3.1261 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=449 \[ -\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (3 c e x (-14 A c e-b B e+16 B c d)+7 A c e (8 c d-7 b e)-B \left (b^2 e^2-60 b c d e+64 c^2 d^2\right )\right )}{35 c e^4}-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (56 A c e (2 c d-b e)-B \left (-b^2 e^2-72 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) (-14 A c e-b B e+16 B c d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt{d+e x}} \]

[Out]

(-2*Sqrt[d + e*x]*(7*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 60*b*c*d*e + b^2*e^
2) + 3*c*e*(16*B*c*d - b*B*e - 14*A*c*e)*x)*Sqrt[b*x + c*x^2])/(35*c*e^4) + (2*(
8*B*d - 7*A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(7*e^2*Sqrt[d + e*x]) + (2*Sqrt[-b]*
(5*b*c*e*(8*B*d - 7*A*e)*(2*c*d - b*e) - (16*B*c*d - b*B*e - 14*A*c*e)*(8*c^2*d^
2 - 3*b*c*d*e - 2*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*c^(3/2)*e^5*Sqrt[1 + (e*x)/d
]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(56*A*c*e*(2*c*d - b*e) - B*(12
8*c^2*d^2 - 72*b*c*d*e - b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*E
llipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*c^(3/2)*e^5*Sqrt[
d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.47745, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (3 c e x (-14 A c e-b B e+16 B c d)+7 A c e (8 c d-7 b e)-B \left (b^2 e^2-60 b c d e+64 c^2 d^2\right )\right )}{35 c e^4}-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (56 A c e (2 c d-b e)-B \left (-b^2 e^2-72 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) (-14 A c e-b B e+16 B c d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(7*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 60*b*c*d*e + b^2*e^
2) + 3*c*e*(16*B*c*d - b*B*e - 14*A*c*e)*x)*Sqrt[b*x + c*x^2])/(35*c*e^4) + (2*(
8*B*d - 7*A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(7*e^2*Sqrt[d + e*x]) + (2*Sqrt[-b]*
(5*b*c*e*(8*B*d - 7*A*e)*(2*c*d - b*e) - (16*B*c*d - b*B*e - 14*A*c*e)*(8*c^2*d^
2 - 3*b*c*d*e - 2*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[Ar
cSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*c^(3/2)*e^5*Sqrt[1 + (e*x)/d
]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(56*A*c*e*(2*c*d - b*e) - B*(12
8*c^2*d^2 - 72*b*c*d*e - b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*E
llipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*c^(3/2)*e^5*Sqrt[
d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 154.78, size = 450, normalized size = 1. \[ - \frac{4 \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (\frac{7 A e}{2} - 4 B d - \frac{B e x}{2}\right )}{7 e^{2} \sqrt{d + e x}} + \frac{8 \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (\frac{5 b c e \left (7 A e - 8 B d\right )}{4} + \frac{3 c e x \left (14 A c e + B b e - 16 B c d\right )}{4} + \left (\frac{b e}{4} - c d\right ) \left (14 A c e + B b e - 16 B c d\right )\right )}{35 c e^{4}} - \frac{2 \sqrt{x} \left (- d\right )^{\frac{3}{2}} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (- 56 A b c e^{2} + 112 A c^{2} d e + B b^{2} e^{2} + 72 B b c d e - 128 B c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{35 c e^{\frac{11}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (- 5 b c e \left (7 A e - 8 B d\right ) \left (b e - 2 c d\right ) + \left (2 b^{2} e^{2} + 3 b c d e - 8 c^{2} d^{2}\right ) \left (14 A c e + B b e - 16 B c d\right )\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{35 c^{\frac{3}{2}} e^{5} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(b*x + c*x**2)**(3/2)*(7*A*e/2 - 4*B*d - B*e*x/2)/(7*e**2*sqrt(d + e*x)) + 8*
sqrt(d + e*x)*sqrt(b*x + c*x**2)*(5*b*c*e*(7*A*e - 8*B*d)/4 + 3*c*e*x*(14*A*c*e
+ B*b*e - 16*B*c*d)/4 + (b*e/4 - c*d)*(14*A*c*e + B*b*e - 16*B*c*d))/(35*c*e**4)
- 2*sqrt(x)*(-d)**(3/2)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(b*e - c*d)*(-56*A*b*c*
