\(\int \frac {d+e x}{(b x+c x^2)^{9/2}} \, dx\) [139]

Optimal result

Integrand size = 19, antiderivative size = 145 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=-\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}+\frac {24 (2 c d-b e) (b+2 c x)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{35 b^6 \left (b x+c x^2\right )^{3/2}}+\frac {1024 c^2 (2 c d-b e) (b+2 c x)}{35 b^8 \sqrt {b x+c x^2}} \]

[Out]

-2/7*(b*d+(-b*e+2*c*d)*x)/b^2/(c*x^2+b*x)^(7/2)+24/35*(-b*e+2*c*d)*(2*c*x+b)/b^4/(c*x^2+b*x)^(5/2)-128/35*c*(-
b*e+2*c*d)*(2*c*x+b)/b^6/(c*x^2+b*x)^(3/2)+1024/35*c^2*(-b*e+2*c*d)*(2*c*x+b)/b^8/(c*x^2+b*x)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {652, 628, 627} \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=\frac {1024 c^2 (b+2 c x) (2 c d-b e)}{35 b^8 \sqrt {b x+c x^2}}-\frac {128 c (b+2 c x) (2 c d-b e)}{35 b^6 \left (b x+c x^2\right )^{3/2}}+\frac {24 (b+2 c x) (2 c d-b e)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac {2 (x (2 c d-b e)+b d)}{7 b^2 \left (b x+c x^2\right )^{7/2}} \]

[In]

Int[(d + e*x)/(b*x + c*x^2)^(9/2),x]

[Out]

(-2*(b*d + (2*c*d - b*e)*x))/(7*b^2*(b*x + c*x^2)^(7/2)) + (24*(2*c*d - b*e)*(b + 2*c*x))/(35*b^4*(b*x + c*x^2
)^(5/2)) - (128*c*(2*c*d - b*e)*(b + 2*c*x))/(35*b^6*(b*x + c*x^2)^(3/2)) + (1024*c^2*(2*c*d - b*e)*(b + 2*c*x
))/(35*b^8*Sqrt[b*x + c*x^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}-\frac {(12 (2 c d-b e)) \int \frac {1}{\left (b x+c x^2\right )^{7/2}} \, dx}{7 b^2} \\ & = -\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}+\frac {24 (2 c d-b e) (b+2 c x)}{35 b^4 \left (b x+c x^2\right )^{5/2}}+\frac {(192 c (2 c d-b e)) \int \frac {1}{\left (b x+c x^2\right )^{5/2}} \, dx}{35 b^4} \\ & = -\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}+\frac {24 (2 c d-b e) (b+2 c x)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{35 b^6 \left (b x+c x^2\right )^{3/2}}-\frac {\left (512 c^2 (2 c d-b e)\right ) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{35 b^6} \\ & = -\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}+\frac {24 (2 c d-b e) (b+2 c x)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{35 b^6 \left (b x+c x^2\right )^{3/2}}+\frac {1024 c^2 (2 c d-b e) (b+2 c x)}{35 b^8 \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=-\frac {2 \sqrt {x (b+c x)} \left (-2048 c^7 d x^7-560 b^4 c^3 x^3 (d-4 e x)+1024 b c^6 x^6 (-7 d+e x)+4480 b^3 c^4 x^4 (-d+e x)+1792 b^2 c^5 x^5 (-5 d+2 e x)-14 b^6 c x (d+2 e x)+56 b^5 c^2 x^2 (d+5 e x)+b^7 (5 d+7 e x)\right )}{35 b^8 x^4 (b+c x)^4} \]

[In]

Integrate[(d + e*x)/(b*x + c*x^2)^(9/2),x]

[Out]

(-2*Sqrt[x*(b + c*x)]*(-2048*c^7*d*x^7 - 560*b^4*c^3*x^3*(d - 4*e*x) + 1024*b*c^6*x^6*(-7*d + e*x) + 4480*b^3*
c^4*x^4*(-d + e*x) + 1792*b^2*c^5*x^5*(-5*d + 2*e*x) - 14*b^6*c*x*(d + 2*e*x) + 56*b^5*c^2*x^2*(d + 5*e*x) + b
^7*(5*d + 7*e*x)))/(35*b^8*x^4*(b + c*x)^4)

