3.385 \(\int \frac{1}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=494 \[ \frac{7 b e+8 c d}{4 b^2 d^2 x (b+c x)^2 (d+e x)^{3/2}}-\frac{\left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{9/2}}+\frac{c^{9/2} \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}+\frac{c (2 c d-b e) \left (-7 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 (b+c x) (d+e x)^{3/2} (c d-b e)^2}+\frac{c \left (-7 b^2 e^2-3 b c d e+12 c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 (d+e x)^{3/2} (c d-b e)}+\frac{e (2 c d-b e) \left (-35 b^4 e^4+10 b^3 c d e^3+2 b^2 c^2 d^2 e^2-24 b c^3 d^3 e+12 c^4 d^4\right )}{4 b^4 d^4 \sqrt{d+e x} (c d-b e)^4}+\frac{e \left (-35 b^4 e^4+45 b^3 c d e^3+27 b^2 c^2 d^2 e^2-144 b c^3 d^3 e+72 c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{1}{2 b d x^2 (b+c x)^2 (d+e x)^{3/2}} \]

[Out]

(e*(72*c^4*d^4 - 144*b*c^3*d^3*e + 27*b^2*c^2*d^2*e^2 + 45*b^3*c*d*e^3 - 35*b^4*
e^4))/(12*b^4*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) + (c*(12*c^2*d^2 - 3*b*c*d*e -
7*b^2*e^2))/(4*b^3*d^2*(c*d - b*e)*(b + c*x)^2*(d + e*x)^(3/2)) - 1/(2*b*d*x^2*(
b + c*x)^2*(d + e*x)^(3/2)) + (8*c*d + 7*b*e)/(4*b^2*d^2*x*(b + c*x)^2*(d + e*x)
^(3/2)) + (c*(2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e - 7*b^2*e^2))/(4*b^4*d^2*(c*
d - b*e)^2*(b + c*x)*(d + e*x)^(3/2)) + (e*(2*c*d - b*e)*(12*c^4*d^4 - 24*b*c^3*
d^3*e + 2*b^2*c^2*d^2*e^2 + 10*b^3*c*d*e^3 - 35*b^4*e^4))/(4*b^4*d^4*(c*d - b*e)
^4*Sqrt[d + e*x]) - ((48*c^2*d^2 + 60*b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x
]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(9/2)*(48*c^2*d^2 - 156*b*c*d*e + 143*b^2*e^2)*
ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

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Rubi [A]  time = 2.64017, antiderivative size = 494, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{7 b e+8 c d}{4 b^2 d^2 x (b+c x)^2 (d+e x)^{3/2}}-\frac{\left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{9/2}}+\frac{c^{9/2} \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}+\frac{c (2 c d-b e) \left (-7 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 (b+c x) (d+e x)^{3/2} (c d-b e)^2}+\frac{c \left (-7 b^2 e^2-3 b c d e+12 c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 (d+e x)^{3/2} (c d-b e)}+\frac{e (2 c d-b e) \left (-35 b^4 e^4+10 b^3 c d e^3+2 b^2 c^2 d^2 e^2-24 b c^3 d^3 e+12 c^4 d^4\right )}{4 b^4 d^4 \sqrt{d+e x} (c d-b e)^4}+\frac{e \left (-35 b^4 e^4+45 b^3 c d e^3+27 b^2 c^2 d^2 e^2-144 b c^3 d^3 e+72 c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{1}{2 b d x^2 (b+c x)^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(e*(72*c^4*d^4 - 144*b*c^3*d^3*e + 27*b^2*c^2*d^2*e^2 + 45*b^3*c*d*e^3 - 35*b^4*
