\(\int \frac {(d+e x)^m}{(c d^2+2 c d e x+c e^2 x^2)^{3/2}} \, dx\) [1097]

Optimal result

Integrand size = 32, antiderivative size = 45 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {(d+e x)^{1+m}}{e (2-m) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[Out]

-(e*x+d)^(1+m)/e/(2-m)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {658, 32} \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=-\frac {(d+e x)^{m+1}}{e (2-m) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]

[In]

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

[Out]

-((d + e*x)^(1 + m)/(e*(2 - m)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 658

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^p/(d
+ e*x)^(2*p), Int[(d + e*x)^(m + 2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !
IntegerQ[p] && EqQ[2*c*d - b*e, 0] &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 \int (d+e x)^{-3+m} \, dx}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \\ & = -\frac {(d+e x)^{1+m}}{e (2-m) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.69 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {(d+e x)^{1+m}}{e (-2+m) \left (c (d+e x)^2\right )^{3/2}} \]

[In]

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

[Out]

(d + e*x)^(1 + m)/(e*(-2 + m)*(c*(d + e*x)^2)^(3/2))

Maple [A] (verified)

Time = 2.47 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {\left (e x +d \right )^{m}}{c \left (e x +d \right ) \sqrt {c \left (e x +d \right )^{2}}\, e \left (-2+m \right )}\)	\(38\)
gosper	\(\frac {\left (e x +d \right )^{1+m}}{e \left (-2+m \right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}\)	\(41\)

[In]

int((e*x+d)^m/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/c/(e*x+d)/(c*(e*x+d)^2)^(1/2)/e/(-2+m)*(e*x+d)^m

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (40) = 80\).

Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.71 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (e x + d\right )}^{m}}{c^{2} d^{3} e m - 2 \, c^{2} d^{3} e + {\left (c^{2} e^{4} m - 2 \, c^{2} e^{4}\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} m - 2 \, c^{2} d e^{3}\right )} x^{2} + 3 \, {\left (c^{2} d^{2} e^{2} m - 2 \, c^{2} d^{2} e^{2}\right )} x} \]

[In]

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

[Out]

sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^m/(c^2*d^3*e*m - 2*c^2*d^3*e + (c^2*e^4*m - 2*c^2*e^4)*x^3 + 3*(
c^2*d*e^3*m - 2*c^2*d*e^3)*x^2 + 3*(c^2*d^2*e^2*m - 2*c^2*d^2*e^2)*x)

Sympy [F]

\[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{m}}{\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral((d + e*x)**m/(c*(d + e*x)**2)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {{\left (e x + d\right )}^{m} \sqrt {c}}{c^{2} e^{3} {\left (m - 2\right )} x^{2} + 2 \, c^{2} d e^{2} {\left (m - 2\right )} x + c^{2} d^{2} e {\left (m - 2\right )}} \]

[In]

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

[Out]

(e*x + d)^m*sqrt(c)/(c^2*e^3*(m - 2)*x^2 + 2*c^2*d*e^2*(m - 2)*x + c^2*d^2*e*(m - 2))

Giac [F]

\[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x+d)^m/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^m/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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