3.1269 \(\int \frac{A+B x}{(d+e x)^{5/2} \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=369 \[ -\frac{2 \sqrt{b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{3 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 A e (2 c d-b e)-B d (b e+c d)) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d^2 e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}+\frac{2 \sqrt{b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (B d-A e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d e \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)} \]

[Out]

(2*(B*d - A*e)*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) - (2*(2*A*e*
(2*c*d - b*e) - B*d*(c*d + b*e))*Sqrt[b*x + c*x^2])/(3*d^2*(c*d - b*e)^2*Sqrt[d
+ e*x]) + (2*Sqrt[-b]*Sqrt[c]*(2*A*e*(2*c*d - b*e) - B*d*(c*d + b*e))*Sqrt[x]*Sq
rt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e
)/(c*d)])/(3*d^2*e*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[
-b]*Sqrt[c]*(B*d - A*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*d*e*(c*d - b*e)*Sqrt[d + e*x]
*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.27793, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 \sqrt{b x+c x^2} (2 A e (2 c d-b e)-B d (b e+c d))}{3 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 A e (2 c d-b e)-B d (b e+c d)) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d^2 e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}+\frac{2 \sqrt{b x+c x^2} (B d-A e)}{3 d (d+e x)^{3/2} (c d-b e)}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (B d-A e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 d e \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^(5/2)*Sqrt[b*x + c*x^2]),x]

[Out]

(2*(B*d - A*e)*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) - (2*(2*A*e*
(2*c*d - b*e) - B*d*(c*d + b*e))*Sqrt[b*x + c*x^2])/(3*d^2*(c*d - b*e)^2*Sqrt[d
+ e*x]) + (2*Sqrt[-b]*Sqrt[c]*(2*A*e*(2*c*d - b*e) - B*d*(c*d + b*e))*Sqrt[x]*Sq
rt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e
)/(c*d)])/(3*d^2*e*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[
-b]*Sqrt[c]*(B*d - A*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*d*e*(c*d - b*e)*Sqrt[d + e*x]
*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 167.835, size = 338, normalized size = 0.92 \[ \frac{2 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (A e - B d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 d e \sqrt{d + e x} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (2 A b e^{2} - 4 A c d e + B b d e + B c d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 d^{2} e \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right )^{2} \sqrt{b x + c x^{2}}} + \frac{2 \left (A e - B d\right ) \sqrt{b x + c x^{2}}}{3 d \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} + \frac{2 \sqrt{b x + c x^{2}} \left (2 A b e^{2} - 4 A c d e + B b d e + B c d^{2}\right )}{3 d^{2} \sqrt{d + e x} \left (b e - c d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c)*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(A*e - B*d)*elliptic_
f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*d*e*sqrt(d + e*x)*(b*e - c*d)*sq
rt(b*x + c*x**2)) - 2*sqrt(c)*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(2*
A*b*e**2 - 4*A*c*d*e + B*b*d*e + B*c*d**2)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(
-b)), b*e/(c*d))/(3*d**2*e*sqrt(1 + e*x/d)*(b*e - c*d)**2*sqrt(b*x + c*x**2)) +
2*(A*e - B*d)*sqrt(b*x + c*x**2)/(3*d*(d + e*x)**(3/2)*(b*e - c*d)) + 2*sqrt(b*x
+ c*x**2)*(2*A*b*e**2 - 4*A*c*d*e + B*b*d*e + B*c*d**2)/(3*d**2*sqrt(d + e*x)*(
b*e - c*d)**2)

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Mathematica [C]  time = 3.08034, size = 347, normalized size = 0.94 \[ \frac{2 \left (b e x (b+c x) \left (A e (b e (3 d+2 e x)-c d (5 d+4 e x))+B d \left (b e^2 x+c d (2 d+e x)\right )\right )-c \sqrt{\frac{b}{c}} (d+e x) \left (-i e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) (3 A c d-b (2 A e+B d)) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (2 A e (b e-2 c d)+B d (b e+c d)) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) (2 A e (b e-2 c d)+B d (b e+c d))\right )\right )}{3 b d^2 e \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^(5/2)*Sqrt[b*x + c*x^2]),x]

[Out]

(2*(b*e*x*(b + c*x)*(B*d*(b*e^2*x + c*d*(2*d + e*x)) + A*e*(b*e*(3*d + 2*e*x) -
c*d*(5*d + 4*e*x))) - Sqrt[b/c]*c*(d + e*x)*(Sqrt[b/c]*(2*A*e*(-2*c*d + b*e) + B
*d*(c*d + b*e))*(b + c*x)*(d + e*x) + I*b*e*(2*A*e*(-2*c*d + b*e) + B*d*(c*d + b
*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/S
qrt[x]], (c*d)/(b*e)] - I*e*(c*d - b*e)*(3*A*c*d - b*(B*d + 2*A*e))*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)])))/(3*b*d^2*e*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))

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Maple [B]  time = 0.056, size = 1635, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x*(c*x+b))^(1/2)*(4*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*
c^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-B*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*d^3*e*((c*x+b)/b)^(1/2)*(-(e*
x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+B*x^3*b*c^2*d*e^3+A*EllipticF(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c
*d))^(1/2)*(-c*x/b)^(1/2)-A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b
*c^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-6*A*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*d^2*e^2*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-4*A*x^3*c^3*d*e^3-A*x^2*b*c^2*d*e^3+
B*x^2*b^2*c*d*e^3+B*x^2*b*c^2*d^2*e^2-5*A*b*c^2*d^2*e^2*x+2*B*b*c^2*d^3*e*x+B*El
lipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*d^2*e^2*((c*x+b)/b)^(1/2)*(
-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*b*c^2*d^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)+2*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*e^4*((c*x+b
)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*A*EllipticE(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))
^(1/2)*(-c*x/b)^(1/2)+B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2
*d^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*A*b^2*c*d*e
^3*x+B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*d*e^3*((c*x+b)/b
)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+B*x^3*c^3*d^2*e^2+2*A*x^2*b^
2*c*e^4-5*A*x^2*c^3*d^2*e^2+2*B*x^2*c^3*d^3*e+2*A*x^3*b*c^2*e^4-A*EllipticF(((c*
x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^2*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)
*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-6*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*x*b^2*c*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^
(1/2)+4*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^2*d^2*e^2*((c
*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-B*EllipticF(((c*x+b)/
b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b
*e-c*d))^(1/2)*(-c*x/b)^(1/2)+B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2
))*x*b*c^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-B
*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^2*d^3*e*((c*x+b)/b)^(1
/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+A*EllipticF(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*x*b^2*c*d*e^3*((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/(e*x+d)^(3/2)/d^2/e/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^(5/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)/((e^2*x^2 + 2*d*e*x + d^2)*sqrt(c*x^2 + b*x)*sqrt(e*x + d)),
x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/(sqrt(x*(b + c*x))*(d + e*x)**(5/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^(5/2)), x)