3.414 \(\int \frac{(d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=395 \[ \frac{4 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 c^2}-\frac{4 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{b^2 c} \]

[Out]

(-2*(d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*Sqrt[b*x + c*x^2]) + (4*e*(3*c
^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*b^2*c^2) + (
2*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(b^2*c) + (2*(2*c*d - b*e)*
(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*Elli
pticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sq
rt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (4*d*(c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 2
*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*
Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x +
c*x^2])

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Rubi [A]  time = 1.46126, antiderivative size = 395, 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 \[ \frac{4 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 c^2}-\frac{4 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 (d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{b^2 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^(5/2)*(b*d + (2*c*d - b*e)*x))/(b^2*Sqrt[b*x + c*x^2]) + (4*e*(3*c
^2*d^2 - 3*b*c*d*e + 2*b^2*e^2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*b^2*c^2) + (
2*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(b^2*c) + (2*(2*c*d - b*e)*
(3*c^2*d^2 - 3*b*c*d*e + 8*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*Elli
pticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sq
rt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (4*d*(c*d - b*e)*(3*c^2*d^2 - 3*b*c*d*e + 2
*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*
Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(3/2)*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x +
c*x^2])

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Rubi in Sympy [A]  time = 169.748, size = 379, normalized size = 0.96 \[ - \frac{2 \sqrt{x} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) \left (8 b^{2} e^{2} - 3 b c d e + 3 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 c^{\frac{5}{2}} \left (- b\right )^{\frac{3}{2}} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} - \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b d - x \left (b e - 2 c d\right )\right )}{b^{2} \sqrt{b x + c x^{2}}} - \frac{2 e \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{b^{2} c} + \frac{4 e \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (2 b^{2} e^{2} - 3 b c d e + 3 c^{2} d^{2}\right )}{3 b^{2} c^{2}} - \frac{4 \sqrt{x} \left (- d\right )^{\frac{3}{2}} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (2 b^{2} e^{2} - 3 b c d e + 3 c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{3 b^{2} c^{2} \sqrt{e} \sqrt{d + e x} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(c*x**2+b*x)**(3/2),x)

[Out]

-2*sqrt(x)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(b*e - 2*c*d)*(8*b**2*e**2 - 3*b*c*d*e
+ 3*c**2*d**2)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*c**(5/2)
*(-b)**(3/2)*sqrt(1 + e*x/d)*sqrt(b*x + c*x**2)) - 2*(d + e*x)**(5/2)*(b*d - x*(
b*e - 2*c*d))/(b**2*sqrt(b*x + c*x**2)) - 2*e*(d + e*x)**(3/2)*(b*e - 2*c*d)*sqr
t(b*x + c*x**2)/(b**2*c) + 4*e*sqrt(d + e*x)*sqrt(b*x + c*x**2)*(2*b**2*e**2 - 3
*b*c*d*e + 3*c**2*d**2)/(3*b**2*c**2) - 4*sqrt(x)*(-d)**(3/2)*sqrt(1 + c*x/b)*sq
rt(1 + e*x/d)*(b*e - c*d)*(2*b**2*e**2 - 3*b*c*d*e + 3*c**2*d**2)*elliptic_f(asi
n(sqrt(e)*sqrt(x)/sqrt(-d)), c*d/(b*e))/(3*b**2*c**2*sqrt(e)*sqrt(d + e*x)*sqrt(
b*x + c*x**2))

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Mathematica [C]  time = 3.33725, size = 360, normalized size = 0.91 \[ -\frac{2 \left (b (d+e x) \left (-b^2 e^3 x (b+c x)+3 c^2 d^3 (b+c x)+3 x (c d-b e)^3\right )-\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^3 e^3+23 b^2 c d e^2-18 b c^2 d^2 e+3 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right )\right )\right )}{3 b^3 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(b*(d + e*x)*(3*(c*d - b*e)^3*x + 3*c^2*d^3*(b + c*x) - b^2*e^3*x*(b + c*x))
- Sqrt[b/c]*(Sqrt[b/c]*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*b^3*e^3)
*(b + c*x)*(d + e*x) + I*b*e*(6*c^3*d^3 - 9*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 8*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*(3*c^3*d^3 - 18*b*c^2*d^2*e + 23*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)])))/(3*b^3*c^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.092, size = 863, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(c*x^2+b*x)^(3/2),x)

[Out]

2/3*(4*((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^4*c*d*e^3-10*((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^2*d^2*e^2+12*((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^2*c^3*d^3*e-6*((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*c^4*d^4+8*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d)
)^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*e^
4-27*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e^3+28*((c*x+b)/b)^(1/2)*(-(e*x+d)
*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(
1/2))*b^3*c^2*d^2*e^2-15*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)
^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3*e+6*((c*x+
b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1
/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^4+x^3*b^2*c^3*e^4+4*x^2*b^3*c^2*e^4-8*x^2*b^2
*c^3*d*e^3+9*x^2*b*c^4*d^2*e^2-6*x^2*c^5*d^3*e+4*x*b^3*c^2*d*e^3-9*x*b^2*c^3*d^2
*e^2+6*x*b*c^4*d^3*e-6*x*c^5*d^4-3*b*c^4*d^4)/x*(x*(c*x+b))^(1/2)/c^4/(c*x+b)/b^
2/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2), x)