3.413 \(\int \frac{1}{(d+e x)^{7/2} \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=403 \[ -\frac{2 e \sqrt{b x+c x^2} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )}{15 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{8 e \sqrt{b x+c x^2} (2 c d-b e)}{15 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{8 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2}}{5 d (d+e x)^{5/2} (c d-b e)} \]

[Out]

(-2*e*Sqrt[b*x + c*x^2])/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - (8*e*(2*c*d - b*e)*
Sqrt[b*x + c*x^2])/(15*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) - (2*e*(23*c^2*d^2 - 2
3*b*c*d*e + 8*b^2*e^2)*Sqrt[b*x + c*x^2])/(15*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) +
(2*Sqrt[-b]*Sqrt[c]*(23*c^2*d^2 - 23*b*c*d*e + 8*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)])/(
15*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (8*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^2*(c*d - b*e)^2*Sqrt[d + e*x]*Sqrt[b
*x + c*x^2])

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Rubi [A]  time = 1.38482, antiderivative size = 403, 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 \[ -\frac{2 e \sqrt{b x+c x^2} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )}{15 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{8 e \sqrt{b x+c x^2} (2 c d-b e)}{15 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{8 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2}}{5 d (d+e x)^{5/2} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*Sqrt[b*x + c*x^2])/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - (8*e*(2*c*d - b*e)*
Sqrt[b*x + c*x^2])/(15*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) - (2*e*(23*c^2*d^2 - 2
3*b*c*d*e + 8*b^2*e^2)*Sqrt[b*x + c*x^2])/(15*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) +
(2*Sqrt[-b]*Sqrt[c]*(23*c^2*d^2 - 23*b*c*d*e + 8*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)])/(
15*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (8*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^2*(c*d - b*e)^2*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(1/(e*x+d)**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.53998, size = 381, normalized size = 0.95 \[ -\frac{2 \left (b e x (b+c x) \left ((d+e x)^2 \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )+3 d^2 (c d-b e)^2+4 d (d+e x) (2 c d-b e) (c d-b e)\right )-c \sqrt{\frac{b}{c}} (d+e x)^2 \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )+i x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^3 e^3+27 b^2 c d e^2-34 b c^2 d^2 e+15 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )\right )\right )}{15 b d^3 \sqrt{x (b+c x)} (d+e x)^{5/2} (c d-b e)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*e*x*(b + c*x)*(3*d^2*(c*d - b*e)^2 + 4*d*(c*d - b*e)*(2*c*d - b*e)*(d + e
*x) + (23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*(d + e*x)^2) - Sqrt[b/c]*c*(d + e*x)
^2*(Sqrt[b/c]*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*(b + c*x)*(d + e*x) + I*b*e*
(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*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*(15*c^3*d^3 - 34*b*c^
2*d^2*e + 27*b^2*c*d*e^2 - 8*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)])))/(15*b*d^3*(c*d - b*e)^
3*Sqrt[x*(b + c*x)]*(d + e*x)^(5/2))

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Maple [B]  time = 0.052, size = 1912, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(x*(c*x+b))^(1/2)*(92*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)-46
*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)+8*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)-24*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)+16*Elli
pticF(((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)-31*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)+46*EllipticE(((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)-23*Ellipt
icE(((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)+4*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)-12*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)+8*Ellipt
icF(((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)-62*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)-23*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)+8*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)+8*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)+8*Ell
ipticE(((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)+15*x*b^3*c*d^2*e^3+16*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)-12*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)-3
1*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^4-35*x^3*b*c^
3*d^2*e^3+20*x^2*b^3*c*d*e^4-43*x^2*b^2*c^2*d^2*e^3+13*x^2*b*c^3*d^3*e^2-41*x*b^
2*c^2*d^3*e^2+34*x*b*c^3*d^4*e+46*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)
+4*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)-23*x^4*b*c^3*d*e^4+8*x^4*b^2*c
^2*e^5+23*x^4*c^4*d^2*e^3+8*x^3*b^3*c*e^5+54*x^3*c^4*d^3*e^2+34*x^2*c^4*d^4*e)/(
c*x+b)/x/(b*e-c*d)^3/(e*x+d)^(5/2)/c/d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)),x, algorithm="maxima")

[Out]

integrate(1/(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. \[{\rm integral}\left (\frac{1}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

integral(1/((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*sqrt(c*x^2 + b*x)*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(1/(e*x+d)**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^2 + b*x)*(e*x + d)^(7/2)), x)