e^4))/(12*b^4*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) + (c*(12*c^2*d^2 - 3*b*c*d*e -
7*b^2*e^2))/(4*b^3*d^2*(c*d - b*e)*(b + c*x)^2*(d + e*x)^(3/2)) - 1/(2*b*d*x^2*(
b + c*x)^2*(d + e*x)^(3/2)) + (8*c*d + 7*b*e)/(4*b^2*d^2*x*(b + c*x)^2*(d + e*x)
^(3/2)) + (c*(2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e - 7*b^2*e^2))/(4*b^4*d^2*(c*
d - b*e)^2*(b + c*x)*(d + e*x)^(3/2)) + (e*(2*c*d - b*e)*(12*c^4*d^4 - 24*b*c^3*
d^3*e + 2*b^2*c^2*d^2*e^2 + 10*b^3*c*d*e^3 - 35*b^4*e^4))/(4*b^4*d^4*(c*d - b*e)
^4*Sqrt[d + e*x]) - ((48*c^2*d^2 + 60*b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x
]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(9/2)*(48*c^2*d^2 - 156*b*c*d*e + 143*b^2*e^2)*
ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.92626, size = 285, normalized size = 0.58 \[ \frac{1}{12} \left (-\frac{3 \left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^5 d^{9/2}}+\frac{3 c^{9/2} \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^5 (c d-b e)^{9/2}}+\sqrt{d+e x} \left (\frac{3 c^5 (12 c d-23 b e)}{b^4 (b+c x) (c d-b e)^4}+\frac{33 b e+36 c d}{b^4 d^4 x}-\frac{6 c^5}{b^3 (b+c x)^2 (b e-c d)^3}-\frac{6}{b^3 d^3 x^2}+\frac{72 e^5 (b e-2 c d)}{d^4 (d+e x) (c d-b e)^4}-\frac{8 e^5}{d^3 (d+e x)^2 (c d-b e)^3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(Sqrt[d + e*x]*(-6/(b^3*d^3*x^2) + (36*c*d + 33*b*e)/(b^4*d^4*x) - (6*c^5)/(b^3*
(-(c*d) + b*e)^3*(b + c*x)^2) + (3*c^5*(12*c*d - 23*b*e))/(b^4*(c*d - b*e)^4*(b
+ c*x)) - (8*e^5)/(d^3*(c*d - b*e)^3*(d + e*x)^2) + (72*e^5*(-2*c*d + b*e))/(d^4
*(c*d - b*e)^4*(d + e*x))) - (3*(48*c^2*d^2 + 60*b*c*d*e + 35*b^2*e^2)*ArcTanh[S
qrt[d + e*x]/Sqrt[d]])/(b^5*d^(9/2)) + (3*c^(9/2)*(48*c^2*d^2 - 156*b*c*d*e + 14
3*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^5*(c*d - b*e)^(9
/2)))/12

