3.1357 \(\int \frac{(b d+2 c d x)^{9/2}}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=273 \[ -\frac{14 d^{9/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c \sqrt{a+b x+c x^2}}+\frac{14 d^{9/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c \sqrt{a+b x+c x^2}}+\frac{28}{45} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}+\frac{4}{9} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2} \]

[Out]

(28*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/45 + (4*d*(b*
d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2])/9 + (14*(b^2 - 4*a*c)^(11/4)*d^(9/2)*S
qrt[-((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])/(15*c*Sqrt[a + b*x + c*x^2]) - (14*(b^2 -
4*a*c)^(11/4)*d^(9/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[Arc
Sin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(15*c*Sqrt[a + b*x
+ c*x^2])

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Rubi [A]  time = 0.850666, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{14 d^{9/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c \sqrt{a+b x+c x^2}}+\frac{14 d^{9/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c \sqrt{a+b x+c x^2}}+\frac{28}{45} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}+\frac{4}{9} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(9/2)/Sqrt[a + b*x + c*x^2],x]

[Out]

(28*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/45 + (4*d*(b*
d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2])/9 + (14*(b^2 - 4*a*c)^(11/4)*d^(9/2)*S
qrt[-((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])/(15*c*Sqrt[a + b*x + c*x^2]) - (14*(b^2 -
4*a*c)^(11/4)*d^(9/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[Arc
Sin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(15*c*Sqrt[a + b*x
+ c*x^2])

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Rubi in Sympy [A]  time = 147.205, size = 264, normalized size = 0.97 \[ \frac{28 d^{3} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{45} + \frac{4 d \left (b d + 2 c d x\right )^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}}{9} + \frac{14 d^{\frac{9}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{15 c \sqrt{a + b x + c x^{2}}} - \frac{14 d^{\frac{9}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{15 c \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

28*d**3*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)*sqrt(a + b*x + c*x**2)/45 + 4*d*(
b*d + 2*c*d*x)**(7/2)*sqrt(a + b*x + c*x**2)/9 + 14*d**(9/2)*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)/(15*c*sqrt(a + b*x + c*x**2)) - 14*d**
(9/2)*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)/(15*c*sqrt(a
+ b*x + c*x**2))

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Mathematica [C]  time = 1.40713, size = 205, normalized size = 0.75 \[ \frac{2 (d (b+2 c x))^{9/2} \left (8 (a+x (b+c x)) \left (c \left (5 c x^2-7 a\right )+3 b^2+5 b c x\right )+\frac{21 i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{c \left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}\right )}{45 (b+2 c x)^3 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(9/2)/Sqrt[a + b*x + c*x^2],x]

[Out]

(2*(d*(b + 2*c*x))^(9/2)*(8*(a + x*(b + c*x))*(3*b^2 + 5*b*c*x + c*(-7*a + 5*c*x
^2)) + ((21*I)*(b^2 - 4*a*c)^2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(Ellip
ticE[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1] - EllipticF[I*ArcSin
h[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(c*(-((b + 2*c*x)/Sqrt[b^2 - 4*
a*c]))^(3/2))))/(45*(b + 2*c*x)^3*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.047, size = 703, normalized size = 2.6 \[{\frac{{d}^{4}}{45\,c \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 320\,{c}^{6}{x}^{6}+960\,b{c}^{5}{x}^{5}+1344\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{3}{c}^{3}-1008\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{b}^{2}{c}^{2}+252\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{4}c-21\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{6}-128\,{x}^{4}a{c}^{5}+1232\,{x}^{4}{b}^{2}{c}^{4}-256\,{x}^{3}ab{c}^{4}+864\,{b}^{3}{c}^{3}{x}^{3}-448\,{x}^{2}{a}^{2}{c}^{4}+32\,{x}^{2}a{b}^{2}{c}^{3}+320\,{x}^{2}{b}^{4}{c}^{2}-448\,{a}^{2}b{c}^{3}x+160\,a{b}^{3}{c}^{2}x+48\,{b}^{5}cx-112\,{a}^{2}{b}^{2}{c}^{2}+48\,a{b}^{4}c \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

1/45*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^4*(320*c^6*x^6+960*b*c^5*x^5+1344
*((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)*Elliptic
E(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^3-1008*((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+252*((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-21*((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^6-128*x^4*a*c^5+1232*x^4*b^2*c^4-256*x^3*a*b*
c^4+864*b^3*c^3*x^3-448*x^2*a^2*c^4+32*x^2*a*b^2*c^3+320*x^2*b^4*c^2-448*a^2*b*c
^3*x+160*a*b^3*c^2*x+48*b^5*c*x-112*a^2*b^2*c^2+48*a*b^4*c)/c/(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. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} \sqrt{2 \, c d x + b d}}{\sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

integral((16*c^4*d^4*x^4 + 32*b*c^3*d^4*x^3 + 24*b^2*c^2*d^4*x^2 + 8*b^3*c*d^4*x
+ b^4*d^4)*sqrt(2*c*d*x + b*d)/sqrt(c*x^2 + b*x + a), 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((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(9/2)/sqrt(c*x^2 + b*x + a), x)