3.1349 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{7/2}} \, dx\)

Optimal. Leaf size=310 \[ \frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{20 c^4 d^{7/2} \sqrt{a+b x+c x^2}}-\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{20 c^4 d^{7/2} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{20 c^3 d^5}-\frac{\left (a+b x+c x^2\right )^{3/2}}{2 c^2 d^3 \sqrt{b d+2 c d x}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{5 c d (b d+2 c d x)^{5/2}} \]

[Out]

(3*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(20*c^3*d^5) - (a + b*x + c*x^2)
^(3/2)/(2*c^2*d^3*Sqrt[b*d + 2*c*d*x]) - (a + b*x + c*x^2)^(5/2)/(5*c*d*(b*d + 2
*c*d*x)^(5/2)) - (3*(b^2 - 4*a*c)^(7/4)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*
c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(
20*c^4*d^(7/2)*Sqrt[a + b*x + c*x^2]) + (3*(b^2 - 4*a*c)^(7/4)*Sqrt[-((c*(a + b*
x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^
(1/4)*Sqrt[d])], -1])/(20*c^4*d^(7/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.957887, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{20 c^4 d^{7/2} \sqrt{a+b x+c x^2}}-\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{20 c^4 d^{7/2} \sqrt{a+b x+c x^2}}+\frac{3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{20 c^3 d^5}-\frac{\left (a+b x+c x^2\right )^{3/2}}{2 c^2 d^3 \sqrt{b d+2 c d x}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{5 c d (b d+2 c d x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(20*c^3*d^5) - (a + b*x + c*x^2)
^(3/2)/(2*c^2*d^3*Sqrt[b*d + 2*c*d*x]) - (a + b*x + c*x^2)^(5/2)/(5*c*d*(b*d + 2
*c*d*x)^(5/2)) - (3*(b^2 - 4*a*c)^(7/4)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*
c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(
20*c^4*d^(7/2)*Sqrt[a + b*x + c*x^2]) + (3*(b^2 - 4*a*c)^(7/4)*Sqrt[-((c*(a + b*
x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^
(1/4)*Sqrt[d])], -1])/(20*c^4*d^(7/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 169.652, size = 298, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{5 c d \left (b d + 2 c d x\right )^{\frac{5}{2}}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{2 c^{2} d^{3} \sqrt{b d + 2 c d x}} + \frac{3 \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{20 c^{3} d^{5}} - \frac{3 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{20 c^{4} d^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}} + \frac{3 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{20 c^{4} d^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x + c*x**2)**(5/2)/(5*c*d*(b*d + 2*c*d*x)**(5/2)) - (a + b*x + c*x**2)**
(3/2)/(2*c**2*d**3*sqrt(b*d + 2*c*d*x)) + 3*(b*d + 2*c*d*x)**(3/2)*sqrt(a + b*x
+ c*x**2)/(20*c**3*d**5) - 3*sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**2))*(-4*a*c +
b**2)**(7/4)*elliptic_e(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4
))), -1)/(20*c**4*d**(7/2)*sqrt(a + b*x + c*x**2)) + 3*sqrt(c*(a + b*x + c*x**2)
/(4*a*c - b**2))*(-4*a*c + b**2)**(7/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(sqr
t(d)*(-4*a*c + b**2)**(1/4))), -1)/(20*c**4*d**(7/2)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 2.60763, size = 253, normalized size = 0.82 \[ \frac{c (a+x (b+c x)) \left (4 c^2 \left (-a^2-12 a c x^2+c^2 x^4\right )+2 b^2 c \left (9 c x^2-5 a\right )+8 b c^2 x \left (c x^2-6 a\right )+3 b^4+14 b^3 c x\right )-\frac{3 i \left (b^2-4 a c\right ) (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}}{20 c^4 d \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(a + x*(b + c*x))*(3*b^4 + 14*b^3*c*x + 8*b*c^2*x*(-6*a + c*x^2) + 2*b^2*c*(-
5*a + 9*c*x^2) + 4*c^2*(-a^2 - 12*a*c*x^2 + c^2*x^4)) - ((3*I)*(b^2 - 4*a*c)*(b
+ 2*c*x)^4*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(EllipticE[I*ArcSinh[Sqrt[
-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - EllipticF[I*ArcSinh[Sqrt[-((b + 2*c*x)
/Sqrt[b^2 - 4*a*c])]], -1]))/(-((b + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(3/2))/(20*c^4*d
*(d*(b + 2*c*x))^(5/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.041, size = 1362, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(192*EllipticE(1/2*((b+2*c*x+(-4*a*
c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^2*a^2*c^4*((b+2*c*x+(
-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+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)-96*EllipticE(1/2*((b
+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x^2*a*b^2*
c^3*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+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)+12*El
lipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1
/2))*x^2*b^4*c^2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*
x+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)+8*c^6*x^6+192*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(
1/2))^(1/2)*2^(1/2),2^(1/2))*x*a^2*b*c^3*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b
^2)^(1/2))^(1/2)*(-(2*c*x+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)-96*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(
-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x*a*b^3*c^2*((b+2*c*x+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+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)+12*EllipticE(1/2*((b+2*c*x+(-4*a*c+
b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*x*b^5*c*((b+2*c*x+(-4*a*c
+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+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)+24*b*c^5*x^5+48*((b+2*c*x+
(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+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)*EllipticE(1/2*((b+2
*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*a^2*b^2*c^2-
24*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+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)*Ellipt
icE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))
*a*b^4*c+3*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+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
)*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),
2^(1/2))*b^6-88*x^4*a*c^5+52*x^4*b^2*c^4-176*x^3*a*b*c^4+64*b^3*c^3*x^3-104*x^2*
a^2*c^4-80*x^2*a*b^2*c^3+34*x^2*b^4*c^2-104*a^2*b*c^3*x+8*a*b^3*c^2*x+6*b^5*c*x-
8*a^3*c^3-20*a^2*b^2*c^2+6*a*b^4*c)/d^4/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)/
(2*c*x+b)^2/c^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{{\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} \sqrt{2 \, c d x + b d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(7/2),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)/((8*c^3*d^3*x^3 + 12*b*c^2*d^3*x^2 + 6*b^2*c*d^3*x + b^3*d^3)*sqrt(2*c*
d*x + b*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+a)**(5/2)/(2*c*d*x+b*d)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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