3.1322 \(\int (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=279 \[ \frac{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c^2 \sqrt{a+b x+c x^2}}-\frac{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c^2 \sqrt{a+b x+c x^2}}-\frac{2 d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{45 c}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{9 c d} \]

[Out]

(-2*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(45*c) + ((b*d
+ 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2])/(9*c*d) - ((b^2 - 4*a*c)^(11/4)*d^(5/2)*
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])/(15*c^2*Sqrt[a + b*x + c*x^2]) + ((b^2 -
4*a*c)^(11/4)*d^(5/2)*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^2*Sqrt[a + b*
x + c*x^2])

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Rubi [A]  time = 0.880401, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c^2 \sqrt{a+b x+c x^2}}-\frac{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{15 c^2 \sqrt{a+b x+c x^2}}-\frac{2 d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}{45 c}+\frac{\sqrt{a+b x+c x^2} (b d+2 c d x)^{7/2}}{9 c d} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(45*c) + ((b*d
+ 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2])/(9*c*d) - ((b^2 - 4*a*c)^(11/4)*d^(5/2)*
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])/(15*c^2*Sqrt[a + b*x + c*x^2]) + ((b^2 -
4*a*c)^(11/4)*d^(5/2)*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^2*Sqrt[a + b*
x + c*x^2])

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Rubi in Sympy [A]  time = 147.676, size = 264, normalized size = 0.95 \[ - \frac{2 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}{45 c} + \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}} \sqrt{a + b x + c x^{2}}}{9 c d} - \frac{d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{15 c^{2} \sqrt{a + b x + c x^{2}}} + \frac{d^{\frac{5}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{11}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{15 c^{2} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)*sqrt(a + b*x + c*x**2)/(45*c) + (b*d
+ 2*c*d*x)**(7/2)*sqrt(a + b*x + c*x**2)/(9*c*d) - d**(5/2)*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**2*sqrt(a + b*x + c*x**2)) + d**
(5/2)*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**2*sqrt
(a + b*x + c*x**2))

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Mathematica [C]  time = 1.48361, size = 208, normalized size = 0.75 \[ \frac{(d (b+2 c x))^{5/2} \left (\frac{c (a+x (b+c x)) \left (4 c \left (2 a+5 c x^2\right )+3 b^2+20 b c x\right )}{b+2 c x}+\frac{3 i \left (b^2-4 a c\right )^{3/2} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{\left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{5/2}}\right )}{45 c^2 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(5/2)*((c*(a + x*(b + c*x))*(3*b^2 + 20*b*c*x + 4*c*(2*a + 5*c*
x^2)))/(b + 2*c*x) + ((3*I)*(b^2 - 4*a*c)^(3/2)*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] - E
llipticF[I*ArcSinh[Sqrt[-((b + 2*c*x)/Sqrt[b^2 - 4*a*c])]], -1]))/(-((b + 2*c*x)
/Sqrt[b^2 - 4*a*c]))^(5/2)))/(45*c^2*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.028, size = 703, normalized size = 2.5 \[ -{\frac{{d}^{2}}{90\,{c}^{2} \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( -160\,{c}^{6}{x}^{6}-480\,b{c}^{5}{x}^{5}+192\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{3}{c}^{3}-144\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{b}^{2}{c}^{2}+36\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{4}c-3\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{6}-224\,{x}^{4}a{c}^{5}-544\,{x}^{4}{b}^{2}{c}^{4}-448\,{x}^{3}ab{c}^{4}-288\,{b}^{3}{c}^{3}{x}^{3}-64\,{x}^{2}{a}^{2}{c}^{4}-304\,{x}^{2}a{b}^{2}{c}^{3}-70\,{x}^{2}{b}^{4}{c}^{2}-64\,{a}^{2}b{c}^{3}x-80\,a{b}^{3}{c}^{2}x-6\,{b}^{5}cx-16\,{a}^{2}{b}^{2}{c}^{2}-6\,a{b}^{4}c \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/90*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^2*(-160*c^6*x^6-480*b*c^5*x^5+19
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)*Ellipti
cE(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^3-144*((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+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^6-224*x^4*a*c^5-544*x^4*b^2*c^4-448*x^3*a*b*c^4
-288*b^3*c^3*x^3-64*x^2*a^2*c^4-304*x^2*a*b^2*c^3-70*x^2*b^4*c^2-64*a^2*b*c^3*x-
80*a*b^3*c^2*x-6*b^5*c*x-16*a^2*b^2*c^2-6*a*b^4*c)/c^2/(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. \[ \int{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} \sqrt{c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*(b + 2*c*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. \[ \int{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} \sqrt{c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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