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

Optimal. Leaf size=309 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{16 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} (-7 b e+8 c d-6 c e x)}{5 e^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}} \]

[Out]

(-2*Sqrt[d + e*x]*(8*c*d - 7*b*e - 6*c*e*x)*Sqrt[b*x + c*x^2])/(5*e^3) - (2*(b*x
+ c*x^2)^(3/2))/(e*Sqrt[d + e*x]) + (2*Sqrt[-b]*(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*Sqrt[c]*e^4*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (
16*Sqrt[-b]*d*(c*d - b*e)*(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)])/(5*Sqrt[c]*e^4*S
qrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.03868, antiderivative size = 309, 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 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{16 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{5 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} (-7 b e+8 c d-6 c e x)}{5 e^3}-\frac{2 \left (b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(8*c*d - 7*b*e - 6*c*e*x)*Sqrt[b*x + c*x^2])/(5*e^3) - (2*(b*x
+ c*x^2)^(3/2))/(e*Sqrt[d + e*x]) + (2*Sqrt[-b]*(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*Sqrt[c]*e^4*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (
16*Sqrt[-b]*d*(c*d - b*e)*(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)])/(5*Sqrt[c]*e^4*S
qrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 111.234, size = 289, normalized size = 0.94 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{e \sqrt{d + e x}} + \frac{4 \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (\frac{7 b e}{2} - 4 c d + 3 c e x\right )}{5 e^{3}} - \frac{16 d \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) \left (b e - c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{5 \sqrt{c} e^{4} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{5 \sqrt{c} e^{4} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(b*x + c*x**2)**(3/2)/(e*sqrt(d + e*x)) + 4*sqrt(d + e*x)*sqrt(b*x + c*x**2)*
(7*b*e/2 - 4*c*d + 3*c*e*x)/(5*e**3) - 16*d*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqr
t(1 + e*x/d)*(b*e - 2*c*d)*(b*e - c*d)*elliptic_f(asin(sqrt(c)*sqrt(x)/sqrt(-b))
, b*e/(c*d))/(5*sqrt(c)*e**4*sqrt(d + e*x)*sqrt(b*x + c*x**2)) + 2*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)*ellipt
ic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(5*sqrt(c)*e**4*sqrt(1 + e*x/d)*
sqrt(b*x + c*x**2))

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Mathematica [C]  time = 2.68797, size = 340, normalized size = 1.1 \[ \frac{-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-9 b c d e+8 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 \left (b^3 e^2 (d+e x)+b^2 c e \left (-16 d^2-8 d e x+3 e^2 x^2\right )+b c^2 \left (16 d^3-8 d^2 e x-11 d e^2 x^2+3 e^3 x^3\right )+c^3 x \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{5 c e^4 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b^3*e^2*(d + e*x) + b^2*c*e*(-16*d^2 - 8*d*e*x + 3*e^2*x^2) + c^3*x*(16*d^3
+ 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + b*c^2*(16*d^3 - 8*d^2*e*x - 11*d*e^2*x^2
+ 3*e^3*x^3)) + (2*I)*Sqrt[b/c]*c*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)] - (2*I)*Sqrt[b/c]*c*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[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)
])/(5*c*e^4*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.036, size = 685, normalized size = 2.2 \[ -{\frac{2}{5\,{c}^{2}x \left ( ce{x}^{2}+bex+cdx+bd \right ){e}^{4}}\sqrt{x \left ( cx+b \right ) }\sqrt{ex+d} \left ( 8\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}cd{e}^{2}-24\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}{c}^{2}{d}^{2}e+16\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{3}{d}^{3}+\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{4}{e}^{3}-17\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}cd{e}^{2}+32\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}{c}^{2}{d}^{2}e-16\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{3}{d}^{3}-{x}^{4}{c}^{4}{e}^{3}-3\,{x}^{3}b{c}^{3}{e}^{3}+2\,{x}^{3}{c}^{4}d{e}^{2}-2\,{x}^{2}{b}^{2}{c}^{2}{e}^{3}-5\,{x}^{2}b{c}^{3}d{e}^{2}+8\,{x}^{2}{c}^{4}{d}^{2}e-7\,x{b}^{2}{c}^{2}d{e}^{2}+8\,xb{c}^{3}{d}^{2}e \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(8*((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
*e^2-24*((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^2*d^2*e+16*((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^3*d^3+((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*e^3-17*((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*d*e^2+32*((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^2*d^2*e-
16*((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^3*d^3-x^4*c^4*e^3-3*x^3*b*c^3*e^3+2*x^3*
c^4*d*e^2-2*x^2*b^2*c^2*e^3-5*x^2*b*c^3*d*e^2+8*x^2*c^4*d^2*e-7*x*b^2*c^2*d*e^2+
8*x*b*c^3*d^2*e)/c^2/x/(c*e*x^2+b*e*x+c*d*x+b*d)/e^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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