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

Optimal. Leaf size=537 \[ \frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{693 c e^3}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^4 e^4-7 b^3 c d e^3-12 c e x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+195 b^2 c^2 d^2 e^2-304 b c^3 d^3 e+128 c^4 d^4\right )}{693 c^2 e^5}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^4 e^4+5 b^3 c d e^3+123 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^4 e^4+29 b^3 c d e^3+99 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{5/2} \sqrt{d+e x}}{11 e} \]

[Out]

(2*Sqrt[d + e*x]*(128*c^4*d^4 - 304*b*c^3*d^3*e + 195*b^2*c^2*d^2*e^2 - 7*b^3*c*
d*e^3 - 4*b^4*e^4 - 12*c*e*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*x)*Sq
rt[b*x + c*x^2])/(693*c^2*e^5) + (10*Sqrt[d + e*x]*(16*c^2*d^2 - 23*b*c*d*e + 3*
b^2*e^2 - 7*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(693*c*e^3) + (2*Sqrt[d +
e*x]*(b*x + c*x^2)^(5/2))/(11*e) - (2*Sqrt[-b]*(2*c*d - b*e)*(128*c^4*d^4 - 256*
b*c^3*d^3*e + 99*b^2*c^2*d^2*e^2 + 29*b^3*c*d*e^3 + 8*b^4*e^4)*Sqrt[x]*Sqrt[1 +
(c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)
])/(693*c^(5/2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (4*Sqrt[-b]*d*(c*d -
b*e)*(128*c^4*d^4 - 256*b*c^3*d^3*e + 123*b^2*c^2*d^2*e^2 + 5*b^3*c*d*e^3 + 2*b^
4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqr
t[x])/Sqrt[-b]], (b*e)/(c*d)])/(693*c^(5/2)*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.64846, antiderivative size = 537, 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 \[ \frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{693 c e^3}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^4 e^4-7 b^3 c d e^3-12 c e x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+195 b^2 c^2 d^2 e^2-304 b c^3 d^3 e+128 c^4 d^4\right )}{693 c^2 e^5}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^4 e^4+5 b^3 c d e^3+123 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^4 e^4+29 b^3 c d e^3+99 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{5/2} \sqrt{d+e x}}{11 e} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(128*c^4*d^4 - 304*b*c^3*d^3*e + 195*b^2*c^2*d^2*e^2 - 7*b^3*c*
d*e^3 - 4*b^4*e^4 - 12*c*e*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*x)*Sq
rt[b*x + c*x^2])/(693*c^2*e^5) + (10*Sqrt[d + e*x]*(16*c^2*d^2 - 23*b*c*d*e + 3*
b^2*e^2 - 7*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(693*c*e^3) + (2*Sqrt[d +
e*x]*(b*x + c*x^2)^(5/2))/(11*e) - (2*Sqrt[-b]*(2*c*d - b*e)*(128*c^4*d^4 - 256*
b*c^3*d^3*e + 99*b^2*c^2*d^2*e^2 + 29*b^3*c*d*e^3 + 8*b^4*e^4)*Sqrt[x]*Sqrt[1 +
(c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)
])/(693*c^(5/2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (4*Sqrt[-b]*d*(c*d -
b*e)*(128*c^4*d^4 - 256*b*c^3*d^3*e + 123*b^2*c^2*d^2*e^2 + 5*b^3*c*d*e^3 + 2*b^
4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqr
t[x])/Sqrt[-b]], (b*e)/(c*d)])/(693*c^(5/2)*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 5.02596, size = 557, normalized size = 1.04 \[ \frac{2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (-4 b^4 e^4+b^3 c e^3 (3 e x-7 d)+b^2 c^2 e^2 \left (195 d^2-139 d e x+113 e^2 x^2\right )+b c^3 e \left (-304 d^3+224 d^2 e x-185 d e^2 x^2+161 e^3 x^3\right )+c^4 \left (128 d^4-96 d^3 e x+80 d^2 e^2 x^2-70 d e^3 x^3+63 e^4 x^4\right )\right )+\sqrt{\frac{b}{c}} \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^5 e^5-9 b^4 c d e^4-34 b^3 c^2 d^2 e^3+259 b^2 c^3 d^3 e^2-336 b c^4 d^4 e+128 c^5 d^5\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^5 e^5-13 b^4 c d e^4-41 b^3 c^2 d^2 e^3+454 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (8 b^5 e^5+13 b^4 c d e^4+41 b^3 c^2 d^2 e^3-454 b^2 c^3 d^3 e^2+640 b c^4 d^4 e-256 c^5 d^5\right )\right )\right )}{693 b c^2 e^6 x^3 (b+c x)^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^4*e^4 + b^3*c*e^3*(-7*d
+ 3*e*x) + b^2*c^2*e^2*(195*d^2 - 139*d*e*x + 113*e^2*x^2) + b*c^3*e*(-304*d^3 +
224*d^2*e*x - 185*d*e^2*x^2 + 161*e^3*x^3) + c^4*(128*d^4 - 96*d^3*e*x + 80*d^2
*e^2*x^2 - 70*d*e^3*x^3 + 63*e^4*x^4)) + Sqrt[b/c]*(Sqrt[b/c]*(-256*c^5*d^5 + 64
0*b*c^4*d^4*e - 454*b^2*c^3*d^3*e^2 + 41*b^3*c^2*d^2*e^3 + 13*b^4*c*d*e^4 + 8*b^
5*e^5)*(b + c*x)*(d + e*x) - I*b*e*(256*c^5*d^5 - 640*b*c^4*d^4*e + 454*b^2*c^3*
d^3*e^2 - 41*b^3*c^2*d^2*e^3 - 13*b^4*c*d*e^4 - 8*b^5*e^5)*Sqrt[1 + b/(c*x)]*Sqr
t[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*
b*e*(128*c^5*d^5 - 336*b*c^4*d^4*e + 259*b^2*c^3*d^3*e^2 - 34*b^3*c^2*d^2*e^3 -
9*b^4*c*d*e^4 - 8*b^5*e^5)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF
[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(693*b*c^2*e^6*x^3*(b + c*x)^3*Sq
rt[d + e*x])

