,c\,x^5\,\sqrt {c\,x^2}\,\left (m^2+9\,m+20\right )}{m^3+15\,m^2+74\,m+120}+\frac {2\,a\,b\,c\,x^4\,\sqrt {c\,x^2}\,\left (m^2+10\,m+24\right )}{m^3+15\,m^2+74\,m+120}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x)^m*((a^2*c*x^3*(c*x^2)^(1/2)*(11*m + m^2 + 30))/(74*m + 15*m^2 + m^3 + 120) + (b^2*c*x^5*(c*x^2)^(1/2)*(9
*m + m^2 + 20))/(74*m + 15*m^2 + m^3 + 120) + (2*a*b*c*x^4*(c*x^2)^(1/2)*(10*m + m^2 + 24))/(74*m + 15*m^2 + m
^3 + 120))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {\int \frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx + \int \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx + \int \frac {2 a b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx}{d^{6}} & \text {for}\: m = -6 \\\frac {\int \frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx + \int \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx + \int \frac {2 a b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx}{d^{5}} & \text {for}\: m = -5 \\\frac {\int \frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx + \int \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx + \int \frac {2 a b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx}{d^{4}} & \text {for}\: m = -4 \\\frac {a^{2} c^{\frac {3}{2}} d^{m} m^{2} x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {11 a^{2} c^{\frac {3}{2}} d^{m} m x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {30 a^{2} c^{\frac {3}{2}} d^{m} x x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {2 a b c^{\frac {3}{2}} d^{m} m^{2} x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {20 a b c^{\frac {3}{2}} d^{m} m x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {48 a b c^{\frac {3}{2}} d^{m} x^{2} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {b^{2} c^{\frac {3}{2}} d^{m} m^{2} x^{3} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {9 b^{2} c^{\frac {3}{2}} d^{m} m x^{3} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} + \frac {20 b^{2} c^{\frac {3}{2}} d^{m} x^{3} x^{m} \left (x^{2}\right )^{\frac {3}{2}}}{m^{3} + 15 m^{2} + 74 m + 120} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(c*x**2)**(3/2)*(b*x+a)**2,x)

[Out]

Piecewise(((Integral(a**2*(c*x**2)**(3/2)/x**6, x) + Integral(b**2*(c*x**2)**(3/2)/x**4, x) + Integral(2*a*b*(
c*x**2)**(3/2)/x**5, x))/d**6, Eq(m, -6)), ((Integral(a**2*(c*x**2)**(3/2)/x**5, x) + Integral(b**2*(c*x**2)**
(3/2)/x**3, x) + Integral(2*a*b*(c*x**2)**(3/2)/x**4, x))/d**5, Eq(m, -5)), ((Integral(a**2*(c*x**2)**(3/2)/x*
*4, x) + Integral(b**2*(c*x**2)**(3/2)/x**2, x) + Integral(2*a*b*(c*x**2)**(3/2)/x**3, x))/d**4, Eq(m, -4)), (
a**2*c**(3/2)*d**m*m**2*x*x**m*(x**2)**(3/2)/(m**3 + 15*m**2 + 74*m + 120) + 11*a**2*c**(3/2)*d**m*m*x*x**m*(x
**2)**(3/2)/(m**3 + 15*m**2 + 74*m + 120) + 30*a**2*c**(3/2)*d**m*x*x**m*(x**2)**(3/2)/(m**3 + 15*m**2 + 74*m
+ 120) + 2*a*b*c**(3/2)*d**m*m**2*x**2*x**m*(x**2)**(3/2)/(m**3 + 15*m**2 + 74*m + 120) + 20*a*b*c**(3/2)*d**m
*m*x**2*x**m*(x**2)**(3/2)/(m**3 + 15*m**2 + 74*m + 120) + 48*a*b*c**(3/2)*d**m*x**2*x**m*(x**2)**(3/2)/(m**3
+ 15*m**2 + 74*m + 120) + b**2*c**(3/2)*d**m*m**2*x**3*x**m*(x**2)**(3/2)/(m**3 + 15*m**2 + 74*m + 120) + 9*b*
*2*c**(3/2)*d**m*m*x**3*x**m*(x**2)**(3/2)/(m**3 + 15*m**2 + 74*m + 120) + 20*b**2*c**(3/2)*d**m*x**3*x**m*(x*
*2)**(3/2)/(m**3 + 15*m**2 + 74*m + 120), True))

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