3.391 \(\int \frac{\sqrt{b x+c x^2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=398 \[ \frac{4 \sqrt{b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{15 d^2 e \sqrt{d+e x} (c d-b e)^2}-\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^2 e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d e^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{b x+c x^2} (2 c d-b e)}{15 d e (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{b x+c x^2}}{5 e (d+e x)^{5/2}} \]

[Out]

(-2*Sqrt[b*x + c*x^2])/(5*e*(d + e*x)^(5/2)) + (2*(2*c*d - b*e)*Sqrt[b*x + c*x^2
])/(15*d*e*(c*d - b*e)*(d + e*x)^(3/2)) + (4*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[
b*x + c*x^2])/(15*d^2*e*(c*d - b*e)^2*Sqrt[d + e*x]) - (4*Sqrt[-b]*Sqrt[c]*(c^2*
d^2 - b*c*d*e + b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSi
n[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d^2*e^2*(c*d - b*e)^2*Sqrt[1 +
(e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 +
(c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/
(c*d)])/(15*d*e^2*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.4102, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{4 \sqrt{b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{15 d^2 e \sqrt{d+e x} (c d-b e)^2}-\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^2 e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d e^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{b x+c x^2} (2 c d-b e)}{15 d e (d+e x)^{3/2} (c d-b e)}-\frac{2 \sqrt{b x+c x^2}}{5 e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/(d + e*x)^(7/2),x]

[Out]

(-2*Sqrt[b*x + c*x^2])/(5*e*(d + e*x)^(5/2)) + (2*(2*c*d - b*e)*Sqrt[b*x + c*x^2
])/(15*d*e*(c*d - b*e)*(d + e*x)^(3/2)) + (4*(c^2*d^2 - b*c*d*e + b^2*e^2)*Sqrt[
b*x + c*x^2])/(15*d^2*e*(c*d - b*e)^2*Sqrt[d + e*x]) - (4*Sqrt[-b]*Sqrt[c]*(c^2*
d^2 - b*c*d*e + b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSi
n[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d^2*e^2*(c*d - b*e)^2*Sqrt[1 +
(e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*Sqrt[c]*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 +
(c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/
(c*d)])/(15*d*e^2*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 168.184, size = 354, normalized size = 0.89 \[ \frac{2 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{15 d e^{2} \sqrt{d + e x} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{4 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b^{2} e^{2} - b c d e + c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{15 d^{2} e^{2} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right )^{2} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{b x + c x^{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} + \frac{2 \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{15 d e \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} + \frac{4 \sqrt{b x + c x^{2}} \left (b^{2} e^{2} - b c d e + c^{2} d^{2}\right )}{15 d^{2} e \sqrt{d + e x} \left (b e - c d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/(e*x+d)**(7/2),x)

[Out]

2*sqrt(c)*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(b*e - 2*c*d)*ellipti
c_f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(15*d*e**2*sqrt(d + e*x)*(b*e - c
*d)*sqrt(b*x + c*x**2)) - 4*sqrt(c)*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*
x)*(b**2*e**2 - b*c*d*e + c**2*d**2)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)),
b*e/(c*d))/(15*d**2*e**2*sqrt(1 + e*x/d)*(b*e - c*d)**2*sqrt(b*x + c*x**2)) - 2*
sqrt(b*x + c*x**2)/(5*e*(d + e*x)**(5/2)) + 2*(b*e - 2*c*d)*sqrt(b*x + c*x**2)/(
15*d*e*(d + e*x)**(3/2)*(b*e - c*d)) + 4*sqrt(b*x + c*x**2)*(b**2*e**2 - b*c*d*e
+ c**2*d**2)/(15*d**2*e*sqrt(d + e*x)*(b*e - c*d)**2)

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Mathematica [C]  time = 1.94686, size = 362, normalized size = 0.91 \[ -\frac{2 \left (b e x (b+c x) \left (-b^2 e^3 x (5 d+2 e x)+b c d e \left (-d^2+7 d e x+2 e^2 x^2\right )-c^2 d^2 \left (d^2+6 d e x+2 e^2 x^2\right )\right )+c \sqrt{\frac{b}{c}} (d+e x)^2 \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^2 e^2-3 b c d e+c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 e^2-b c d e+c^2 d^2\right )\right )\right )}{15 b d^2 e^2 \sqrt{x (b+c x)} (d+e x)^{5/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^(7/2),x]

[Out]

(-2*(b*e*x*(b + c*x)*(-(b^2*e^3*x*(5*d + 2*e*x)) - c^2*d^2*(d^2 + 6*d*e*x + 2*e^
2*x^2) + b*c*d*e*(-d^2 + 7*d*e*x + 2*e^2*x^2)) + Sqrt[b/c]*c*(d + e*x)^2*(2*Sqrt
[b/c]*(c^2*d^2 - b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) + (2*I)*b*e*(c^2*d^2 - b
*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSin
h[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqr
t[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]],
(c*d)/(b*e)])))/(15*b*d^2*e^2*(c*d - b*e)^2*Sqrt[x*(b + c*x)]*(d + e*x)^(5/2))

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Maple [B]  time = 0.075, size = 1897, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/(e*x+d)^(7/2),x)

[Out]

2/15*(8*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^3*e^2*((c
*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-4*EllipticE(((c*x+b)/
b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^4*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)+2*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*x*b^3*c*d^2*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-6
*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^3*e^2*((c*x+b)/b
)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*EllipticF(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^4*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^
(1/2)*(-c*x/b)^(1/2)-4*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^
3*c*d*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+4*Ellipt
icE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^2*e^3*((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^2*b*c^3*d^3*e^2*((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^2*b^3*c
*d*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^2*b^2*c^2*d^2*e^3*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*EllipticF(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*x^2*b*c^3*d^3*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)-8*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^
2*e^3*((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*c^3*d^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c
/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+2*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(
1/2))*b*c^3*d^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))*x^2*b^4*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*d^2*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c
*x/b)^(1/2)+4*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d*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))*b^2*c^2*d^4*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)-4*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*b^3*c*d^3*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*x
^3*b^2*c^2*d*e^4-5*x^3*b*c^3*d^2*e^3+5*x^2*b^3*c*d*e^4-7*x^2*b^2*c^2*d^2*e^3+7*x
^2*b*c^3*d^3*e^2+x*b^2*c^2*d^3*e^2+x*b*c^3*d^4*e+4*EllipticE(((c*x+b)/b)^(1/2),(
b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e*((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))*b^3*c*d^3*e
^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-2*x^4*b*c^3*d*e
^4+2*x^4*b^2*c^2*e^5+2*x^4*c^4*d^2*e^3+2*x^3*b^3*c*e^5+6*x^3*c^4*d^3*e^2+x^2*c^4
*d^4*e)*(x*(c*x+b))^(1/2)/c/(b*e-c*d)^2/d^2/(c*x+b)/x/e^2/(e*x+d)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x)/(e*x + d)^(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/(e*x+d)**(7/2),x)

[Out]

Integral(sqrt(x*(b + c*x))/(d + e*x)**(7/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + b*x)/(e*x + d)^(7/2), x)