3.402 \(\int \frac{\left (b x+c x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=401 \[ \frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{21 \sqrt{c} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{21 \sqrt{c} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (51 b^2 e^2-48 c e x (2 c d-b e)-176 b c d e+128 c^2 d^2\right )}{21 e^5}+\frac{10 \left (b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]

[Out]

(2*Sqrt[d + e*x]*(128*c^2*d^2 - 176*b*c*d*e + 51*b^2*e^2 - 48*c*e*(2*c*d - b*e)*
x)*Sqrt[b*x + c*x^2])/(21*e^5) + (10*(16*c*d - 7*b*e + 2*c*e*x)*(b*x + c*x^2)^(3
/2))/(21*e^3*Sqrt[d + e*x]) - (2*(b*x + c*x^2)^(5/2))/(3*e*(d + e*x)^(3/2)) - (2
*Sqrt[-b]*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e + 3*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
)])/(21*Sqrt[c]*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (4*Sqrt[-b]*d*(c*d -
b*e)*(128*c^2*d^2 - 128*b*c*d*e + 27*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)])/(21*Sqrt[c
]*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.3685, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ \frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{21 \sqrt{c} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (3 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{21 \sqrt{c} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (51 b^2 e^2-48 c e x (2 c d-b e)-176 b c d e+128 c^2 d^2\right )}{21 e^5}+\frac{10 \left (b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{21 e^3 \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{3 e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(128*c^2*d^2 - 176*b*c*d*e + 51*b^2*e^2 - 48*c*e*(2*c*d - b*e)*
x)*Sqrt[b*x + c*x^2])/(21*e^5) + (10*(16*c*d - 7*b*e + 2*c*e*x)*(b*x + c*x^2)^(3
/2))/(21*e^3*Sqrt[d + e*x]) - (2*(b*x + c*x^2)^(5/2))/(3*e*(d + e*x)^(3/2)) - (2
*Sqrt[-b]*(2*c*d - b*e)*(128*c^2*d^2 - 128*b*c*d*e + 3*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
)])/(21*Sqrt[c]*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (4*Sqrt[-b]*d*(c*d -
b*e)*(128*c^2*d^2 - 128*b*c*d*e + 27*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)])/(21*Sqrt[c
]*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 74.4673, size = 381, normalized size = 0.95 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} - \frac{20 \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (\frac{7 b e}{2} - 8 c d - c e x\right )}{21 e^{3} \sqrt{d + e x}} + \frac{8 \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (\frac{51 b^{2} e^{2}}{4} - 44 b c d e + 32 c^{2} d^{2} + 12 c e x \left (b e - 2 c d\right )\right )}{21 e^{5}} + \frac{4 \sqrt{x} \left (- d\right )^{\frac{3}{2}} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (27 b^{2} e^{2} - 128 b c d e + 128 c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{21 e^{\frac{13}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) \left (3 b^{2} e^{2} - 128 b c d e + 128 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{21 \sqrt{c} e^{6} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(5/2),x)

[Out]

-2*(b*x + c*x**2)**(5/2)/(3*e*(d + e*x)**(3/2)) - 20*(b*x + c*x**2)**(3/2)*(7*b*
e/2 - 8*c*d - c*e*x)/(21*e**3*sqrt(d + e*x)) + 8*sqrt(d + e*x)*sqrt(b*x + c*x**2
)*(51*b**2*e**2/4 - 44*b*c*d*e + 32*c**2*d**2 + 12*c*e*x*(b*e - 2*c*d))/(21*e**5
) + 4*sqrt(x)*(-d)**(3/2)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(b*e - c*d)*(27*b**2*e
**2 - 128*b*c*d*e + 128*c**2*d**2)*elliptic_f(asin(sqrt(e)*sqrt(x)/sqrt(-d)), c*
d/(b*e))/(21*e**(13/2)*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)*(3*b**2*e**2 - 128*b*c*d*e + 128*c**2*
d**2)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(21*sqrt(c)*e**6*sqr
t(1 + e*x/d)*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 3.87957, size = 442, normalized size = 1.1 \[ \frac{2 (x (b+c x))^{5/2} \left (\frac{e \sqrt{x} (b+c x) \left (b^2 e^2 \left (51 d^2+67 d e x+9 e^2 x^2\right )+b c e \left (-176 d^3-224 d^2 e x-25 d e^2 x^2+9 e^3 x^3\right )+c^2 \left (128 d^4+160 d^3 e x+16 d^2 e^2 x^2-6 d e^3 x^3+3 e^4 x^4\right )\right )}{d+e x}-\frac{(b+c x) (d+e x) \left (-3 b^3 e^3+134 b^2 c d e^2-384 b c^2 d^2 e+256 c^3 d^3\right )}{c \sqrt{x}}+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-3 b^3 e^3+83 b^2 c d e^2-208 b c^2 d^2 e+128 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (3 b^3 e^3-134 b^2 c d e^2+384 b c^2 d^2 e-256 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )\right )}{21 e^6 x^{5/2} (b+c x)^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(5/2),x]

