3.401 \(\int \frac{\left (b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=457 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{63 c e^5}+\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}} \]

[Out]

(-2*Sqrt[d + e*x]*(128*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3 - 3
*c*e*(32*c^2*d^2 - 32*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(63*c*e^5) - (10*
Sqrt[d + e*x]*(16*c*d - 15*b*e - 14*c*e*x)*(b*x + c*x^2)^(3/2))/(63*e^3) - (2*(b
*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (4*Sqrt[-b]*(128*c^4*d^4 - 256*b*c^3*d^3*
e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqr
t[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*c^(3/
2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(2*c*d -
b*e)*(128*c^2*d^2 - 128*b*c*d*e - 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)])/(63*c^(3/2)*
e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.61489, antiderivative size = 457, 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 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-b^3 e^3-3 c e x \left (b^2 e^2-32 b c d e+32 c^2 d^2\right )+111 b^2 c d e^2-240 b c^2 d^2 e+128 c^3 d^3\right )}{63 c e^5}+\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{63 c^{3/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac{2 \left (b x+c x^2\right )^{5/2}}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(128*c^3*d^3 - 240*b*c^2*d^2*e + 111*b^2*c*d*e^2 - b^3*e^3 - 3
*c*e*(32*c^2*d^2 - 32*b*c*d*e + b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(63*c*e^5) - (10*
Sqrt[d + e*x]*(16*c*d - 15*b*e - 14*c*e*x)*(b*x + c*x^2)^(3/2))/(63*e^3) - (2*(b
*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (4*Sqrt[-b]*(128*c^4*d^4 - 256*b*c^3*d^3*
e + 135*b^2*c^2*d^2*e^2 - 7*b^3*c*d*e^3 - b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqr
t[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(63*c^(3/
2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(2*c*d -
b*e)*(128*c^2*d^2 - 128*b*c*d*e - 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)])/(63*c^(3/2)*
e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 159.106, size = 447, normalized size = 0.98 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{e \sqrt{d + e x}} + \frac{20 \sqrt{d + e x} \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (\frac{15 b e}{2} - 8 c d + 7 c e x\right )}{63 e^{3}} + \frac{8 \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (\frac{b^{3} e^{3}}{4} - \frac{111 b^{2} c d e^{2}}{4} + 60 b c^{2} d^{2} e - 32 c^{3} d^{3} + \frac{3 c e x \left (b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{4}\right )}{63 c e^{5}} + \frac{2 d \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) \left (b e - c d\right ) \left (b^{2} e^{2} + 128 b c d e - 128 c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{63 c^{\frac{3}{2}} e^{6} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{4 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b^{4} e^{4} + 7 b^{3} c d e^{3} - 135 b^{2} c^{2} d^{2} e^{2} + 256 b c^{3} d^{3} e - 128 c^{4} d^{4}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{63 c^{\frac{3}{2}} e^{6} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(3/2),x)

[Out]

