3.398 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=476 \[ \frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 d e^4 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 d^2 e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{35 d e^3 (d+e x)^{5/2} (c d-b e)}+\frac{4 \sqrt{b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{35 d^2 e^3 \sqrt{d+e x} (c d-b e)^2}-\frac{2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]

[Out]

(4*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[b*x + c*x^2])/(35*d^2*e^
3*(c*d - b*e)^2*Sqrt[d + e*x]) - (2*(d*(8*c^2*d^2 - 5*b*c*d*e - 2*b^2*e^2) + e*(
14*c^2*d^2 - 14*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(35*d*e^3*(c*d - b*e)*(
d + e*x)^(5/2)) - (2*(b*x + c*x^2)^(3/2))/(7*e*(d + e*x)^(7/2)) - (4*Sqrt[-b]*Sq
rt[c]*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*
Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*d^
2*e^4*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*Sqrt[c]*(
16*c^2*d^2 - 16*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*E
llipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*d*e^4*(c*d - b*e)
*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.63548, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ \frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 d e^4 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 d^2 e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{b x+c x^2} \left (e x \left (b^2 e^2-14 b c d e+14 c^2 d^2\right )+d \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{35 d e^3 (d+e x)^{5/2} (c d-b e)}+\frac{4 \sqrt{b x+c x^2} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{35 d^2 e^3 \sqrt{d+e x} (c d-b e)^2}-\frac{2 \left (b x+c x^2\right )^{3/2}}{7 e (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

(4*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[b*x + c*x^2])/(35*d^2*e^
3*(c*d - b*e)^2*Sqrt[d + e*x]) - (2*(d*(8*c^2*d^2 - 5*b*c*d*e - 2*b^2*e^2) + e*(
14*c^2*d^2 - 14*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(35*d*e^3*(c*d - b*e)*(
d + e*x)^(5/2)) - (2*(b*x + c*x^2)^(3/2))/(7*e*(d + e*x)^(7/2)) - (4*Sqrt[-b]*Sq
rt[c]*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*
Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*d^
2*e^4*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*Sqrt[c]*(
16*c^2*d^2 - 16*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*E
llipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*d*e^4*(c*d - b*e)
*Sqrt[d + e*x]*Sqrt[b*x + c*x^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((c*x**2+b*x)**(3/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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Mathematica [C]  time = 4.58573, size = 479, normalized size = 1.01 \[ -\frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left (d (d+e x)^2 \left (b^2 e^2-19 b c d e+19 c^2 d^2\right ) (c d-b e)-2 (d+e x)^3 \left (b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right )+5 d^3 (c d-b e)^3-8 d^2 (d+e x) (2 c d-b e) (c d-b e)^2\right )+c \sqrt{\frac{b}{c}} (d+e x)^3 \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^3 e^3+3 b^2 c d e^2-13 b c^2 d^2 e+8 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right )\right )\right )}{35 b d^2 e^4 x^2 (b+c x)^2 (d+e x)^{7/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

(-2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(5*d^3*(c*d - b*e)^3 - 8*d^2*(c*d - b*e
)^2*(2*c*d - b*e)*(d + e*x) + d*(c*d - b*e)*(19*c^2*d^2 - 19*b*c*d*e + b^2*e^2)*
(d + e*x)^2 - 2*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(d + e*x)
^3) + Sqrt[b/c]*c*(d + e*x)^3*(2*Sqrt[b/c]*(8*c^3*d^3 - 12*b*c^2*d^2*e + 2*b^2*c
*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) + (2*I)*b*e*(8*c^3*d^3 - 12*b*c^2*d^2*e +
2*b^2*c*d*e^2 + b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I
*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^3*d^3 - 13*b*c^2*d^2*e +
3*b^2*c*d*e^2 + 2*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF
[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(35*b*d^2*e^4*(c*d - b*e)^2*x^2*(
b + c*x)^2*(d + e*x)^(7/2))

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Maple [B]  time = 0.09, size = 3267, normalized size = 6.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/(e*x+d)^(9/2),x)

[Out]

