3.406 \(\int \frac{(d+e x)^{7/2}}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=379 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}+\frac{12 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{5/2}}{7 c} \]

[Out]

(2*e*(71*c^2*d^2 - 71*b*c*d*e + 24*b^2*e^2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(10
5*c^3) + (12*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(35*c^2) + (2*e*
(d + e*x)^(5/2)*Sqrt[b*x + c*x^2])/(7*c) + (16*Sqrt[-b]*(2*c*d - b*e)*(11*c^2*d^
2 - 11*b*c*d*e + 6*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)])/(105*c^(7/2)*Sqrt[1 + (e*x)/d]*S
qrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(71*c^2*d^2 - 71*b*c*d*e + 24*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^(7/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.38719, antiderivative size = 379, 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 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}+\frac{12 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{5/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(71*c^2*d^2 - 71*b*c*d*e + 24*b^2*e^2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(10
5*c^3) + (12*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(35*c^2) + (2*e*
(d + e*x)^(5/2)*Sqrt[b*x + c*x^2])/(7*c) + (16*Sqrt[-b]*(2*c*d - b*e)*(11*c^2*d^
2 - 11*b*c*d*e + 6*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)])/(105*c^(7/2)*Sqrt[1 + (e*x)/d]*S
qrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(71*c^2*d^2 - 71*b*c*d*e + 24*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^(7/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 156.414, size = 359, normalized size = 0.95 \[ \frac{2 e \left (d + e x\right )^{\frac{5}{2}} \sqrt{b x + c x^{2}}}{7 c} - \frac{12 e \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{35 c^{2}} + \frac{2 e \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (24 b^{2} e^{2} - 71 b c d e + 71 c^{2} d^{2}\right )}{105 c^{3}} + \frac{2 d \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (24 b^{2} e^{2} - 71 b c d e + 71 c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{105 c^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{16 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) \left (6 b^{2} e^{2} - 11 b c d e + 11 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{105 c^{\frac{7}{2}} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

2*e*(d + e*x)**(5/2)*sqrt(b*x + c*x**2)/(7*c) - 12*e*(d + e*x)**(3/2)*(b*e - 2*c
*d)*sqrt(b*x + c*x**2)/(35*c**2) + 2*e*sqrt(d + e*x)*sqrt(b*x + c*x**2)*(24*b**2
*e**2 - 71*b*c*d*e + 71*c**2*d**2)/(105*c**3) + 2*d*sqrt(x)*sqrt(-b)*sqrt(1 + c*
x/b)*sqrt(1 + e*x/d)*(b*e - c*d)*(24*b**2*e**2 - 71*b*c*d*e + 71*c**2*d**2)*elli
ptic_f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(105*c**(7/2)*sqrt(d + e*x)*sq
rt(b*x + c*x**2)) - 16*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(b*e - 2*c
*d)*(6*b**2*e**2 - 11*b*c*d*e + 11*c**2*d**2)*elliptic_e(asin(sqrt(c)*sqrt(x)/sq
rt(-b)), b*e/(c*d))/(105*c**(7/2)*sqrt(1 + e*x/d)*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 3.99003, size = 388, normalized size = 1.02 \[ \frac{2 \sqrt{x} \left (e \sqrt{x} (b+c x) (d+e x) \left (24 b^2 e^2-b c e (89 d+18 e x)+c^2 \left (122 d^2+66 d e x+15 e^2 x^2\right )\right )+\frac{8 (b+c x) (d+e x) \left (-6 b^3 e^3+23 b^2 c d e^2-33 b c^2 d^2 e+22 c^3 d^3\right )}{c \sqrt{x}}+8 i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-6 b^3 e^3+23 b^2 c d e^2-33 b c^2 d^2 e+22 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (48 b^4 e^4-208 b^3 c d e^3+353 b^2 c^2 d^2 e^2-298 b c^3 d^3 e+105 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )}{b}\right )}{105 c^3 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*((8*(22*c^3*d^3 - 33*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 6*b^3*e^3)*(b + c
*x)*(d + e*x))/(c*Sqrt[x]) + e*Sqrt[x]*(b + c*x)*(d + e*x)*(24*b^2*e^2 - b*c*e*(
89*d + 18*e*x) + c^2*(122*d^2 + 66*d*e*x + 15*e^2*x^2)) + (8*I)*Sqrt[b/c]*e*(22*
c^3*d^3 - 33*b*c^2*d^2*e + 23*b^2*c*d*e^2 - 6*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1
+ d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + (I*Sqrt[b/c]
*(105*c^4*d^4 - 298*b*c^3*d^3*e + 353*b^2*c^2*d^2*e^2 - 208*b^3*c*d*e^3 + 48*b^4
*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x
]], (c*d)/(b*e)])/b))/(105*c^3*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.063, size = 918, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x)

[Out]

2/105*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(15*x^5*c^5*e^4+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^4*c*d*e^3-95*((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+142*((
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-71*((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+48*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*e^4-232*((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+448*((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^2-440
*((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+176*((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^4-3*x^4*b*c^4*e^4+81*x^4*c^5*d*e^3+6*x^3*b^2*c^3*e^4-26*x^3*b*c^4*d
*e^3+188*x^3*c^5*d^2*e^2+24*x^2*b^3*c^2*e^4-83*x^2*b^2*c^3*d*e^3+99*x^2*b*c^4*d^
2*e^2+122*x^2*c^5*d^3*e+24*x*b^3*c^2*d*e^3-89*x*b^2*c^3*d^2*e^2+122*x*b*c^4*d^3*
e)/c^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. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x),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)/sqrt(c*x^2 + b*
x), 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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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