\(\int (d x)^m (c x^2)^{5/2} (a+b x) \, dx\) [969]

Optimal result

Integrand size = 20, antiderivative size = 65 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {a c^2 (d x)^{6+m} \sqrt {c x^2}}{d^6 (6+m) x}+\frac {b c^2 (d x)^{7+m} \sqrt {c x^2}}{d^7 (7+m) x} \]

[Out]

a*c^2*(d*x)^(6+m)*(c*x^2)^(1/2)/d^6/(6+m)/x+b*c^2*(d*x)^(7+m)*(c*x^2)^(1/2)/d^7/(7+m)/x

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {15, 16, 45} \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {a c^2 \sqrt {c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac {b c^2 \sqrt {c x^2} (d x)^{m+7}}{d^7 (m+7) x} \]

[In]

Int[(d*x)^m*(c*x^2)^(5/2)*(a + b*x),x]

[Out]

(a*c^2*(d*x)^(6 + m)*Sqrt[c*x^2])/(d^6*(6 + m)*x) + (b*c^2*(d*x)^(7 + m)*Sqrt[c*x^2])/(d^7*(7 + 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 45

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*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int x^5 (d x)^m (a+b x) \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int (d x)^{5+m} (a+b x) \, dx}{d^5 x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a (d x)^{5+m}+\frac {b (d x)^{6+m}}{d}\right ) \, dx}{d^5 x} \\ & = \frac {a c^2 (d x)^{6+m} \sqrt {c x^2}}{d^6 (6+m) x}+\frac {b c^2 (d x)^{7+m} \sqrt {c x^2}}{d^7 (7+m) x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {x (d x)^m \left (c x^2\right )^{5/2} (a (7+m)+b (6+m) x)}{(6+m) (7+m)} \]

[In]

Integrate[(d*x)^m*(c*x^2)^(5/2)*(a + b*x),x]

[Out]

(x*(d*x)^m*(c*x^2)^(5/2)*(a*(7 + m) + b*(6 + m)*x))/((6 + m)*(7 + m))

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.62

method	result	size
gosper	\(\frac {x \left (b m x +a m +6 b x +7 a \right ) \left (d x \right )^{m} \left (c \,x^{2}\right )^{\frac {5}{2}}}{\left (7+m \right ) \left (6+m \right )}\)	\(40\)
risch	\(\frac {c^{2} x^{5} \sqrt {c \,x^{2}}\, \left (b m x +a m +6 b x +7 a \right ) \left (d x \right )^{m}}{\left (7+m \right ) \left (6+m \right )}\)	\(45\)

[In]

int((d*x)^m*(c*x^2)^(5/2)*(b*x+a),x,method=_RETURNVERBOSE)

[Out]

x*(b*m*x+a*m+6*b*x+7*a)*(d*x)^m*(c*x^2)^(5/2)/(7+m)/(6+m)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {{\left ({\left (b c^{2} m + 6 \, b c^{2}\right )} x^{6} + {\left (a c^{2} m + 7 \, a c^{2}\right )} x^{5}\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{2} + 13 \, m + 42} \]

[In]

integrate((d*x)^m*(c*x^2)^(5/2)*(b*x+a),x, algorithm="fricas")

[Out]

((b*c^2*m + 6*b*c^2)*x^6 + (a*c^2*m + 7*a*c^2)*x^5)*sqrt(c*x^2)*(d*x)^m/(m^2 + 13*m + 42)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (56) = 112\).

Time = 6.58 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.77 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x) \, dx=\begin {cases} \frac {- \frac {a \left (c x^{2}\right )^{\frac {5}{2}}}{x^{6}} + \frac {b \left (c x^{2}\right )^{\frac {5}{2}} \log {\left (x \right )}}{x^{5}}}{d^{7}} & \text {for}\: m = -7 \\\frac {\frac {a \left (c x^{2}\right )^{\frac {5}{2}} \log {\left (x \right )}}{x^{5}} + \frac {b \left (c x^{2}\right )^{\frac {5}{2}}}{x^{4}}}{d^{6}} & \text {for}\: m = -6 \\\frac {a m x \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{2} + 13 m + 42} + \frac {7 a x \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{2} + 13 m + 42} + \frac {b m x^{2} \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{2} + 13 m + 42} + \frac {6 b x^{2} \left (c x^{2}\right )^{\frac {5}{2}} \left (d x\right )^{m}}{m^{2} + 13 m + 42} & \text {otherwise} \end {cases} \]

[In]

integrate((d*x)**m*(c*x**2)**(5/2)*(b*x+a),x)

[Out]

Piecewise(((-a*(c*x**2)**(5/2)/x**6 + b*(c*x**2)**(5/2)*log(x)/x**5)/d**7, Eq(m, -7)), ((a*(c*x**2)**(5/2)*log
(x)/x**5 + b*(c*x**2)**(5/2)/x**4)/d**6, Eq(m, -6)), (a*m*x*(c*x**2)**(5/2)*(d*x)**m/(m**2 + 13*m + 42) + 7*a*
x*(c*x**2)**(5/2)*(d*x)**m/(m**2 + 13*m + 42) + b*m*x**2*(c*x**2)**(5/2)*(d*x)**m/(m**2 + 13*m + 42) + 6*b*x**
2*(c*x**2)**(5/2)*(d*x)**m/(m**2 + 13*m + 42), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {b c^{\frac {5}{2}} d^{m} x^{7} x^{m}}{m + 7} + \frac {a c^{\frac {5}{2}} d^{m} x^{6} x^{m}}{m + 6} \]

[In]

integrate((d*x)^m*(c*x^2)^(5/2)*(b*x+a),x, algorithm="maxima")

[Out]

b*c^(5/2)*d^m*x^7*x^m/(m + 7) + a*c^(5/2)*d^m*x^6*x^m/(m + 6)

Giac [F(-2)]

Exception generated. \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*x)^m*(c*x^2)^(5/2)*(b*x+a),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.68 \[ \int (d x)^m \left (c x^2\right )^{5/2} (a+b x) \, dx=\frac {c^2\,x^5\,{\left (d\,x\right )}^m\,\sqrt {c\,x^2}\,\left (7\,a+a\,m+6\,b\,x+b\,m\,x\right )}{m^2+13\,m+42} \]

[In]

int((d*x)^m*(c*x^2)^(5/2)*(a + b*x),x)

[Out]

(c^2*x^5*(d*x)^m*(c*x^2)^(1/2)*(7*a + a*m + 6*b*x + b*m*x))/(13*m + m^2 + 42)