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

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

[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)

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Rubi [A]  time = 0.0446737, antiderivative size = 97, normalized size of antiderivative = 1., 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} \[ \frac{a^2 c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{2 a b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x}+\frac{b^2 c \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x} \]

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*}

Mathematica [A]  time = 0.0562466, size = 48, normalized size = 0.49 \[ x \left (c x^2\right )^{3/2} (d x)^m \left (\frac{a^2}{m+4}+\frac{2 a b x}{m+5}+\frac{b^2 x^2}{m+6}\right ) \]

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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Maple [A]  time = 0.004, size = 95, normalized size = 1. \begin{align*}{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x+9\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}+20\,abmx+20\,{b}^{2}{x}^{2}+11\,{a}^{2}m+48\,abx+30\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

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)/(6+m)/(5+m)/(4+m)

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Maxima [A]  time = 1.06926, size = 86, normalized size = 0.89 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.23126, size = 238, normalized size = 2.45 \begin{align*} \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{align*}

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

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]

Exception raised: TypeError

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

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