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Maple [A]  time = 0.049, size = 582, normalized size = 1.2 \[ 6\,{\frac{{e}^{6}b}{{d}^{4} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}-12\,{\frac{{e}^{5}c}{{d}^{3} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}+{\frac{2\,{e}^{5}}{3\,{d}^{3} \left ( be-cd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{23\,{e}^{2}{c}^{6}}{4\,{b}^{3} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{e{c}^{7} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}}}-{\frac{25\,{e}^{3}{c}^{5}}{4\,{b}^{2} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}+{\frac{37\,{e}^{2}{c}^{6}d}{4\,{b}^{3} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-3\,{\frac{e{c}^{7}\sqrt{ex+d}{d}^{2}}{{b}^{4} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}}}-{\frac{143\,{e}^{2}{c}^{5}}{4\,{b}^{3} \left ( be-cd \right ) ^{4}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+39\,{\frac{e{c}^{6}d}{{b}^{4} \left ( be-cd \right ) ^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-12\,{\frac{{c}^{7}{d}^{2}}{{b}^{5} \left ( be-cd \right ) ^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{11}{4\,{b}^{3}{d}^{4}{x}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}c}{e{b}^{4}{d}^{3}{x}^{2}}}-{\frac{13}{4\,{d}^{3}{b}^{3}{x}^{2}}\sqrt{ex+d}}-3\,{\frac{c\sqrt{ex+d}}{e{b}^{4}{d}^{2}{x}^{2}}}-{\frac{35\,{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{9}{2}}}}-15\,{\frac{ce}{{b}^{4}{d}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{c}^{2}}{{b}^{5}{d}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x)

[Out]

6*e^6/d^4/(b*e-c*d)^4/(e*x+d)^(1/2)*b-12*e^5/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*c+2/3
*e^5/d^3/(b*e-c*d)^3/(e*x+d)^(3/2)-23/4*e^2*c^6/b^3/(b*e-c*d)^4/(c*e*x+b*e)^2*(e
*x+d)^(3/2)+3*e*c^7/b^4/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d-25/4*e^3*c^5/b
^2/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)+37/4*e^2*c^6/b^3/(b*e-c*d)^4/(c*e*x+b
*e)^2*(e*x+d)^(1/2)*d-3*e*c^7/b^4/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^2-14
3/4*e^2*c^5/b^3/(b*e-c*d)^4/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d
)*c)^(1/2))+39*e*c^6/b^4/(b*e-c*d)^4/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/
((b*e-c*d)*c)^(1/2))*d-12*c^7/b^5/(b*e-c*d)^4/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+
d)^(1/2)/((b*e-c*d)*c)^(1/2))*d^2+11/4/b^3/d^4/x^2*(e*x+d)^(3/2)+3/e/b^4/d^3/x^2
*(e*x+d)^(3/2)*c-13/4/b^3/d^3/x^2*(e*x+d)^(1/2)-3/e/b^4/d^2/x^2*(e*x+d)^(1/2)*c-
35/4*e^2/b^3/d^(9/2)*arctanh((e*x+d)^(1/2)/d^(1/2))-15*e/b^4/d^(7/2)*arctanh((e*
