3.2511 \(\int (d+e x)^3 \left (a+b x+c x^2\right )^{5/4} \, dx\)

Optimal. Leaf size=448 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{14784 \sqrt{2} c^{21/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right )}{7392 c^5}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right )}{616 c^4}+\frac{e \left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right )}{2574 c^3}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c} \]

[Out]

(-5*(b^2 - 4*a*c)*(2*c*d - b*e)*(44*c^2*d^2 + 17*b^2*e^2 - 4*c*e*(11*b*d + 6*a*e
))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(7392*c^5) + ((2*c*d - b*e)*(44*c^2*d^2
+ 17*b^2*e^2 - 4*c*e*(11*b*d + 6*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/4))/(616
*c^4) + (2*e*(d + e*x)^2*(a + b*x + c*x^2)^(9/4))/(13*c) + (e*(1320*c^2*d^2 + 22
1*b^2*e^2 - 2*c*e*(507*b*d + 88*a*e) + 306*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2
)^(9/4))/(2574*c^3) + (5*(b^2 - 4*a*c)^(9/4)*(2*c*d - b*e)*(44*c^2*d^2 + 17*b^2*
e^2 - 4*c*e*(11*b*d + 6*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*
Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x
^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1
/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(14784*Sqrt[2]*c^(21/4)*(b + 2*c*x))

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Rubi [A]  time = 1.19747, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{14784 \sqrt{2} c^{21/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right )}{7392 c^5}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right )}{616 c^4}+\frac{e \left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right )}{2574 c^3}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c} \]

Warning: Unable to verify antiderivative.

[In]  Int[(d + e*x)^3*(a + b*x + c*x^2)^(5/4),x]

[Out]

(-5*(b^2 - 4*a*c)*(2*c*d - b*e)*(44*c^2*d^2 + 17*b^2*e^2 - 4*c*e*(11*b*d + 6*a*e
))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(7392*c^5) + ((2*c*d - b*e)*(44*c^2*d^2
+ 17*b^2*e^2 - 4*c*e*(11*b*d + 6*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(5/4))/(616
*c^4) + (2*e*(d + e*x)^2*(a + b*x + c*x^2)^(9/4))/(13*c) + (e*(1320*c^2*d^2 + 22
1*b^2*e^2 - 2*c*e*(507*b*d + 88*a*e) + 306*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2
)^(9/4))/(2574*c^3) + (5*(b^2 - 4*a*c)^(9/4)*(2*c*d - b*e)*(44*c^2*d^2 + 17*b^2*
e^2 - 4*c*e*(11*b*d + 6*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*
Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x
^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1
/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(14784*Sqrt[2]*c^(21/4)*(b + 2*c*x))

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Rubi in Sympy [A]  time = 107.687, size = 517, normalized size = 1.15 \[ \frac{2 e \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{9}{4}}}{13 c} + \frac{8 e \left (a + b x + c x^{2}\right )^{\frac{9}{4}} \left (- 11 a c e^{2} + \frac{221 b^{2} e^{2}}{16} - \frac{507 b c d e}{8} + \frac{165 c^{2} d^{2}}{2} - \frac{153 c e x \left (b e - 2 c d\right )}{8}\right )}{1287 c^{3}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}} \left (- 24 a c e^{2} + 17 b^{2} e^{2} - 44 b c d e + 44 c^{2} d^{2}\right )}{616 c^{4}} + \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \sqrt [4]{a + b x + c x^{2}} \left (- 24 a c e^{2} + 17 b^{2} e^{2} - 44 b c d e + 44 c^{2} d^{2}\right )}{7392 c^{5}} - \frac{5 \sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \left (b e - 2 c d\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \left (- 24 a c e^{2} + 17 b^{2} e^{2} - 44 b c d e + 44 c^{2} d^{2}\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{29568 c^{\frac{21}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(c*x**2+b*x+a)**(5/4),x)

[Out]

2*e*(d + e*x)**2*(a + b*x + c*x**2)**(9/4)/(13*c) + 8*e*(a + b*x + c*x**2)**(9/4
)*(-11*a*c*e**2 + 221*b**2*e**2/16 - 507*b*c*d*e/8 + 165*c**2*d**2/2 - 153*c*e*x
*(b*e - 2*c*d)/8)/(1287*c**3) - (b + 2*c*x)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(5
/4)*(-24*a*c*e**2 + 17*b**2*e**2 - 44*b*c*d*e + 44*c**2*d**2)/(616*c**4) + 5*(b
+ 2*c*x)*(-4*a*c + b**2)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(1/4)*(-24*a*c*e**2 +
17*b**2*e**2 - 44*b*c*d*e + 44*c**2*d**2)/(7392*c**5) - 5*sqrt(2)*sqrt(-(-4*a*c
+ b**2 + c*(4*a + 4*b*x + 4*c*x**2))/((4*a*c - b**2)*(2*sqrt(c)*sqrt(a + b*x +
c*x**2)/sqrt(-4*a*c + b**2) + 1)**2))*(-4*a*c + b**2)**(9/4)*(b*e - 2*c*d)*(2*sq
rt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2) + 1)*(-24*a*c*e**2 + 17*b**2*e*
*2 - 44*b*c*d*e + 44*c**2*d**2)*sqrt((b + 2*c*x)**2)*elliptic_f(2*atan(sqrt(2)*c
**(1/4)*(a + b*x + c*x**2)**(1/4)/(-4*a*c + b**2)**(1/4)), 1/2)/(29568*c**(21/4)
*(b + 2*c*x)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2)))

