3.403 \(\int \frac{\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=392 \[ -\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (15 b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 \sqrt{c} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{15 e^5 \sqrt{d+e x}}+\frac{2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^3 (d+e x)^{3/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}} \]

[Out]

(-2*(128*c^2*d^2 - 112*b*c*d*e + 15*b^2*e^2 + 16*c*e*(2*c*d - b*e)*x)*Sqrt[b*x +
c*x^2])/(15*e^5*Sqrt[d + e*x]) + (2*(16*c*d - 5*b*e + 6*c*e*x)*(b*x + c*x^2)^(3
/2))/(15*e^3*(d + e*x)^(3/2)) - (2*(b*x + c*x^2)^(5/2))/(5*e*(d + e*x)^(5/2)) +
(4*Sqrt[-b]*Sqrt[c]*(128*c^2*d^2 - 128*b*c*d*e + 23*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)])
/(15*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(2*c*d - b*e)*(128*c
^2*d^2 - 128*b*c*d*e + 15*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*E
llipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*Sqrt[c]*e^6*Sqrt[
d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.26648, antiderivative size = 392, 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 \[ -\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \left (15 b^2 e^2-128 b c d e+128 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 \sqrt{c} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{15 e^5 \sqrt{d+e x}}+\frac{2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^3 (d+e x)^{3/2}}-\frac{2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(5/2)/(d + e*x)^(7/2),x]

[Out]

(-2*(128*c^2*d^2 - 112*b*c*d*e + 15*b^2*e^2 + 16*c*e*(2*c*d - b*e)*x)*Sqrt[b*x +
c*x^2])/(15*e^5*Sqrt[d + e*x]) + (2*(16*c*d - 5*b*e + 6*c*e*x)*(b*x + c*x^2)^(3
/2))/(15*e^3*(d + e*x)^(3/2)) - (2*(b*x + c*x^2)^(5/2))/(5*e*(d + e*x)^(5/2)) +
(4*Sqrt[-b]*Sqrt[c]*(128*c^2*d^2 - 128*b*c*d*e + 23*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)])
/(15*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(2*c*d - b*e)*(128*c
^2*d^2 - 128*b*c*d*e + 15*b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*E
llipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*Sqrt[c]*e^6*Sqrt[
d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 119.516, size = 376, normalized size = 0.96 \[ \frac{4 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (23 b^{2} e^{2} - 128 b c d e + 128 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{15 e^{6} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} - \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} - \frac{4 \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (\frac{5 b e}{2} - 8 c d - 3 c e x\right )}{15 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{8 \sqrt{b x + c x^{2}} \left (\frac{15 b^{2} e^{2}}{4} - 28 b c d e + 32 c^{2} d^{2} - 4 c e x \left (b e - 2 c d\right )\right )}{15 e^{5} \sqrt{d + e x}} + \frac{2 \sqrt{x} \sqrt{- d} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) \left (15 b^{2} e^{2} - 128 b c d e + 128 c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{15 e^{\frac{13}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(7/2),x)

[Out]

4*sqrt(c)*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(23*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))/(15*
e**6*sqrt(1 + e*x/d)*sqrt(b*x + c*x**2)) - 2*(b*x + c*x**2)**(5/2)/(5*e*(d + e*x
)**(5/2)) - 4*(b*x + c*x**2)**(3/2)*(5*b*e/2 - 8*c*d - 3*c*e*x)/(15*e**3*(d + e*
x)**(3/2)) - 8*sqrt(b*x + c*x**2)*(15*b**2*e**2/4 - 28*b*c*d*e + 32*c**2*d**2 -
4*c*e*x*(b*e - 2*c*d))/(15*e**5*sqrt(d + e*x)) + 2*sqrt(x)*sqrt(-d)*sqrt(1 + c*x
/b)*sqrt(1 + e*x/d)*(b*e - 2*c*d)*(15*b**2*e**2 - 128*b*c*d*e + 128*c**2*d**2)*e
lliptic_f(asin(sqrt(e)*sqrt(x)/sqrt(-d)), c*d/(b*e))/(15*e**(13/2)*sqrt(d + e*x)
*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 2.51506, size = 401, normalized size = 1.02 \[ \frac{2 (x (b+c x))^{5/2} \left (\frac{2 (b+c x) (d+e x) \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right )}{\sqrt{x}}-i c e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (31 b^2 e^2-144 b c d e+128 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 i c e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-\frac{e \sqrt{x} (b+c x) \left (b^2 e^2 \left (15 d^2+35 d e x+23 e^2 x^2\right )-b c e \left (112 d^3+256 d^2 e x+161 d e^2 x^2+11 e^3 x^3\right )+c^2 \left (128 d^4+288 d^3 e x+176 d^2 e^2 x^2+10 d e^3 x^3-3 e^4 x^4\right )\right )}{(d+e x)^2}\right )}{15 e^6 x^{5/2} (b+c x)^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(5/2)/(d + e*x)^(7/2),x]