x+d)^(1/2)/d^(1/2))*c-12/b^5/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 15.6469, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/24*(3*((48*c^8*d^6*e - 156*b*c^7*d^5*e^2 + 143*b^2*c^6*d^4*e^3)*x^5 + (48*c^8
*d^7 - 60*b*c^7*d^6*e - 169*b^2*c^6*d^5*e^2 + 286*b^3*c^5*d^4*e^3)*x^4 + (96*b*c
^7*d^7 - 264*b^2*c^6*d^6*e + 130*b^3*c^5*d^5*e^2 + 143*b^4*c^4*d^4*e^3)*x^3 + (4
8*b^2*c^6*d^7 - 156*b^3*c^5*d^6*e + 143*b^4*c^4*d^5*e^2)*x^2)*sqrt(e*x + d)*sqrt
(d)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*s
qrt(c/(c*d - b*e)))/(c*x + b)) + 3*((48*c^8*d^6*e - 132*b*c^7*d^5*e^2 + 83*b^2*c
^6*d^4*e^3 + 28*b^3*c^5*d^3*e^4 + 18*b^4*c^4*d^2*e^5 - 80*b^5*c^3*d*e^6 + 35*b^6
*c^2*e^7)*x^5 + (48*c^8*d^7 - 36*b*c^7*d^6*e - 181*b^2*c^6*d^5*e^2 + 194*b^3*c^5
*d^4*e^3 + 74*b^4*c^4*d^3*e^4 - 44*b^5*c^3*d^2*e^5 - 125*b^6*c^2*d*e^6 + 70*b^7*
c*e^7)*x^4 + (96*b*c^7*d^7 - 216*b^2*c^6*d^6*e + 34*b^3*c^5*d^5*e^2 + 139*b^4*c^
4*d^4*e^3 + 64*b^5*c^3*d^3*e^4 - 142*b^6*c^2*d^2*e^5 - 10*b^7*c*d*e^6 + 35*b^8*e
^7)*x^3 + (48*b^2*c^6*d^7 - 132*b^3*c^5*d^6*e + 83*b^4*c^4*d^5*e^2 + 28*b^5*c^3*
d^4*e^3 + 18*b^6*c^2*d^3*e^4 - 80*b^7*c*d^2*e^5 + 35*b^8*d*e^6)*x^2)*sqrt(e*x +
d)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x) - 2*(6*b^4*c^4*d^7 - 24*b^5*
c^3*d^6*e + 36*b^6*c^2*d^5*e^2 - 24*b^7*c*d^4*e^3 + 6*b^8*d^3*e^4 - 3*(24*b*c^7*
d^5*e^2 - 60*b^2*c^6*d^4*e^3 + 28*b^3*c^5*d^3*e^4 + 18*b^4*c^4*d^2*e^5 - 80*b^5*
c^3*d*e^6 + 35*b^6*c^2*e^7)*x^5 - (144*b*c^7*d^6*e - 252*b^2*c^6*d^5*e^2 - 105*b
^3*c^5*d^4*e^3 + 240*b^4*c^4*d^3*e^4 - 212*b^5*c^3*d^2*e^5 - 340*b^6*c^2*d*e^6 +
210*b^7*c*e^7)*x^4 - (72*b*c^7*d^7 + 36*b^2*c^6*d^6*e - 438*b^3*c^5*d^5*e^2 + 2
55*b^4*c^4*d^4*e^3 + 180*b^5*c^3*d^3*e^4 - 565*b^6*c^2*d^2*e^5 + 40*b^7*c*d*e^6
+ 105*b^8*e^7)*x^3 - (108*b^2*c^6*d^7 - 225*b^3*c^5*d^6*e + 180*b^5*c^3*d^4*e^3
- 30*b^6*c^2*d^3*e^4 - 278*b^7*c*d^2*e^5 + 140*b^8*d*e^6)*x^2 - 3*(8*b^3*c^5*d^7
- 25*b^4*c^4*d^6*e + 20*b^5*c^3*d^5*e^2 + 10*b^6*c^2*d^4*e^3 - 20*b^7*c*d^3*e^4
+ 7*b^8*d^2*e^5)*x)*sqrt(d))/(((b^5*c^6*d^8*e - 4*b^6*c^5*d^7*e^2 + 6*b^7*c^4*d
^6*e^3 - 4*b^8*c^3*d^5*e^4 + b^9*c^2*d^4*e^5)*x^5 + (b^5*c^6*d^9 - 2*b^6*c^5*d^8
*e - 2*b^7*c^4*d^7*e^2 + 8*b^8*c^3*d^6*e^3 - 7*b^9*c^2*d^5*e^4 + 2*b^10*c*d^4*e^
5)*x^4 + (2*b^6*c^5*d^9 - 7*b^7*c^4*d^8*e + 8*b^8*c^3*d^7*e^2 - 2*b^9*c^2*d^6*e^
3 - 2*b^10*c*d^5*e^4 + b^11*d^4*e^5)*x^3 + (b^7*c^4*d^9 - 4*b^8*c^3*d^8*e + 6*b^
9*c^2*d^7*e^2 - 4*b^10*c*d^6*e^3 + b^11*d^5*e^4)*x^2)*sqrt(e*x + d)*sqrt(d)), 1/