e**2 + 112*A*c**2*d*e + B*b**2*e**2 + 72*B*b*c*d*e - 128*B*c**2*d**2)*elliptic_f
(asin(sqrt(e)*sqrt(x)/sqrt(-d)), c*d/(b*e))/(35*c*e**(11/2)*sqrt(d + e*x)*sqrt(b
*x + c*x**2)) - 2*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(-5*b*c*e*(7*A*
e - 8*B*d)*(b*e - 2*c*d) + (2*b**2*e**2 + 3*b*c*d*e - 8*c**2*d**2)*(14*A*c*e + B
*b*e - 16*B*c*d))*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(35*c**(
3/2)*e**5*sqrt(1 + e*x/d)*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 6.30827, size = 514, normalized size = 1.14 \[ \frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left ((d+e x) \left (7 A c e (2 b e-3 c d)+B \left (b^2 e^2-25 b c d e+29 c^2 d^2\right )\right )+c e x (d+e x) (7 A c e+8 b B e-13 B c d)+35 c d (B d-A e) (c d-b e)+5 B c^2 e^2 x^2 (d+e x)\right )+\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (7 A c e (8 c d-b e)+2 B \left (b^2 e^2+6 b c d e-32 c^2 d^2\right )\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (7 A c e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (2 b^3 e^3+11 b^2 c d e^2-136 b c^2 d^2 e+128 c^3 d^3\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (7 A c e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (2 b^3 e^3+11 b^2 c d e^2-136 b c^2 d^2 e+128 c^3 d^3\right )\right )\right )\right )}{35 b c e^5 x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(35*c*d*(B*d - A*e)*(c*d - b*e) + (7*A*c
*e*(-3*c*d + 2*b*e) + B*(29*c^2*d^2 - 25*b*c*d*e + b^2*e^2))*(d + e*x) + c*e*(-1
3*B*c*d + 8*b*B*e + 7*A*c*e)*x*(d + e*x) + 5*B*c^2*e^2*x^2*(d + e*x)) + Sqrt[b/c
]*(Sqrt[b/c]*(7*A*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*(128*c^3*d^3 - 136
*b*c^2*d^2*e + 11*b^2*c*d*e^2 + 2*b^3*e^3))*(b + c*x)*(d + e*x) + I*b*e*(7*A*c*e
*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*(128*c^3*d^3 - 136*b*c^2*d^2*e + 11*b^2
*c*d*e^2 + 2*b^3*e^3))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*A
rcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(7*A*c*e*(8*c*d - b*
e) + 2*B*(-32*c^2*d^2 + 6*b*c*d*e + b^2*e^2))*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)])))/(35*b*c*e^5*x^
2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [B]  time = 0.048, size = 1610, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(-B*((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^3-71*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2*e^2+200*B*((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-9*B*((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
^3-264*B*((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+128*B*((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+7*A*((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*e^4+25*B*x^3
*b*c^4*d*e^3-13*B*x^4*b*c^4*e^4-9*B*x^3*b^2*c^3*e^4-16*B*x^3*c^5*d^2*e^2-14*A*x^
2*b^2*c^3*e^4+56*A*x^2*c^5*d^2*e^2-B*x^2*b^3*c^2*e^4-64*B*x^2*c^5*d^3*e-35*A*x^2
*b*c^4*d*e^3+16*B*x^2*b^2*c^3*d*e^3+44*B*x^2*b*c^4*d^2*e^2-49*A*x*b^2*c^3*d*e^3+
56*A*x*b*c^4*d^2*e^2-B*x*b^3*c^2*d*e^3+60*B*x*b^2*c^3*d^2*e^2-64*B*x*b*c^4*d^3*e
-2*B*((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^4-128*B*((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*B*x^4*c^5*d*e^3-21*A*x^3*b*c^4*e^4+14*A*x^3*c^5*d*e^3-7*A*x^4*c
^5*e^4-5*B*x^5*c^5*e^4+56*A*((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*e^3-168*A
*((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^2*e^2+112*A*((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^3*e-119*A*((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*e^3+224*A*
((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^2*e^2-112*A*((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^3*e+147*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3
*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))/c^3/
e^5/x/(c*e*x^2+b*e*x+c*d*x+b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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