[Out]

(2*(x*(b + c*x))^(5/2)*(-(((256*c^3*d^3 - 384*b*c^2*d^2*e + 134*b^2*c*d*e^2 - 3*
b^3*e^3)*(b + c*x)*(d + e*x))/(c*Sqrt[x])) + (e*Sqrt[x]*(b + c*x)*(b^2*e^2*(51*d
^2 + 67*d*e*x + 9*e^2*x^2) + b*c*e*(-176*d^3 - 224*d^2*e*x - 25*d*e^2*x^2 + 9*e^
3*x^3) + c^2*(128*d^4 + 160*d^3*e*x + 16*d^2*e^2*x^2 - 6*d*e^3*x^3 + 3*e^4*x^4))
)/(d + e*x) + I*Sqrt[b/c]*e*(-256*c^3*d^3 + 384*b*c^2*d^2*e - 134*b^2*c*d*e^2 +
3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/S
qrt[x]], (c*d)/(b*e)] + I*Sqrt[b/c]*e*(128*c^3*d^3 - 208*b*c^2*d^2*e + 83*b^2*c*
d*e^2 - 3*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqr
t[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(21*e^6*x^(5/2)*(b + c*x)^3*Sqrt[d + e*x])

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Maple [B]  time = 0.049, size = 1692, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(5/2)/(e*x+d)^(5/2),x)

[Out]

-2/21*(x*(c*x+b))^(1/2)*(-640*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*x*b^2*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)
+256*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^4*d^4*e*((c*x+b)/b
)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+54*EllipticF(((c*x+b)/b)^(1/
2),(b*e/(b*e-c*d))^(1/2))*x*b^4*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)-310*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b
^3*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)+512
*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*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)-256*EllipticF(((c*x+b)/b)^(1
/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^4*d^4*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d)
)^(1/2)*(-c*x/b)^(1/2)-137*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*
b^4*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)-12*x^5
*b*c^4*e^5+6*x^5*c^5*d*e^4-18*x^4*b^2*c^3*e^5-16*x^4*c^5*d^2*e^3-9*x^3*b^3*c^2*e
^5-137*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*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)-160*x^3*c^5*d^3*e^2-128*x^
2*c^5*d^4*e-3*x^6*c^5*e^5+31*x^4*b*c^4*d*e^4+3*EllipticE(((c*x+b)/b)^(1/2),(b*e/
(b*e-c*d))^(1/2))*x*b^5*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)+3*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*d*e^4*((c*x+b
)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+256*EllipticE(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))
^(1/2)*(-c*x/b)^(1/2)-256*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c
^4*d^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+518*Ellipti
cE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*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)-640*EllipticE(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*b^2*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)+54*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*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)-310*EllipticF(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*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)+512*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*
d))^(1/2))*b^2*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)+518*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*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)+173*x^2*b^2*c^3*d^
2*e^3+16*x^2*b*c^4*d^3*e^2-51*x*b^3*c^2*d^2*e^3+176*x*b^2*c^3*d^3*e^2-128*x*b*c^
4*d^4*e-42*x^3*b^2*c^3*d*e^4+208*x^3*b*c^4*d^2*e^3-67*x^2*b^3*c^2*d*e^4)/(c*x+b)
/x/(e*x+d)^(3/2)/c^2/e^6

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x}}{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*sqrt(c*x^2 + b*x)/((e^2*x^2 + 2*d*e*x +
d^2)*sqrt(e*x + d)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)/(d + e*x)**(5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(5/2), x)