24*(6*((48*c^8*d^6*e - 156*b*c^7*d^5*e^2 + 143*b^2*c^6*d^4*e^3)*x^5 + (48*c^8*d^
7 - 60*b*c^7*d^6*e - 169*b^2*c^6*d^5*e^2 + 286*b^3*c^5*d^4*e^3)*x^4 + (96*b*c^7*
d^7 - 264*b^2*c^6*d^6*e + 130*b^3*c^5*d^5*e^2 + 143*b^4*c^4*d^4*e^3)*x^3 + (48*b
^2*c^6*d^7 - 156*b^3*c^5*d^6*e + 143*b^4*c^4*d^5*e^2)*x^2)*sqrt(e*x + d)*sqrt(d)
*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)
) + 3*((48*c^8*d^6*e - 132*b*c^7*d^5*e^2 + 83*b^2*c^6*d^4*e^3 + 28*b^3*c^5*d^3*e
^4 + 18*b^4*c^4*d^2*e^5 - 80*b^5*c^3*d*e^6 + 35*b^6*c^2*e^7)*x^5 + (48*c^8*d^7 -
36*b*c^7*d^6*e - 181*b^2*c^6*d^5*e^2 + 194*b^3*c^5*d^4*e^3 + 74*b^4*c^4*d^3*e^4
- 44*b^5*c^3*d^2*e^5 - 125*b^6*c^2*d*e^6 + 70*b^7*c*e^7)*x^4 + (96*b*c^7*d^7 -
216*b^2*c^6*d^6*e + 34*b^3*c^5*d^5*e^2 + 139*b^4*c^4*d^4*e^3 + 64*b^5*c^3*d^3*e^
4 - 142*b^6*c^2*d^2*e^5 - 10*b^7*c*d*e^6 + 35*b^8*e^7)*x^3 + (48*b^2*c^6*d^7 - 1
32*b^3*c^5*d^6*e + 83*b^4*c^4*d^5*e^2 + 28*b^5*c^3*d^4*e^3 + 18*b^6*c^2*d^3*e^4
- 80*b^7*c*d^2*e^5 + 35*b^8*d*e^6)*x^2)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) -
2*sqrt(e*x + d)*d)/x) - 2*(6*b^4*c^4*d^7 - 24*b^5*c^3*d^6*e + 36*b^6*c^2*d^5*e^
2 - 24*b^7*c*d^4*e^3 + 6*b^8*d^3*e^4 - 3*(24*b*c^7*d^5*e^2 - 60*b^2*c^6*d^4*e^3
+ 28*b^3*c^5*d^3*e^4 + 18*b^4*c^4*d^2*e^5 - 80*b^5*c^3*d*e^6 + 35*b^6*c^2*e^7)*x
^5 - (144*b*c^7*d^6*e - 252*b^2*c^6*d^5*e^2 - 105*b^3*c^5*d^4*e^3 + 240*b^4*c^4*
d^3*e^4 - 212*b^5*c^3*d^2*e^5 - 340*b^6*c^2*d*e^6 + 210*b^7*c*e^7)*x^4 - (72*b*c
^7*d^7 + 36*b^2*c^6*d^6*e - 438*b^3*c^5*d^5*e^2 + 255*b^4*c^4*d^4*e^3 + 180*b^5*
c^3*d^3*e^4 - 565*b^6*c^2*d^2*e^5 + 40*b^7*c*d*e^6 + 105*b^8*e^7)*x^3 - (108*b^2
*c^6*d^7 - 225*b^3*c^5*d^6*e + 180*b^5*c^3*d^4*e^3 - 30*b^6*c^2*d^3*e^4 - 278*b^
7*c*d^2*e^5 + 140*b^8*d*e^6)*x^2 - 3*(8*b^3*c^5*d^7 - 25*b^4*c^4*d^6*e + 20*b^5*
c^3*d^5*e^2 + 10*b^6*c^2*d^4*e^3 - 20*b^7*c*d^3*e^4 + 7*b^8*d^2*e^5)*x)*sqrt(d))
/(((b^5*c^6*d^8*e - 4*b^6*c^5*d^7*e^2 + 6*b^7*c^4*d^6*e^3 - 4*b^8*c^3*d^5*e^4 +
b^9*c^2*d^4*e^5)*x^5 + (b^5*c^6*d^9 - 2*b^6*c^5*d^8*e - 2*b^7*c^4*d^7*e^2 + 8*b^
8*c^3*d^6*e^3 - 7*b^9*c^2*d^5*e^4 + 2*b^10*c*d^4*e^5)*x^4 + (2*b^6*c^5*d^9 - 7*b
^7*c^4*d^8*e + 8*b^8*c^3*d^7*e^2 - 2*b^9*c^2*d^6*e^3 - 2*b^10*c*d^5*e^4 + b^11*d
^4*e^5)*x^3 + (b^7*c^4*d^9 - 4*b^8*c^3*d^8*e + 6*b^9*c^2*d^7*e^2 - 4*b^10*c*d^6*
e^3 + b^11*d^5*e^4)*x^2)*sqrt(e*x + d)*sqrt(d)), 1/24*(3*((48*c^8*d^6*e - 156*b*
c^7*d^5*e^2 + 143*b^2*c^6*d^4*e^3)*x^5 + (48*c^8*d^7 - 60*b*c^7*d^6*e - 169*b^2*