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.24

method	result	size
gosper	\(-\frac {2 x \left (c x +b \right ) \left (1024 b \,c^{6} e \,x^{7}-2048 c^{7} d \,x^{7}+3584 b^{2} c^{5} e \,x^{6}-7168 b \,c^{6} d \,x^{6}+4480 b^{3} c^{4} e \,x^{5}-8960 b^{2} c^{5} d \,x^{5}+2240 b^{4} c^{3} e \,x^{4}-4480 b^{3} c^{4} d \,x^{4}+280 b^{5} c^{2} e \,x^{3}-560 b^{4} c^{3} d \,x^{3}-28 b^{6} c e \,x^{2}+56 b^{5} c^{2} d \,x^{2}+7 b^{7} e x -14 b^{6} c d x +5 d \,b^{7}\right )}{35 b^{8} \left (c \,x^{2}+b x \right )^{\frac {9}{2}}}\)	\(180\)
trager	\(-\frac {2 \left (1024 b \,c^{6} e \,x^{7}-2048 c^{7} d \,x^{7}+3584 b^{2} c^{5} e \,x^{6}-7168 b \,c^{6} d \,x^{6}+4480 b^{3} c^{4} e \,x^{5}-8960 b^{2} c^{5} d \,x^{5}+2240 b^{4} c^{3} e \,x^{4}-4480 b^{3} c^{4} d \,x^{4}+280 b^{5} c^{2} e \,x^{3}-560 b^{4} c^{3} d \,x^{3}-28 b^{6} c e \,x^{2}+56 b^{5} c^{2} d \,x^{2}+7 b^{7} e x -14 b^{6} c d x +5 d \,b^{7}\right ) \sqrt {c \,x^{2}+b x}}{35 b^{8} \left (c x +b \right )^{4} x^{4}}\)	\(184\)
risch	\(-\frac {2 \left (c x +b \right ) \left (462 b \,c^{2} e \,x^{3}-1024 c^{3} d \,x^{3}-56 b^{2} c e \,x^{2}+162 b \,c^{2} d \,x^{2}+7 b^{3} e x -34 b^{2} c d x +5 d \,b^{3}\right )}{35 b^{8} x^{3} \sqrt {x \left (c x +b \right )}}-\frac {2 c^{3} \left (562 x^{3} b \,c^{3} e -1024 x^{3} c^{4} d +1792 x^{2} b^{2} c^{2} e -3234 x^{2} b \,c^{3} d +1925 x \,b^{3} c e -3430 x \,b^{2} c^{2} d +700 b^{4} e -1225 b^{3} c d \right ) x}{35 \sqrt {x \left (c x +b \right )}\, \left (c^{3} x^{3}+3 c^{2} b \,x^{2}+3 c \,b^{2} x +b^{3}\right ) b^{8}}\)	\(205\)
default	\(d \left (-\frac {2 \left (2 c x +b \right )}{7 b^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}-\frac {24 c \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{7 b^{2}}\right )+e \left (-\frac {1}{7 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{7 b^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}-\frac {24 c \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{7 b^{2}}\right )}{2 c}\right )\)	\(237\)

[In]

int((e*x+d)/(c*x^2+b*x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/35*x*(c*x+b)*(1024*b*c^6*e*x^7-2048*c^7*d*x^7+3584*b^2*c^5*e*x^6-7168*b*c^6*d*x^6+4480*b^3*c^4*e*x^5-8960*b
^2*c^5*d*x^5+2240*b^4*c^3*e*x^4-4480*b^3*c^4*d*x^4+280*b^5*c^2*e*x^3-560*b^4*c^3*d*x^3-28*b^6*c*e*x^2+56*b^5*c
^2*d*x^2+7*b^7*e*x-14*b^6*c*d*x+5*b^7*d)/b^8/(c*x^2+b*x)^(9/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.54 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=-\frac {2 \, {\left (5 \, b^{7} d - 1024 \, {\left (2 \, c^{7} d - b c^{6} e\right )} x^{7} - 3584 \, {\left (2 \, b c^{6} d - b^{2} c^{5} e\right )} x^{6} - 4480 \, {\left (2 \, b^{2} c^{5} d - b^{3} c^{4} e\right )} x^{5} - 2240 \, {\left (2 \, b^{3} c^{4} d - b^{4} c^{3} e\right )} x^{4} - 280 \, {\left (2 \, b^{4} c^{3} d - b^{5} c^{2} e\right )} x^{3} + 28 \, {\left (2 \, b^{5} c^{2} d - b^{6} c e\right )} x^{2} - 7 \, {\left (2 \, b^{6} c d - b^{7} e\right )} x\right )} \sqrt {c x^{2} + b x}}{35 \, {\left (b^{8} c^{4} x^{8} + 4 \, b^{9} c^{3} x^{7} + 6 \, b^{10} c^{2} x^{6} + 4 \, b^{11} c x^{5} + b^{12} x^{4}\right )}} \]