[Out]

(2*(x*(b + c*x))^(5/2)*((2*(128*c^2*d^2 - 128*b*c*d*e + 23*b^2*e^2)*(b + c*x)*(d
+ e*x))/Sqrt[x] - (e*Sqrt[x]*(b + c*x)*(b^2*e^2*(15*d^2 + 35*d*e*x + 23*e^2*x^2
) - b*c*e*(112*d^3 + 256*d^2*e*x + 161*d*e^2*x^2 + 11*e^3*x^3) + c^2*(128*d^4 +
288*d^3*e*x + 176*d^2*e^2*x^2 + 10*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x)^2 + (2*I)*
Sqrt[b/c]*c*e*(128*c^2*d^2 - 128*b*c*d*e + 23*b^2*e^2)*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]*
c*e*(128*c^2*d^2 - 144*b*c*d*e + 31*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]
*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(15*e^6*x^(5/2)*(b + c
*x)^3*Sqrt[d + e*x])

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Maple [B]  time = 0.049, size = 2170, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(5/2)/(e*x+d)^(7/2),x)

[Out]

2/15*(x*(c*x+b))^(1/2)*(15*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^
2*b^4*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+15*Ellip
ticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*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)-1024*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*x*b^2*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*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*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)-316*EllipticF
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*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)+768*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*
e-c*d))^(1/2))*x*b^2*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))*x*b*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)+302*EllipticE(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*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)-512*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e
-c*d))^(1/2))*x^2*b^2*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)+256*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*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)-158*Ellip
ticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*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)+384*EllipticF(((c*x+b)/b)^(1/2),(b*e
/(b*e-c*d))^(1/2))*x^2*b^2*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)-256*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*
b*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)+604*
EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*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)+256*EllipticE(((c*x+b)/b)^(1/2)
,(b*e/(b*e-c*d))^(1/2))*b*c^3*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^3*d^5
*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-46*EllipticE(((c*
x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^4*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(
b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-46*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1
/2))*b^4*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)+1
4*x^5*b*c^3*e^5-10*x^5*c^4*d*e^4-15*x*b^3*c*d^2*e^3-92*EllipticE(((c*x+b)/b)^(1/
2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(
1/2)*(-c*x/b)^(1/2)+384*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c
^2*d^4*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*x^6*c^4
*e^5+302*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*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)+30*EllipticF(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)+126*x^3*b^2*c^2*d*e^4+80*x^3*b*c^3*d^2*e^3-35*x^2*b^3*c*
d*e^4+241*x^2*b^2*c^2*d^2*e^3-176*x^2*b*c^3*d^3*e^2+112*x*b^2*c^2*d^3*e^2-128*x*
b*c^3*d^4*e-512*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e
*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-158*EllipticF(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*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)+151*x^4*b*c^3*d*e^4-12*x^4*b^2*c^2*e^5-176*x^4
*c^4*d^2*e^3-23*x^3*b^3*c*e^5-288*x^3*c^4*d^3*e^2-128*x^2*c^4*d^4*e)/(c*x+b)/x/(
e*x+d)^(5/2)/e^6/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(7/2),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^3*x^3 + 3*d*e^2*x
^2 + 3*d^2*e*x + d^3)*sqrt(e*x + d)), 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((c*x**2+b*x)**(5/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(5/2)/(e*x + d)^(7/2), x)