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Maple [B]  time = 0.031, size = 1441, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x)

[Out]

-2/693*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(8*((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^7*e
^6-256*((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^6*d^6+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^6*d^6+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^5*c^2*d^2*e^4+31*x^5*b*c^6
*d*e^5+7*x*b^4*c^3*d^2*e^4-195*x*b^3*c^4*d^3*e^3+304*x*b^2*c^5*d^4*e^2-128*x*b*c
^6*d^5*e+28*((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*c^2*d^2*e^4+768*((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^5*d^5*e-758*((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^4*d^4
*e^2+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^6*c*d*e^5+1094*((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^4*d^4*e^2+236*((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^3*d^3*e^3-8
96*((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^5*d^5*e+5*((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*c*d*e^5-63*x^7*c^7*e^6-495*((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^3*d^3
*e^3+50*x^4*b^2*c^5*d*e^5-49*x^4*b*c^6*d^2*e^4+30*x^3*b^3*c^4*d*e^5-95*x^3*b^2*c
^5*d^2*e^4+96*x^3*b*c^6*d^3*e^3+8*x^2*b^4*c^3*d*e^5-49*x^2*b^3*c^4*d^2*e^4-115*x
^2*b^2*c^5*d^3*e^3+272*x^2*b*c^6*d^4*e^2+4*x*b^5*c^2*d*e^5-224*x^6*b*c^6*e^6+7*x
^6*c^7*d*e^5-274*x^5*b^2*c^5*e^6-10*x^5*c^7*d^2*e^4-116*x^4*b^3*c^4*e^6+16*x^4*c
^7*d^3*e^3+x^3*b^4*c^3*e^6-32*x^3*c^7*d^4*e^2+4*x^2*b^5*c^2*e^6-128*x^2*c^7*d^5*
e)/c^4/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. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}{\sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + b^2*x^2)*sqrt(c*x^2 + b*x)/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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.795628, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done