[In]

integrate((e*x+d)/(c*x^2+b*x)^(9/2),x, algorithm="fricas")

[Out]

-2/35*(5*b^7*d - 1024*(2*c^7*d - b*c^6*e)*x^7 - 3584*(2*b*c^6*d - b^2*c^5*e)*x^6 - 4480*(2*b^2*c^5*d - b^3*c^4
*e)*x^5 - 2240*(2*b^3*c^4*d - b^4*c^3*e)*x^4 - 280*(2*b^4*c^3*d - b^5*c^2*e)*x^3 + 28*(2*b^5*c^2*d - b^6*c*e)*
x^2 - 7*(2*b^6*c*d - b^7*e)*x)*sqrt(c*x^2 + b*x)/(b^8*c^4*x^8 + 4*b^9*c^3*x^7 + 6*b^10*c^2*x^6 + 4*b^11*c*x^5
+ b^12*x^4)

Sympy [F]

\[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=\int \frac {d + e x}{\left (x \left (b + c x\right )\right )^{\frac {9}{2}}}\, dx \]

[In]

integrate((e*x+d)/(c*x**2+b*x)**(9/2),x)

[Out]

Integral((d + e*x)/(x*(b + c*x))**(9/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (129) = 258\).

Time = 0.19 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.01 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=-\frac {4 \, c d x}{7 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b^{2}} + \frac {96 \, c^{2} d x}{35 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{4}} - \frac {512 \, c^{3} d x}{35 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{6}} + \frac {4096 \, c^{4} d x}{35 \, \sqrt {c x^{2} + b x} b^{8}} + \frac {2 \, e x}{7 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b} - \frac {48 \, c e x}{35 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3}} + \frac {256 \, c^{2} e x}{35 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5}} - \frac {2048 \, c^{3} e x}{35 \, \sqrt {c x^{2} + b x} b^{7}} - \frac {2 \, d}{7 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b} + \frac {48 \, c d}{35 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3}} - \frac {256 \, c^{2} d}{35 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5}} + \frac {2048 \, c^{3} d}{35 \, \sqrt {c x^{2} + b x} b^{7}} - \frac {24 \, e}{35 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2}} + \frac {128 \, c e}{35 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4}} - \frac {1024 \, c^{2} e}{35 \, \sqrt {c x^{2} + b x} b^{6}} \]

[In]

integrate((e*x+d)/(c*x^2+b*x)^(9/2),x, algorithm="maxima")

[Out]

