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

Optimal. Leaf size=145 \[ \frac{1024 c^2 (b+2 c x) (2 c d-b e)}{35 b^8 \sqrt{b x+c x^2}}-\frac{128 c (b+2 c x) (2 c d-b e)}{35 b^6 \left (b x+c x^2\right )^{3/2}}+\frac{24 (b+2 c x) (2 c d-b e)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac{2 (x (2 c d-b e)+b d)}{7 b^2 \left (b x+c x^2\right )^{7/2}} \]

[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])

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Rubi [A]  time = 0.0448613, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {638, 614, 613} \[ \frac{1024 c^2 (b+2 c x) (2 c d-b e)}{35 b^8 \sqrt{b x+c x^2}}-\frac{128 c (b+2 c x) (2 c d-b e)}{35 b^6 \left (b x+c x^2\right )^{3/2}}+\frac{24 (b+2 c x) (2 c d-b e)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac{2 (x (2 c d-b e)+b d)}{7 b^2 \left (b x+c x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[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 638

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)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), 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]

Rule 614

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 613

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]

Rubi steps

\begin{align*} \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{(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]  time = 0.0613502, size = 150, normalized size = 1.03 \[ -\frac{2 \sqrt{x (b+c x)} \left (56 b^5 c^2 x^2 (d+5 e x)-560 b^4 c^3 x^3 (d-4 e x)+4480 b^3 c^4 x^4 (e x-d)+1792 b^2 c^5 x^5 (2 e x-5 d)-14 b^6 c x (d+2 e x)+b^7 (5 d+7 e x)+1024 b c^6 x^6 (e x-7 d)-2048 c^7 d x^7\right )}{35 b^8 x^4 (b+c x)^4} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.007, size = 180, normalized size = 1.2 \begin{align*} -{\frac{2\,x \left ( cx+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}ce{x}^{2}+56\,{b}^{5}{c}^{2}d{x}^{2}+7\,{b}^{7}ex-14\,{b}^{6}cdx+5\,d{b}^{7} \right ) }{35\,{b}^{8}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [B]  time = 1.07145, size = 394, normalized size = 2.72 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.98347, size = 463, normalized size = 3.19 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x}{\left (x \left (b + c x\right )\right )^{\frac{9}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.18417, size = 309, normalized size = 2.13 \begin{align*} \frac{{\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} c^{4}} + \frac{7 \,{\left (2 \, b c^{6} d - b^{2} c^{5} e\right )}}{b^{8} c^{4}}\right )} + \frac{35 \,{\left (2 \, b^{2} c^{5} d - b^{3} c^{4} e\right )}}{b^{8} c^{4}}\right )} x + \frac{35 \,{\left (2 \, b^{3} c^{4} d - b^{4} c^{3} e\right )}}{b^{8} c^{4}}\right )} x + \frac{35 \,{\left (2 \, b^{4} c^{3} d - b^{5} c^{2} e\right )}}{b^{8} c^{4}}\right )} x - \frac{7 \,{\left (2 \, b^{5} c^{2} d - b^{6} c e\right )}}{b^{8} c^{4}}\right )} x + \frac{7 \,{\left (2 \, b^{6} c d - b^{7} e\right )}}{b^{8} c^{4}}\right )} x - \frac{5 \, d}{b c^{4}}}{105 \,{\left (c x^{2} + b x\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*((4*(2*(8*(2*(4*x*(2*(2*c^7*d - b*c^6*e)*x/(b^8*c^4) + 7*(2*b*c^6*d - b^2*c^5*e)/(b^8*c^4)) + 35*(2*b^2*
c^5*d - b^3*c^4*e)/(b^8*c^4))*x + 35*(2*b^3*c^4*d - b^4*c^3*e)/(b^8*c^4))*x + 35*(2*b^4*c^3*d - b^5*c^2*e)/(b^
8*c^4))*x - 7*(2*b^5*c^2*d - b^6*c*e)/(b^8*c^4))*x + 7*(2*b^6*c*d - b^7*e)/(b^8*c^4))*x - 5*d/(b*c^4))/(c*x^2
+ b*x)^(7/2)