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

Optimal. Leaf size=354 \[ \frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 d e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 \sqrt{b x+c x^2} \left (e x \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+c d^2 (8 c d-7 b e)\right )}{5 d e^3 (d+e x)^{3/2} (c d-b e)}-\frac{16 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}} \]

[Out]

(-2*(c*d^2*(8*c*d - 7*b*e) + e*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x +
c*x^2])/(5*d*e^3*(c*d - b*e)*(d + e*x)^(3/2)) - (2*(b*x + c*x^2)^(3/2))/(5*e*(d
+ e*x)^(5/2)) + (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[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (
b*e)/(c*d)])/(5*d*e^4*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (16*Sqr
t[-b]*Sqrt[c]*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Elliptic
F[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(5*e^4*Sqrt[d + e*x]*Sqrt[b*
x + c*x^2])

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Rubi [A]  time = 1.11881, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 d e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 \sqrt{b x+c x^2} \left (e x \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+c d^2 (8 c d-7 b e)\right )}{5 d e^3 (d+e x)^{3/2} (c d-b e)}-\frac{16 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(c*d^2*(8*c*d - 7*b*e) + e*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x +
c*x^2])/(5*d*e^3*(c*d - b*e)*(d + e*x)^(3/2)) - (2*(b*x + c*x^2)^(3/2))/(5*e*(d
+ e*x)^(5/2)) + (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[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (
b*e)/(c*d)])/(5*d*e^4*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (16*Sqr
t[-b]*Sqrt[c]*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Elliptic
F[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(5*e^4*Sqrt[d + e*x]*Sqrt[b*
x + c*x^2])

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Rubi in Sympy [A]  time = 124.935, size = 325, normalized size = 0.92 \[ - \frac{2 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{5 d e^{4} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} + \frac{16 c \sqrt{x} \sqrt{- d} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{5 e^{\frac{9}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} - \frac{4 \sqrt{b x + c x^{2}} \left (\frac{c d^{2} \left (7 b e - 8 c d\right )}{2} - \frac{e x \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{2}\right )}{5 d e^{3} \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(c)*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(b**2*e**2 - 16*b*c*d*
e + 16*c**2*d**2)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(5*d*e**
4*sqrt(1 + e*x/d)*(b*e - c*d)*sqrt(b*x + c*x**2)) + 16*c*sqrt(x)*sqrt(-d)*sqrt(1
+ c*x/b)*sqrt(1 + e*x/d)*(b*e - 2*c*d)*elliptic_f(asin(sqrt(e)*sqrt(x)/sqrt(-d)
), c*d/(b*e))/(5*e**(9/2)*sqrt(d + e*x)*sqrt(b*x + c*x**2)) - 2*(b*x + c*x**2)**
(3/2)/(5*e*(d + e*x)**(5/2)) - 4*sqrt(b*x + c*x**2)*(c*d**2*(7*b*e - 8*c*d)/2 -
e*x*(b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)/2)/(5*d*e**3*(d + e*x)**(3/2)*(b*e -
c*d))

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Mathematica [C]  time = 2.38143, size = 369, normalized size = 1.04 \[ -\frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left (b^2 e^4 x^2-b c d e \left (7 d^2+16 d e x+11 e^2 x^2\right )+c^2 d^2 \left (8 d^2+18 d e x+11 e^2 x^2\right )\right )-c \sqrt{\frac{b}{c}} (d+e x)^2 \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-9 b c d e+8 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )\right )}{5 b d e^4 x^2 (b+c x)^2 (d+e x)^{5/2} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(b^2*e^4*x^2 - b*c*d*e*(7*d^2 + 16*d*e*
x + 11*e^2*x^2) + c^2*d^2*(8*d^2 + 18*d*e*x + 11*e^2*x^2)) - Sqrt[b/c]*c*(d + e*
x)^2*(Sqrt[b/c]*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) + I*b*e*
(16*c^2*d^2 - 16*b*c*d*e + 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*b*e*(8*c^2*d^2 - 9*b*c*
d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[S
qrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(5*b*d*e^4*(c*d - b*e)*x^2*(b + c*x)^2*(d + e
*x)^(5/2))

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Maple [B]  time = 0.044, size = 1885, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(x*(c*x+b))^(1/2)*(64*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)-32*
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)+16*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)-48*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)+32*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)-17*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)+32*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)-16*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)+8*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)-24*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)+16*Ellip
ticF(((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)-34*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)-16*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)+16*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)+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)+Ellip
ticE(((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)+2*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)-24*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)-17*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)-11*x^3*b^2*c^2*d*e^4-5*x^3*b*c^3*d^2*e^3-16*x^2*b^2*c
^2*d^2*e^3+11*x^2*b*c^3*d^3*e^2-7*x*b^2*c^2*d^3*e^2+8*x*b*c^3*d^4*e+32*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)+8*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)-11*x^4*b*c^3*d*e^4+x^4*b^2*c^2*e^5+11*x^4*c^4*d^2*e^3+x^3*b^3*c*e^5+18*x^
3*c^4*d^3*e^2+8*x^2*c^4*d^4*e)/(c*x+b)/x/(b*e-c*d)/c/(e*x+d)^(5/2)/e^4/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x)^(3/2)/((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. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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