3.2443 \(\int \sqrt{d x} \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=616 \[ \frac{2 \sqrt{d x} \left (3 c x \left (308 a^2 c^2-181 a b^2 c+24 b^4\right )+b \left (108 a^2 c^2-151 a b^2 c+24 b^4\right )\right ) \sqrt{a+b x+c x^2}}{9009 c^3}-\frac{4 d x \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right ) \sqrt{a+b x+c x^2}}{9009 c^{7/2} \sqrt{d x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{\sqrt [4]{a} d \sqrt{x} \left (\sqrt{a} b \sqrt{c} \left (708 a^2 c^2-241 a b^2 c+24 b^4\right )+2 \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right )\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{9009 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt [4]{a} d \sqrt{x} \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{9009 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}-\frac{10 \sqrt{d x} \left (14 c x \left (3 b^2-11 a c\right )+3 b \left (6 b^2-19 a c\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac{2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}+\frac{10 b \sqrt{d x} \left (a+b x+c x^2\right )^{5/2}}{143 c} \]

[Out]

(-4*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*d*x*Sqrt[a + b*x + c*
x^2])/(9009*c^(7/2)*Sqrt[d*x]*(Sqrt[a] + Sqrt[c]*x)) + (2*Sqrt[d*x]*(b*(24*b^4 -
151*a*b^2*c + 108*a^2*c^2) + 3*c*(24*b^4 - 181*a*b^2*c + 308*a^2*c^2)*x)*Sqrt[a
+ b*x + c*x^2])/(9009*c^3) - (10*Sqrt[d*x]*(3*b*(6*b^2 - 19*a*c) + 14*c*(3*b^2
- 11*a*c)*x)*(a + b*x + c*x^2)^(3/2))/(9009*c^2) + (10*b*Sqrt[d*x]*(a + b*x + c*
x^2)^(5/2))/(143*c) + (2*(d*x)^(3/2)*(a + b*x + c*x^2)^(5/2))/(13*d) + (4*a^(1/4
)*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*d*Sqrt[x]*(Sqrt[a] + Sq
rt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(
1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(9009*c^(15/4)*Sqrt[d*x]*S
qrt[a + b*x + c*x^2]) - (a^(1/4)*(Sqrt[a]*b*Sqrt[c]*(24*b^4 - 241*a*b^2*c + 708*
a^2*c^2) + 2*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3))*d*Sqrt[x]*(
Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2
*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(9009*c^(15/4)
*Sqrt[d*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 2.09101, antiderivative size = 616, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{2 \sqrt{d x} \left (3 c x \left (308 a^2 c^2-181 a b^2 c+24 b^4\right )+b \left (108 a^2 c^2-151 a b^2 c+24 b^4\right )\right ) \sqrt{a+b x+c x^2}}{9009 c^3}-\frac{4 d x \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right ) \sqrt{a+b x+c x^2}}{9009 c^{7/2} \sqrt{d x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{\sqrt [4]{a} d \sqrt{x} \left (\sqrt{a} b \sqrt{c} \left (708 a^2 c^2-241 a b^2 c+24 b^4\right )+2 \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right )\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{9009 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt [4]{a} d \sqrt{x} \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{9009 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}-\frac{10 \sqrt{d x} \left (14 c x \left (3 b^2-11 a c\right )+3 b \left (6 b^2-19 a c\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac{2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}+\frac{10 b \sqrt{d x} \left (a+b x+c x^2\right )^{5/2}}{143 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-4*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*d*x*Sqrt[a + b*x + c*
x^2])/(9009*c^(7/2)*Sqrt[d*x]*(Sqrt[a] + Sqrt[c]*x)) + (2*Sqrt[d*x]*(b*(24*b^4 -
151*a*b^2*c + 108*a^2*c^2) + 3*c*(24*b^4 - 181*a*b^2*c + 308*a^2*c^2)*x)*Sqrt[a
+ b*x + c*x^2])/(9009*c^3) - (10*Sqrt[d*x]*(3*b*(6*b^2 - 19*a*c) + 14*c*(3*b^2
- 11*a*c)*x)*(a + b*x + c*x^2)^(3/2))/(9009*c^2) + (10*b*Sqrt[d*x]*(a + b*x + c*
x^2)^(5/2))/(143*c) + (2*(d*x)^(3/2)*(a + b*x + c*x^2)^(5/2))/(13*d) + (4*a^(1/4
)*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*d*Sqrt[x]*(Sqrt[a] + Sq
rt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(
1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(9009*c^(15/4)*Sqrt[d*x]*S
qrt[a + b*x + c*x^2]) - (a^(1/4)*(Sqrt[a]*b*Sqrt[c]*(24*b^4 - 241*a*b^2*c + 708*
a^2*c^2) + 2*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3))*d*Sqrt[x]*(
Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2
*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(9009*c^(15/4)
*Sqrt[d*x]*Sqrt[a + 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((d*x)**(1/2)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 8.31525, size = 708, normalized size = 1.15 \[ \frac{\sqrt{d x} \left (2 c \sqrt{x} \left (b c^2 \left (708 a^2+3071 a c x^2+1701 c^2 x^4\right )+77 c^3 x \left (31 a^2+28 a c x^2+9 c^2 x^4\right )+b^3 c \left (15 c x^2-241 a\right )+3 b^2 c^2 x \left (54 a+371 c x^2\right )+24 b^5-18 b^4 c x\right ) (a+x (b+c x))-\frac{4 \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right ) (a+x (b+c x))}{\sqrt{x}}+\frac{i x \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}+\frac{i x \left (924 a^3 c^3 \sqrt{b^2-4 a c}-1632 a^3 b c^3+1192 a^2 b^3 c^2-951 a^2 b^2 c^2 \sqrt{b^2-4 a c}-292 a b^5 c-24 b^6 \sqrt{b^2-4 a c}+268 a b^4 c \sqrt{b^2-4 a c}+24 b^7\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}\right )}{9009 c^4 \sqrt{x} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d*x]*((-4*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*(a + x*(b
+ c*x)))/Sqrt[x] + 2*c*Sqrt[x]*(a + x*(b + c*x))*(24*b^5 - 18*b^4*c*x + b^3*c*(