-4/7*c*d*x/((c*x^2 + b*x)^(7/2)*b^2) + 96/35*c^2*d*x/((c*x^2 + b*x)^(5/2)*b^4) - 512/35*c^3*d*x/((c*x^2 + b*x)
^(3/2)*b^6) + 4096/35*c^4*d*x/(sqrt(c*x^2 + b*x)*b^8) + 2/7*e*x/((c*x^2 + b*x)^(7/2)*b) - 48/35*c*e*x/((c*x^2
+ b*x)^(5/2)*b^3) + 256/35*c^2*e*x/((c*x^2 + b*x)^(3/2)*b^5) - 2048/35*c^3*e*x/(sqrt(c*x^2 + b*x)*b^7) - 2/7*d
/((c*x^2 + b*x)^(7/2)*b) + 48/35*c*d/((c*x^2 + b*x)^(5/2)*b^3) - 256/35*c^2*d/((c*x^2 + b*x)^(3/2)*b^5) + 2048
/35*c^3*d/(sqrt(c*x^2 + b*x)*b^7) - 24/35*e/((c*x^2 + b*x)^(5/2)*b^2) + 128/35*c*e/((c*x^2 + b*x)^(3/2)*b^4) -
1024/35*c^2*e/(sqrt(c*x^2 + b*x)*b^6)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.37 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, x {\left (\frac {2 \, {\left (2 \, c^{7} d - b c^{6} e\right )} x}{b^{8}} + \frac {7 \, {\left (2 \, b c^{6} d - b^{2} c^{5} e\right )}}{b^{8}}\right )} + \frac {35 \, {\left (2 \, b^{2} c^{5} d - b^{3} c^{4} e\right )}}{b^{8}}\right )} x + \frac {35 \, {\left (2 \, b^{3} c^{4} d - b^{4} c^{3} e\right )}}{b^{8}}\right )} x + \frac {35 \, {\left (2 \, b^{4} c^{3} d - b^{5} c^{2} e\right )}}{b^{8}}\right )} x - \frac {7 \, {\left (2 \, b^{5} c^{2} d - b^{6} c e\right )}}{b^{8}}\right )} x + \frac {7 \, {\left (2 \, b^{6} c d - b^{7} e\right )}}{b^{8}}\right )} x - \frac {5 \, d}{b}\right )}}{35 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}}} \]

[In]

integrate((e*x+d)/(c*x^2+b*x)^(9/2),x, algorithm="giac")

[Out]

2/35*((4*(2*(8*(2*(4*x*(2*(2*c^7*d - b*c^6*e)*x/b^8 + 7*(2*b*c^6*d - b^2*c^5*e)/b^8) + 35*(2*b^2*c^5*d - b^3*c
^4*e)/b^8)*x + 35*(2*b^3*c^4*d - b^4*c^3*e)/b^8)*x + 35*(2*b^4*c^3*d - b^5*c^2*e)/b^8)*x - 7*(2*b^5*c^2*d - b^
6*c*e)/b^8)*x + 7*(2*b^6*c*d - b^7*e)/b^8)*x - 5*d/b)/(c*x^2 + b*x)^(7/2)

Mupad [B] (verification not implemented)

Time = 10.48 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=\frac {\frac {2048\,c^3\,d-1024\,b\,c^2\,e}{35\,b^7}+\frac {2\,c\,x\,\left (2048\,c^3\,d-1024\,b\,c^2\,e\right )}{35\,b^8}}{\sqrt {c\,x^2+b\,x}}-\frac {\frac {256\,c^2\,d-128\,b\,c\,e}{35\,b^5}+\frac {2\,c\,x\,\left (256\,c^2\,d-128\,b\,c\,e\right )}{35\,b^6}}{{\left (c\,x^2+b\,x\right )}^{3/2}}-\frac {\frac {2\,d}{7\,b}-x\,\left (\frac {2\,e}{7\,b}-\frac {4\,c\,d}{7\,b^2}\right )}{{\left (c\,x^2+b\,x\right )}^{7/2}}-\frac {\frac {24\,b\,e-48\,c\,d}{35\,b^3}+\frac {2\,c\,x\,\left (24\,b\,e-48\,c\,d\right )}{35\,b^4}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \]

[In]

int((d + e*x)/(b*x + c*x^2)^(9/2),x)

[Out]

((2048*c^3*d - 1024*b*c^2*e)/(35*b^7) + (2*c*x*(2048*c^3*d - 1024*b*c^2*e))/(35*b^8))/(b*x + c*x^2)^(1/2) - ((
256*c^2*d - 128*b*c*e)/(35*b^5) + (2*c*x*(256*c^2*d - 128*b*c*e))/(35*b^6))/(b*x + c*x^2)^(3/2) - ((2*d)/(7*b)
- x*((2*e)/(7*b) - (4*c*d)/(7*b^2)))/(b*x + c*x^2)^(7/2) - ((24*b*e - 48*c*d)/(35*b^3) + (2*c*x*(24*b*e - 48*
c*d))/(35*b^4))/(b*x + c*x^2)^(5/2)