c^6*d^5*e^2 + 286*b^3*c^5*d^4*e^3)*x^4 + (96*b*c^7*d^7 - 264*b^2*c^6*d^6*e + 130
*b^3*c^5*d^5*e^2 + 143*b^4*c^4*d^4*e^3)*x^3 + (48*b^2*c^6*d^7 - 156*b^3*c^5*d^6*
e + 143*b^4*c^4*d^5*e^2)*x^2)*sqrt(e*x + d)*sqrt(-d)*sqrt(c/(c*d - b*e))*log((c*
e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b))
+ 6*((48*c^8*d^6*e - 132*b*c^7*d^5*e^2 + 83*b^2*c^6*d^4*e^3 + 28*b^3*c^5*d^3*e^4
+ 18*b^4*c^4*d^2*e^5 - 80*b^5*c^3*d*e^6 + 35*b^6*c^2*e^7)*x^5 + (48*c^8*d^7 - 3
6*b*c^7*d^6*e - 181*b^2*c^6*d^5*e^2 + 194*b^3*c^5*d^4*e^3 + 74*b^4*c^4*d^3*e^4 -
44*b^5*c^3*d^2*e^5 - 125*b^6*c^2*d*e^6 + 70*b^7*c*e^7)*x^4 + (96*b*c^7*d^7 - 21
6*b^2*c^6*d^6*e + 34*b^3*c^5*d^5*e^2 + 139*b^4*c^4*d^4*e^3 + 64*b^5*c^3*d^3*e^4
- 142*b^6*c^2*d^2*e^5 - 10*b^7*c*d*e^6 + 35*b^8*e^7)*x^3 + (48*b^2*c^6*d^7 - 132
*b^3*c^5*d^6*e + 83*b^4*c^4*d^5*e^2 + 28*b^5*c^3*d^4*e^3 + 18*b^6*c^2*d^3*e^4 -
80*b^7*c*d^2*e^5 + 35*b^8*d*e^6)*x^2)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*sqrt
(-d))) - 2*(6*b^4*c^4*d^7 - 24*b^5*c^3*d^6*e + 36*b^6*c^2*d^5*e^2 - 24*b^7*c*d^4
*e^3 + 6*b^8*d^3*e^4 - 3*(24*b*c^7*d^5*e^2 - 60*b^2*c^6*d^4*e^3 + 28*b^3*c^5*d^3
*e^4 + 18*b^4*c^4*d^2*e^5 - 80*b^5*c^3*d*e^6 + 35*b^6*c^2*e^7)*x^5 - (144*b*c^7*
d^6*e - 252*b^2*c^6*d^5*e^2 - 105*b^3*c^5*d^4*e^3 + 240*b^4*c^4*d^3*e^4 - 212*b^
5*c^3*d^2*e^5 - 340*b^6*c^2*d*e^6 + 210*b^7*c*e^7)*x^4 - (72*b*c^7*d^7 + 36*b^2*
c^6*d^6*e - 438*b^3*c^5*d^5*e^2 + 255*b^4*c^4*d^4*e^3 + 180*b^5*c^3*d^3*e^4 - 56
5*b^6*c^2*d^2*e^5 + 40*b^7*c*d*e^6 + 105*b^8*e^7)*x^3 - (108*b^2*c^6*d^7 - 225*b
^3*c^5*d^6*e + 180*b^5*c^3*d^4*e^3 - 30*b^6*c^2*d^3*e^4 - 278*b^7*c*d^2*e^5 + 14
0*b^8*d*e^6)*x^2 - 3*(8*b^3*c^5*d^7 - 25*b^4*c^4*d^6*e + 20*b^5*c^3*d^5*e^2 + 10
*b^6*c^2*d^4*e^3 - 20*b^7*c*d^3*e^4 + 7*b^8*d^2*e^5)*x)*sqrt(-d))/(((b^5*c^6*d^8
*e - 4*b^6*c^5*d^7*e^2 + 6*b^7*c^4*d^6*e^3 - 4*b^8*c^3*d^5*e^4 + b^9*c^2*d^4*e^5
)*x^5 + (b^5*c^6*d^9 - 2*b^6*c^5*d^8*e - 2*b^7*c^4*d^7*e^2 + 8*b^8*c^3*d^6*e^3 -
7*b^9*c^2*d^5*e^4 + 2*b^10*c*d^4*e^5)*x^4 + (2*b^6*c^5*d^9 - 7*b^7*c^4*d^8*e +
8*b^8*c^3*d^7*e^2 - 2*b^9*c^2*d^6*e^3 - 2*b^10*c*d^5*e^4 + b^11*d^4*e^5)*x^3 + (
b^7*c^4*d^9 - 4*b^8*c^3*d^8*e + 6*b^9*c^2*d^7*e^2 - 4*b^10*c*d^6*e^3 + b^11*d^5*
e^4)*x^2)*sqrt(e*x + d)*sqrt(-d)), 1/12*(3*((48*c^8*d^6*e - 156*b*c^7*d^5*e^2 +
143*b^2*c^6*d^4*e^3)*x^5 + (48*c^8*d^7 - 60*b*c^7*d^6*e - 169*b^2*c^6*d^5*e^2 +
286*b^3*c^5*d^4*e^3)*x^4 + (96*b*c^7*d^7 - 264*b^2*c^6*d^6*e + 130*b^3*c^5*d^5*e