-241*a + 15*c*x^2) + 3*b^2*c^2*x*(54*a + 371*c*x^2) + 77*c^3*x*(31*a^2 + 28*a*c*
x^2 + 9*c^2*x^4) + b*c^2*(708*a^2 + 3071*a*c*x^2 + 1701*c^2*x^4)) + (I*(24*b^6 -
268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 +
(4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x
- Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a
*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + S
qrt[b^2 - 4*a*c])] + (I*(24*b^7 - 292*a*b^5*c + 1192*a^2*b^3*c^2 - 1632*a^3*b*c^
3 - 24*b^6*Sqrt[b^2 - 4*a*c] + 268*a*b^4*c*Sqrt[b^2 - 4*a*c] - 951*a^2*b^2*c^2*S
qrt[b^2 - 4*a*c] + 924*a^3*c^3*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2
- 4*a*c])*x)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*
x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b +
Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]))/
(9009*c^4*Sqrt[x]*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.083, size = 2810, normalized size = 4.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9009*(d*x)^(1/2)*(708*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(
1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b
^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2))
)^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b
^2)^(1/2)*a^3*b*c^3-4046*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(
1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b
^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2))
)^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^2*b^4*c
^2+728*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-
4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*
EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2
)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a*b^6*c+831*((b+2*c*x+(-4*a
*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*
a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-
4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^2*b^4*c^2-72*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(
-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/
2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(
b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1
/2))^(1/2))*a*b^6*c-3096*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(
1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b
^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2))
)^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^3*b^2*c
^3+9456*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(
-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)
*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/
2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^3*b^2*c^3-5698*x^6*a*c^7
-9086*x^4*a^2*c^6-4774*x^2*a^3*c^5-2256*x^5*b^3*c^5-48*x^2*b^6*c^2+6*x^4*b^4*c^4
-4788*x^7*b*c^7-5628*x^6*b^2*c^6-12*x^3*b^5*c^3+3696*((b+2*c*x+(-4*a*c+b^2)^(1/2
))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/
2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(
1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+
b^2)^(1/2))^(1/2))*a^4*c^4-1386*x^8*c^8+1848*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-
4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2
)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b
+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/
2))^(1/2))*(-4*a*c+b^2)^(1/2)*a^3*b*c^3-1902*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-
4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2
)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b
+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/
2))^(1/2))*(-4*a*c+b^2)^(1/2)*a^2*b^3*c^2+536*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(
-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/
2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(
b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1
/2))^(1/2))*(-4*a*c+b^2)^(1/2)*a*b^5*c-48*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a
*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(
-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-
4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))
^(1/2))*(-4*a*c+b^2)^(1/2)*b^7-7392*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2
)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(
b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+
b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)
)*a^4*c^4-48*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*
c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^
(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*
2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*b^8-241*((b+2*c*x+(-4
*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-
4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+
(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(
1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*a^2*b^3*c^2+24*((b+2*c*x+(-4
*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-
4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+
(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(
1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*a*b^5*c-13856*x^5*a*b*c^6-86
92*x^4*a*b^2*c^5-12332*x^3*a^2*b*c^5+128*x^3*a*b^3*c^4-1740*x^2*a^2*b^2*c^4+518*
x^2*a*b^4*c^3-1416*x*a^3*b*c^4+482*x*a^2*b^3*c^3-48*x*a*b^5*c^2)/(c*x^2+b*x+a)^(
1/2)/c^5/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 1.24946, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done