2/35*(x*(c*x+b))^(1/2)*(-8*x^3*b^3*c^2*d^2*e^5-32*x^4*b^2*c^3*d^2*e^5-18*x^4*b*c
^4*d^3*e^4-32*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b^2*c^3*d^3
*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+16*EllipticF(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b*c^4*d^4*e^3*((c*x+b)/b)^(1/2)*(-(
e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+6*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e
-c*d))^(1/2))*x^2*b^4*c*d^2*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(
-c*x/b)^(1/2)-84*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c^2*
d^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+120*Ellipt
icE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^3*d^4*e^3*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-48*EllipticE(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*x^2*b*c^4*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^
(1/2)*(-c*x/b)^(1/2)+3*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^
4*c*d^2*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+45*Ell
ipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c^2*d^3*e^4*((c*x+b)/b)^
(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*EllipticF(((c*x+b)/b)^(1/2),
(b*e/(b*e-c*d))^(1/2))*x*b^4*c*d^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^
(1/2)*(-c*x/b)^(1/2)+7*x^3*b^4*c*d*e^6+29*x^4*c^5*d^4*e^3+26*x^3*c^5*d^5*e^2+8*x
^2*c^5*d^6*e-30*x^3*b^2*c^3*d^3*e^4-7*x^3*b*c^4*d^4*e^3+4*x^2*b^3*c^2*d^3*e^4-35
*x^2*b^2*c^3*d^4*e^3+15*x^2*b*c^4*d^5*e^2+x*b^3*c^2*d^4*e^3-11*x*b^2*c^3*d^5*e^2
+8*x*b*c^4*d^6*e+2*x^5*b^3*c^2*e^7+16*x^5*c^5*d^3*e^4+2*x^4*b^4*c*e^7+6*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^5*d*e^6*((c*x+b)/b)^(1/2)*(-(e*
x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+6*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*x*b^5*d^2*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)+2*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^4*e^3*((c*x
+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-28*EllipticE(((c*x+b)/b
)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*
e-c*d))^(1/2)*(-c*x/b)^(1/2)+40*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2
))*b^2*c^3*d^6*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+E
llipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^4*e^3*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+15*EllipticF(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*b^3*c^2*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2)-32*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*
d^6*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-96*EllipticF
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^3*d^4*e^3*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+48*EllipticF(((c*x+b)/b)^(1/2),(b*e/
(b*e-c*d))^(1/2))*x^2*b*c^4*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2)+6*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*c*d
^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-84*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c^2*d^4*e^3*((c*x+b)/b)^(1/2)*(
-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+120*EllipticE(((c*x+b)/b)^(1/2),(b*e/
(b*e-c*d))^(1/2))*x*b^2*c^3*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2)-48*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^4*
d^6*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+45*EllipticF
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c^2*d^4*e^3*((c*x+b)/b)^(1/2)*(-
(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-96*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*x*b^2*c^3*d^5*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)
*(-c*x/b)^(1/2)+48*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^4*d^
6*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*EllipticE(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b^4*c*d*e^6*((c*x+b)/b)^(1/2)*(-(e*x+
d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-28*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*
d))^(1/2))*x^3*b^3*c^2*d^2*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-
c*x/b)^(1/2)+40*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b^2*c^3*d
^3*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-16*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b*c^4*d^4*e^3*((c*x+b)/b)^(1/2)*(
-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e
-c*d))^(1/2))*x^3*b^4*c*d*e^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c
*x/b)^(1/2)+15*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^3*b^3*c^2*d^
2*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*x^5*b^2*c^
3*d*e^6-24*x^5*b*c^4*d^2*e^5+11*x^4*b^3*c^2*d*e^6+2*EllipticE(((c*x+b)/b)^(1/2),
(b*e/(b*e-c*d))^(1/2))*x^3*b^5*e^7*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)+2*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*d^3*e^
4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-16*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^7*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b
*e-c*d))^(1/2)*(-c*x/b)^(1/2)+16*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/
2))*b*c^4*d^7*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2))/(c*
x+b)/x/(b*e-c*d)^2/c/(e*x+d)^(7/2)/e^4/d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{{\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x)^(3/2)/((e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x
+ d^4)*sqrt(e*x + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(9/2), x)