3.1348 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{3/2}} \, dx\)

Optimal. Leaf size=316 \[ -\frac{\left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{12 c^3 d^3}+\frac{5 \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}} \]

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(12*c^3*d^3) + (5*(
b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(18*c^2*d^3) - (a + b*x + c*x^2)^(
5/2)/(c*d*Sqrt[b*d + 2*c*d*x]) + ((b^2 - 4*a*c)^(11/4)*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)*Sq
rt[d])], -1])/(12*c^4*d^(3/2)*Sqrt[a + b*x + c*x^2]) - ((b^2 - 4*a*c)^(11/4)*Sqr
t[-((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])/(12*c^4*d^(3/2)*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.979817, antiderivative size = 316, 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 \[ -\frac{\left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}+\frac{\left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{12 c^4 d^{3/2} \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{12 c^3 d^3}+\frac{5 \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{18 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{c d \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(3/2),x]

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(12*c^3*d^3) + (5*(
b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(18*c^2*d^3) - (a + b*x + c*x^2)^(
5/2)/(c*d*Sqrt[b*d + 2*c*d*x]) + ((b^2 - 4*a*c)^(11/4)*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)*Sq
rt[d])], -1])/(12*c^4*d^(3/2)*Sqrt[a + b*x + c*x^2]) - ((b^2 - 4*a*c)^(11/4)*Sqr
t[-((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])/(12*c^4*d^(3/2)*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 166.827, size = 301, normalized size = 0.95 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{c d \sqrt{b d + 2 c d x}} + \frac{5 \left (b d + 2 c d x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{18 c^{2} d^{3}} - \frac{\left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{12 c^{3} d^{3}} + \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{12 c^{4} d^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} - \frac{\sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{12 c^{4} d^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}} \]

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)**(3/2),x)

[Out]

-(a + b*x + c*x**2)**(5/2)/(c*d*sqrt(b*d + 2*c*d*x)) + 5*(b*d + 2*c*d*x)**(3/2)*
(a + b*x + c*x**2)**(3/2)/(18*c**2*d**3) - (-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2
)*sqrt(a + b*x + c*x**2)/(12*c**3*d**3) + sqrt(c*(a + b*x + c*x**2)/(4*a*c - b**
2))*(-4*a*c + b**2)**(11/4)*elliptic_e(asin(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c
+ b**2)**(1/4))), -1)/(12*c**4*d**(3/2)*sqrt(a + b*x + c*x**2)) - sqrt(c*(a + b
*x + c*x**2)/(4*a*c - b**2))*(-4*a*c + b**2)**(11/4)*elliptic_f(asin(sqrt(b*d +
2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(12*c**4*d**(3/2)*sqrt(a + b*x +
c*x**2))

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Mathematica [C]  time = 1.80055, size = 246, normalized size = 0.78 \[ \frac{c (a+x (b+c x)) \left (4 c^2 \left (-9 a^2+4 a c x^2+c^2 x^4\right )+2 b^2 c \left (11 a+c x^2\right )+8 b c^2 x \left (2 a+c x^2\right )-3 b^4-2 b^3 c x\right )+3 i \left (b^2-4 a c\right )^3 \sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{36 c^4 d \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^(3/2),x]

[Out]

(c*(a + x*(b + c*x))*(-3*b^4 - 2*b^3*c*x + 8*b*c^2*x*(2*a + c*x^2) + 2*b^2*c*(11
*a + c*x^2) + 4*c^2*(-9*a^2 + 4*a*c*x^2 + c^2*x^4)) + (3*I)*(b^2 - 4*a*c)^3*Sqrt
[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(E
llipticE[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - EllipticF[I*Ar
cSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(36*c^4*d*Sqrt[d*(b + 2*c*x
)]*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.04, size = 700, normalized size = 2.2 \[{\frac{1}{72\,{c}^{4}{d}^{2} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) }\sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 8\,{c}^{6}{x}^{6}+24\,b{c}^{5}{x}^{5}+192\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{3}{c}^{3}-144\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{b}^{2}{c}^{2}+36\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{4}c-3\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{6}+40\,{x}^{4}a{c}^{5}+20\,{x}^{4}{b}^{2}{c}^{4}+80\,{x}^{3}ab{c}^{4}-40\,{x}^{2}{a}^{2}{c}^{4}+80\,{x}^{2}a{b}^{2}{c}^{3}-10\,{x}^{2}{b}^{4}{c}^{2}-40\,{a}^{2}b{c}^{3}x+40\,a{b}^{3}{c}^{2}x-6\,{b}^{5}cx-72\,{a}^{3}{c}^{3}+44\,{a}^{2}{b}^{2}{c}^{2}-6\,a{b}^{4}c \right ) } \]

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)^(3/2),x)

[Out]

1/72*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)*(8*c^6*x^6+24*b*c^5*x^5+192*((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^3*c^3-1
44*((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^2*b^2*c^2+36*((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*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+40*x^4*a*c^5+20*x^4*b^2*c^4+80*x^3*a*b*c^4-40*x^2*a^2*
c^4+80*x^2*a*b^2*c^3-10*x^2*b^4*c^2-40*a^2*b*c^3*x+40*a*b^3*c^2*x-6*b^5*c*x-72*a
^3*c^3+44*a^2*b^2*c^2-6*a*b^4*c)/d^2/c^4/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(3/2), x)

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

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)^(3/2),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)/(2*c*d*x + b*d)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral((a + b*x + c*x**2)**(5/2)/(d*(b + 2*c*x))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^(3/2), x)