3.386 \(\int (d+e x)^{3/2} \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=362 \[ \frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{105 c^2 e}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

[Out]

(2*Sqrt[d + e*x]*(3*c^2*d^2 + 9*b*c*d*e - 4*b^2*e^2 + 12*c*e*(2*c*d - b*e)*x)*Sq
rt[b*x + c*x^2])/(105*c^2*e) + (2*e*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(7*c) - (
2*Sqrt[-b]*(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]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])
/(105*c^(5/2)*e^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (4*Sqrt[-b]*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)])/(105*c^(5/2)*e^2*
Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.15247, antiderivative size = 362, 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 \[ \frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{105 c^2 e}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(3*c^2*d^2 + 9*b*c*d*e - 4*b^2*e^2 + 12*c*e*(2*c*d - b*e)*x)*Sq
rt[b*x + c*x^2])/(105*c^2*e) + (2*e*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(7*c) - (
2*Sqrt[-b]*(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]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])
/(105*c^(5/2)*e^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (4*Sqrt[-b]*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)])/(105*c^(5/2)*e^2*
Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 135.527, size = 343, normalized size = 0.95 \[ \frac{2 e \sqrt{d + e x} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{7 c} - \frac{8 \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (b^{2} e^{2} - \frac{9 b c d e}{4} - \frac{3 c^{2} d^{2}}{4} + 3 c e x \left (b e - 2 c d\right )\right )}{105 c^{2} e} + \frac{4 \sqrt{x} \left (- d\right )^{\frac{3}{2}} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (2 b^{2} e^{2} - 3 b c d e + 3 c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{105 c^{2} e^{\frac{5}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) \left (8 b^{2} e^{2} - 3 b c d e + 3 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{105 c^{\frac{5}{2}} e^{2} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(c*x**2+b*x)**(1/2),x)

[Out]

2*e*sqrt(d + e*x)*(b*x + c*x**2)**(3/2)/(7*c) - 8*sqrt(d + e*x)*sqrt(b*x + c*x**
2)*(b**2*e**2 - 9*b*c*d*e/4 - 3*c**2*d**2/4 + 3*c*e*x*(b*e - 2*c*d))/(105*c**2*e
) + 4*sqrt(x)*(-d)**(3/2)*sqrt(1 + c*x/b)*sqrt(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(asin(sqrt(e)*sqrt(x)/sqrt(-d)), c*d/(b*
e))/(105*c**2*e**(5/2)*sqrt(d + e*x)*sqrt(b*x + c*x**2)) + 2*sqrt(x)*sqrt(-b)*sq
rt(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))/(105*c**(5/2)*e**2*sqrt(
1 + e*x/d)*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 2.99292, size = 372, normalized size = 1.03 \[ \frac{2 \left (b e x (b+c x) (d+e x) \left (-4 b^2 e^2+3 b c e (3 d+e x)+3 c^2 \left (d^2+8 d e x+5 e^2 x^2\right )\right )-\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^3 e^3+23 b^2 c d e^2-18 b c^2 d^2 e+3 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-8 b^3 e^3+19 b^2 c d e^2-9 b c^2 d^2 e+6 c^3 d^3\right )\right )\right )}{105 b c^2 e^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^2*e^2 + 3*b*c*e*(3*d + e*x) + 3*c^2*(d^2 + 8
*d*e*x + 5*e^2*x^2)) - 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)*Ellipti
cF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(105*b*c^2*e^2*Sqrt[x*(b + c*x)
]*Sqrt[d + e*x])

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Maple [B]  time = 0.083, size = 920, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x)

[Out]

-2/105*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(-15*x^5*c^5*e^4+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)*Ellipt
icE(((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^4-18*x^4*b*c^4*e^4-39*x^4*c^5*d*e^3+x^3*b^2*c^3*e^4-51*x^3*b*c^4*d*e^3-27*
x^3*c^5*d^2*e^2+4*x^2*b^3*c^2*e^4-8*x^2*b^2*c^3*d*e^3-36*x^2*b*c^4*d^2*e^2-3*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^2-3*x*b*c^4*d^3*e)/e^2/x/(c*e*x^2
+b*e*x+c*d*x+b*d)/c^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)*(c*x**2+b*x)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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