^2 + 143*b^4*c^4*d^4*e^3)*x^3 + (48*b^2*c^6*d^7 - 156*b^3*c^5*d^6*e + 143*b^4*c^
4*d^5*e^2)*x^2)*sqrt(e*x + d)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*
sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) + 3*((48*c^8*d^6*e - 132*b*c^7*d^5*e^2 +
83*b^2*c^6*d^4*e^3 + 28*b^3*c^5*d^3*e^4 + 18*b^4*c^4*d^2*e^5 - 80*b^5*c^3*d*e^6
+ 35*b^6*c^2*e^7)*x^5 + (48*c^8*d^7 - 36*b*c^7*d^6*e - 181*b^2*c^6*d^5*e^2 + 19
4*b^3*c^5*d^4*e^3 + 74*b^4*c^4*d^3*e^4 - 44*b^5*c^3*d^2*e^5 - 125*b^6*c^2*d*e^6
+ 70*b^7*c*e^7)*x^4 + (96*b*c^7*d^7 - 216*b^2*c^6*d^6*e + 34*b^3*c^5*d^5*e^2 + 1
39*b^4*c^4*d^4*e^3 + 64*b^5*c^3*d^3*e^4 - 142*b^6*c^2*d^2*e^5 - 10*b^7*c*d*e^6 +
35*b^8*e^7)*x^3 + (48*b^2*c^6*d^7 - 132*b^3*c^5*d^6*e + 83*b^4*c^4*d^5*e^2 + 28
*b^5*c^3*d^4*e^3 + 18*b^6*c^2*d^3*e^4 - 80*b^7*c*d^2*e^5 + 35*b^8*d*e^6)*x^2)*sq
rt(e*x + d)*arctan(d/(sqrt(e*x + d)*sqrt(-d))) - (6*b^4*c^4*d^7 - 24*b^5*c^3*d^6
*e + 36*b^6*c^2*d^5*e^2 - 24*b^7*c*d^4*e^3 + 6*b^8*d^3*e^4 - 3*(24*b*c^7*d^5*e^2
- 60*b^2*c^6*d^4*e^3 + 28*b^3*c^5*d^3*e^4 + 18*b^4*c^4*d^2*e^5 - 80*b^5*c^3*d*e
^6 + 35*b^6*c^2*e^7)*x^5 - (144*b*c^7*d^6*e - 252*b^2*c^6*d^5*e^2 - 105*b^3*c^5*
d^4*e^3 + 240*b^4*c^4*d^3*e^4 - 212*b^5*c^3*d^2*e^5 - 340*b^6*c^2*d*e^6 + 210*b^
7*c*e^7)*x^4 - (72*b*c^7*d^7 + 36*b^2*c^6*d^6*e - 438*b^3*c^5*d^5*e^2 + 255*b^4*
c^4*d^4*e^3 + 180*b^5*c^3*d^3*e^4 - 565*b^6*c^2*d^2*e^5 + 40*b^7*c*d*e^6 + 105*b
^8*e^7)*x^3 - (108*b^2*c^6*d^7 - 225*b^3*c^5*d^6*e + 180*b^5*c^3*d^4*e^3 - 30*b^
6*c^2*d^3*e^4 - 278*b^7*c*d^2*e^5 + 140*b^8*d*e^6)*x^2 - 3*(8*b^3*c^5*d^7 - 25*b
^4*c^4*d^6*e + 20*b^5*c^3*d^5*e^2 + 10*b^6*c^2*d^4*e^3 - 20*b^7*c*d^3*e^4 + 7*b^
8*d^2*e^5)*x)*sqrt(-d))/(((b^5*c^6*d^8*e - 4*b^6*c^5*d^7*e^2 + 6*b^7*c^4*d^6*e^3
- 4*b^8*c^3*d^5*e^4 + b^9*c^2*d^4*e^5)*x^5 + (b^5*c^6*d^9 - 2*b^6*c^5*d^8*e - 2
*b^7*c^4*d^7*e^2 + 8*b^8*c^3*d^6*e^3 - 7*b^9*c^2*d^5*e^4 + 2*b^10*c*d^4*e^5)*x^4
+ (2*b^6*c^5*d^9 - 7*b^7*c^4*d^8*e + 8*b^8*c^3*d^7*e^2 - 2*b^9*c^2*d^6*e^3 - 2*
b^10*c*d^5*e^4 + b^11*d^4*e^5)*x^3 + (b^7*c^4*d^9 - 4*b^8*c^3*d^8*e + 6*b^9*c^2*
d^7*e^2 - 4*b^10*c*d^6*e^3 + b^11*d^5*e^4)*x^2)*sqrt(e*x + d)*sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (b + c x\right )^{3} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Integral(1/(x**3*(b + c*x)**3*(d + e*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.412253, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Done