3.1352 \(\int \frac{(b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=174 \[ \frac{10 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c \sqrt{a+b x+c x^2}}+\frac{20}{21} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{4}{7} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2} \]

[Out]

(20*(b^2 - 4*a*c)*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/21 + (4*d*(b*d
+ 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/7 + (10*(b^2 - 4*a*c)^(9/4)*d^(7/2)*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])/(21*c*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.442246, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{10 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c \sqrt{a+b x+c x^2}}+\frac{20}{21} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{4}{7} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(20*(b^2 - 4*a*c)*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/21 + (4*d*(b*d
+ 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/7 + (10*(b^2 - 4*a*c)^(9/4)*d^(7/2)*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])/(21*c*Sqrt[a + b*x + c*x^2])

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Rubi in Sympy [A]  time = 99.8804, size = 168, normalized size = 0.97 \[ \frac{20 d^{3} \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \sqrt{a + b x + c x^{2}}}{21} + \frac{4 d \left (b d + 2 c d x\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{2}}}{7} + \frac{10 d^{\frac{7}{2}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{21 c \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

20*d**3*(-4*a*c + b**2)*sqrt(b*d + 2*c*d*x)*sqrt(a + b*x + c*x**2)/21 + 4*d*(b*d
+ 2*c*d*x)**(5/2)*sqrt(a + b*x + c*x**2)/7 + 10*d**(7/2)*sqrt(c*(a + b*x + c*x*
*2)/(4*a*c - b**2))*(-4*a*c + b**2)**(9/4)*elliptic_f(asin(sqrt(b*d + 2*c*d*x)/(
sqrt(d)*(-4*a*c + b**2)**(1/4))), -1)/(21*c*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 0.806902, size = 176, normalized size = 1.01 \[ \frac{(d (b+2 c x))^{7/2} \left (\frac{16 (a+x (b+c x)) \left (c \left (3 c x^2-5 a\right )+2 b^2+3 b c x\right )}{(b+2 c x)^3}+\frac{10 i \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{(b+2 c x)^2}} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{-\sqrt{b^2-4 a c}}}{\sqrt{b+2 c x}}\right )\right |-1\right )}{c \sqrt{-\sqrt{b^2-4 a c}} (b+2 c x)^{5/2}}\right )}{21 \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

((d*(b + 2*c*x))^(7/2)*((16*(a + x*(b + c*x))*(2*b^2 + 3*b*c*x + c*(-5*a + 3*c*x
^2)))/(b + 2*c*x)^3 + ((10*I)*(b^2 - 4*a*c)^2*Sqrt[(c*(a + x*(b + c*x)))/(b + 2*
c*x)^2]*EllipticF[I*ArcSinh[Sqrt[-Sqrt[b^2 - 4*a*c]]/Sqrt[b + 2*c*x]], -1])/(c*S
qrt[-Sqrt[b^2 - 4*a*c]]*(b + 2*c*x)^(5/2))))/(21*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.048, size = 567, normalized size = 3.3 \[{\frac{{d}^{3}}{21\,c \left ( 2\,{x}^{3}{c}^{2}+3\,{x}^{2}bc+2\,acx+{b}^{2}x+ab \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 96\,{c}^{5}{x}^{5}+80\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{a}^{2}{c}^{2}-40\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}a{b}^{2}c+5\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}{b}^{4}+240\,b{c}^{4}{x}^{4}-64\,{x}^{3}a{c}^{4}+256\,{x}^{3}{b}^{2}{c}^{3}-96\,{x}^{2}ab{c}^{3}+144\,{x}^{2}{b}^{3}{c}^{2}-160\,{a}^{2}{c}^{3}x+32\,a{b}^{2}{c}^{2}x+32\,{b}^{4}cx-80\,{a}^{2}b{c}^{2}+32\,ac{b}^{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

1/21*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^3*(96*c^5*x^5+80*((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)*EllipticF(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))*(-4*a*c+b^2)^(1/2)
*a^2*c^2-40*((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)*EllipticF(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))*(-4*a*c+b^2)^(1/2)*a*b^2*c+5*((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)*EllipticF(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))*(-4*a*c+b^2)^(1/2)*b^4+240*b*c^4*x^4-64*x^3
*a*c^4+256*x^3*b^2*c^3-96*x^2*a*b*c^3+144*x^2*b^3*c^2-160*a^2*c^3*x+32*a*b^2*c^2
*x+32*b^4*c*x-80*a^2*b*c^2+32*a*c*b^3)/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. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((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)/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)**(7/2)/(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. \[ \int \frac{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}}{\sqrt{c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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