3.2429 \(\int (d x)^{5/2} \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=504 \[ -\frac{4 d^3 x \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \sqrt{a+b x+c x^2}}{315 c^{7/2} \sqrt{d x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{\sqrt [4]{a} d^3 \sqrt{x} \left (42 a^2 c^2-72 a b^2 c+\sqrt{a} b \sqrt{c} \left (8 b^2-27 a c\right )+16 b^4\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt [4]{a} d^3 \sqrt{x} \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{2 d^2 \sqrt{d x} \left (3 c x \left (8 b^2-7 a c\right )+b \left (3 a c+8 b^2\right )\right ) \sqrt{a+b x+c x^2}}{315 c^3}-\frac{4 b d^2 \sqrt{d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c} \]

[Out]

(-4*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*d^3*x*Sqrt[a + b*x + c*x^2])/(315*c^(7/2)*
Sqrt[d*x]*(Sqrt[a] + Sqrt[c]*x)) + (2*d^2*Sqrt[d*x]*(b*(8*b^2 + 3*a*c) + 3*c*(8*
b^2 - 7*a*c)*x)*Sqrt[a + b*x + c*x^2])/(315*c^3) - (4*b*d^2*Sqrt[d*x]*(a + b*x +
c*x^2)^(3/2))/(21*c^2) + (2*d*(d*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*c) + (4*a
^(1/4)*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*d^3*Sqrt[x]*(Sqrt[a] + Sqrt[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])/(315*c^(15/4)*Sqrt[d*x]*Sqrt[a + b*x + c
*x^2]) - (a^(1/4)*(16*b^4 - 72*a*b^2*c + 42*a^2*c^2 + Sqrt[a]*b*Sqrt[c]*(8*b^2 -
27*a*c))*d^3*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sq
rt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[
c]))/4])/(315*c^(15/4)*Sqrt[d*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.52408, antiderivative size = 504, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{4 d^3 x \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \sqrt{a+b x+c x^2}}{315 c^{7/2} \sqrt{d x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{\sqrt [4]{a} d^3 \sqrt{x} \left (42 a^2 c^2-72 a b^2 c+\sqrt{a} b \sqrt{c} \left (8 b^2-27 a c\right )+16 b^4\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt [4]{a} d^3 \sqrt{x} \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{2 d^2 \sqrt{d x} \left (3 c x \left (8 b^2-7 a c\right )+b \left (3 a c+8 b^2\right )\right ) \sqrt{a+b x+c x^2}}{315 c^3}-\frac{4 b d^2 \sqrt{d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-4*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*d^3*x*Sqrt[a + b*x + c*x^2])/(315*c^(7/2)*
Sqrt[d*x]*(Sqrt[a] + Sqrt[c]*x)) + (2*d^2*Sqrt[d*x]*(b*(8*b^2 + 3*a*c) + 3*c*(8*
b^2 - 7*a*c)*x)*Sqrt[a + b*x + c*x^2])/(315*c^3) - (4*b*d^2*Sqrt[d*x]*(a + b*x +
c*x^2)^(3/2))/(21*c^2) + (2*d*(d*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*c) + (4*a
^(1/4)*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*d^3*Sqrt[x]*(Sqrt[a] + Sqrt[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])/(315*c^(15/4)*Sqrt[d*x]*Sqrt[a + b*x + c
*x^2]) - (a^(1/4)*(16*b^4 - 72*a*b^2*c + 42*a^2*c^2 + Sqrt[a]*b*Sqrt[c]*(8*b^2 -
27*a*c))*d^3*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sq
rt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[
c]))/4])/(315*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)**(5/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 6.04131, size = 594, normalized size = 1.18 \[ \frac{(d x)^{5/2} \left (-\frac{4 \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) (a+x (b+c x))}{\sqrt{x}}+\frac{i x \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}-\frac{i x \left (21 a^2 c^2 \sqrt{b^2-4 a c}-48 a^2 b c^2+44 a b^3 c-36 a b^2 c \sqrt{b^2-4 a c}+8 b^4 \sqrt{b^2-4 a c}-8 b^5\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}+2 c \sqrt{x} \left (b c \left (5 c x^2-27 a\right )+7 c^2 x \left (2 a+5 c x^2\right )+8 b^3-6 b^2 c x\right ) (a+x (b+c x))\right )}{315 c^4 x^{5/2} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((d*x)^(5/2)*((-4*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*(a + x*(b + c*x)))/Sqrt[x] +
2*c*Sqrt[x]*(a + x*(b + c*x))*(8*b^3 - 6*b^2*c*x + b*c*(-27*a + 5*c*x^2) + 7*c^
2*x*(2*a + 5*c*x^2)) + (I*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*(-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 + Sqrt[b^2 - 4*a*c])] - (I*(-8*b^5 + 44*a*b^3*c - 48*a^2*b*c^2 + 8*b^
4*Sqrt[b^2 - 4*a*c] - 36*a*b^2*c*Sqrt[b^2 - 4*a*c] + 21*a^2*c^2*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 + Sq
rt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/S
qrt[a/(b + Sqrt[b^2 - 4*a*c])]))/(315*c^4*x^(5/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.071, size = 2062, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*d^2*(d*x)^(1/2)*(70*c^6*x^6+80*b*c^5*x^5+84*((-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))*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2
)^(1/2)))^(1/2)*a^3*c^3-117*((-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))*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a^2*b^
2*c^2+27*((-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))*((b+2
*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*a^2*b*
c^2+24*((-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))*((b+2*c
*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a*b^4*c-8*((-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)*Elli
pticF(((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+2*c*x+(-4*a*c+b^2)^(1/2))/(
b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*a*b^3*c-168*((-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)*Ellipti
cE(((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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(
-4*a*c+b^2)^(1/2)))^(1/2)*a^3*c^3+330*((-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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1
/2)*a^2*b^2*c^2+42*((-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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1
/2)*a^2*b*c^2-136*((-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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a*b^4*c-72*((-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+2*c*x+(-4*a*c+b
^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*a*b^3*c+16*((-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+2*c*x+(-4*a*c+b^2)^
(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*b^6+16*((-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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2
)))^(1/2)*(-4*a*c+b^2)^(1/2)*b^5+98*x^4*a*c^5-2*x^4*b^2*c^4-16*x^3*a*b*c^4+4*b^3
*c^3*x^3+28*x^2*a^2*c^4-66*x^2*a*b^2*c^3+16*x^2*b^4*c^2-54*a^2*b*c^3*x+16*a*b^3*
c^2*x)/x/(c*x^2+b*x+a)^(1/2)/c^5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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