-2*(b*x + c*x**2)**(5/2)/(e*sqrt(d + e*x)) + 20*sqrt(d + e*x)*(b*x + c*x**2)**(3
/2)*(15*b*e/2 - 8*c*d + 7*c*e*x)/(63*e**3) + 8*sqrt(d + e*x)*sqrt(b*x + c*x**2)*
(b**3*e**3/4 - 111*b**2*c*d*e**2/4 + 60*b*c**2*d**2*e - 32*c**3*d**3 + 3*c*e*x*(
b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)/4)/(63*c*e**5) + 2*d*sqrt(x)*sqrt(-b)*sqr
t(1 + c*x/b)*sqrt(1 + e*x/d)*(b*e - 2*c*d)*(b*e - c*d)*(b**2*e**2 + 128*b*c*d*e
- 128*c**2*d**2)*elliptic_f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(63*c**(3
/2)*e**6*sqrt(d + e*x)*sqrt(b*x + c*x**2)) - 4*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*
sqrt(d + e*x)*(b**4*e**4 + 7*b**3*c*d*e**3 - 135*b**2*c**2*d**2*e**2 + 256*b*c**
3*d**3*e - 128*c**4*d**4)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/
(63*c**(3/2)*e**6*sqrt(1 + e*x/d)*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 4.06668, size = 498, normalized size = 1.09 \[ \frac{2 (x (b+c x))^{5/2} \left (-e \sqrt{x} (b+c x) \left (-b^3 e^3 (d+e x)+3 b^2 c e^2 \left (37 d^2+11 d e x-5 e^2 x^2\right )-b c^2 e \left (240 d^3+64 d^2 e x-31 d e^2 x^2+19 e^3 x^3\right )+c^3 \left (128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right )\right )+\frac{2 (b+c x) (d+e x) \left (-b^4 e^4-7 b^3 c d e^3+135 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )}{c \sqrt{x}}+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^4 e^4+13 b^3 c d e^3-159 b^2 c^2 d^2 e^2+272 b c^3 d^3 e-128 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^4 e^4+7 b^3 c d e^3-135 b^2 c^2 d^2 e^2+256 b c^3 d^3 e-128 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )\right )}{63 c e^6 x^{5/2} (b+c x)^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(2*(x*(b + c*x))^(5/2)*((2*(128*c^4*d^4 - 256*b*c^3*d^3*e + 135*b^2*c^2*d^2*e^2
- 7*b^3*c*d*e^3 - b^4*e^4)*(b + c*x)*(d + e*x))/(c*Sqrt[x]) - e*Sqrt[x]*(b + c*x
)*(-(b^3*e^3*(d + e*x)) + 3*b^2*c*e^2*(37*d^2 + 11*d*e*x - 5*e^2*x^2) - b*c^2*e*
(240*d^3 + 64*d^2*e*x - 31*d*e^2*x^2 + 19*e^3*x^3) + c^3*(128*d^4 + 32*d^3*e*x -
16*d^2*e^2*x^2 + 10*d*e^3*x^3 - 7*e^4*x^4)) - (2*I)*Sqrt[b/c]*e*(-128*c^4*d^4 +
256*b*c^3*d^3*e - 135*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 + b^4*e^4)*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]*e*(-128*c^4*d^4 + 272*b*c^3*d^3*e - 159*b^2*c^2*d^2*e^2 + 13*b^3*c*d*
e^3 + 2*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)]))/(63*c*e^6*x^(5/2)*(b + c*x)^3*Sqrt[d + e*x])

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Maple [B]  time = 0.046, size = 1170, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(5/2)/(e*x+d)^(3/2),x)

[Out]

2/63*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(12*((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*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)*EllipticF(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c*d*e^4+125*((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^2*d^2*e^3+782*((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^3*d^3*e^2-768
*((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^4*d^4*e-128*x^2*c^6*d^4*e+26*x^5*b*c^5*e
^5-10*x^5*c^6*d*e^4+34*x^4*b^2*c^4*e^5+16*x^4*c^6*d^2*e^3+16*x^3*b^3*c^3*e^5-32*
x^3*c^6*d^3*e^2+x^2*b^4*c^2*e^5-41*x^4*b*c^5*d*e^4+640*((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^4*d^4*e-64*x^3*b^2*c^4*d*e^4+80*x^3*b*c^5*d^2*e^3-32*x^2*b^3*c^3*d
*e^4-47*x^2*b^2*c^4*d^2*e^3+2*((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^6*e^5-284*((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^2*d^2*e^3-510*((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^3*d^3*e^2+208*x^2*b*c^5*d^3*e^2+x*b^4*c^2*d*e^4-111*x*b^3*c^3*d^2*e^3+240
*x*b^2*c^4*d^3*e^2-256*((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^5*d^5+256*((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^5*d^5-128*x*b*c^5*d^4*e+7*x^6*c^6*e^5)/c^3/e^6/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. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(3/2),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*x + d)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)/(d + e*x)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(3/2), x)