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Mathematica [C]  time = 2.49242, size = 590, normalized size = 1.32 \[ \frac{195 \sqrt [4]{2} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )-4 c (a+x (b+c x)) \left (-16 b^2 c^2 \left (3419 a^2 e^3-2 a c e \left (5148 d^2+1131 d e x+158 e^2 x^2\right )+c^2 x \left (429 d^3+429 d^2 e x+195 d e^2 x^2+35 e^3 x^3\right )\right )+32 b c^3 \left (a^2 e^2 (5421 d+431 e x)-4 a c \left (858 d^3+429 d^2 e x+156 d e^2 x^2+25 e^3 x^3\right )-3 c^2 x^2 \left (1287 d^3+2717 d^2 e x+2093 d e^2 x^2+567 e^3 x^3\right )\right )-64 c^3 \left (-308 a^3 e^3+a^2 c e \left (3003 d^2+585 d e x+77 e^2 x^2\right )+2 a c^2 x \left (1716 d^3+3003 d^2 e x+2106 d e^2 x^2+539 e^3 x^3\right )+3 c^3 x^3 \left (429 d^3+1001 d^2 e x+819 d e^2 x^2+231 e^3 x^3\right )\right )-52 b^4 c e \left (c \left (495 d^2+117 d e x+17 e^2 x^2\right )-498 a e^2\right )+8 b^3 c^2 \left (c \left (2145 d^3+1287 d^2 e x+507 d e^2 x^2+85 e^3 x^3\right )-26 a e^2 (513 d+43 e x)\right )-3315 b^6 e^3+78 b^5 c e^2 (195 d+17 e x)\right )}{1153152 c^6 (a+x (b+c x))^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(5/4),x]

[Out]

(-4*c*(a + x*(b + c*x))*(-3315*b^6*e^3 + 78*b^5*c*e^2*(195*d + 17*e*x) - 52*b^4*
c*e*(-498*a*e^2 + c*(495*d^2 + 117*d*e*x + 17*e^2*x^2)) - 16*b^2*c^2*(3419*a^2*e
^3 - 2*a*c*e*(5148*d^2 + 1131*d*e*x + 158*e^2*x^2) + c^2*x*(429*d^3 + 429*d^2*e*
x + 195*d*e^2*x^2 + 35*e^3*x^3)) + 8*b^3*c^2*(-26*a*e^2*(513*d + 43*e*x) + c*(21
45*d^3 + 1287*d^2*e*x + 507*d*e^2*x^2 + 85*e^3*x^3)) - 64*c^3*(-308*a^3*e^3 + a^
2*c*e*(3003*d^2 + 585*d*e*x + 77*e^2*x^2) + 3*c^3*x^3*(429*d^3 + 1001*d^2*e*x +
819*d*e^2*x^2 + 231*e^3*x^3) + 2*a*c^2*x*(1716*d^3 + 3003*d^2*e*x + 2106*d*e^2*x
^2 + 539*e^3*x^3)) + 32*b*c^3*(a^2*e^2*(5421*d + 431*e*x) - 4*a*c*(858*d^3 + 429
*d^2*e*x + 156*d*e^2*x^2 + 25*e^3*x^3) - 3*c^2*x^2*(1287*d^3 + 2717*d^2*e*x + 20
93*d*e^2*x^2 + 567*e^3*x^3))) + 195*2^(1/4)*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(44*c^
2*d^2 + 17*b^2*e^2 - 4*c*e*(11*b*d + 6*a*e))*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b
+ Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^(3/4)*Hypergeometric2F1[1/4, 3/
4, 5/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(1153152*c^6*(a
+ x*(b + c*x))^(3/4))

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Maple [F]  time = 0.136, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(c*x^2+b*x+a)^(5/4),x)

[Out]

int((e*x+d)^3*(c*x^2+b*x+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}{\left (e x + d\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^3,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c e^{3} x^{5} +{\left (3 \, c d e^{2} + b e^{3}\right )} x^{4} + a d^{3} +{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{3} +{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{2} +{\left (b d^{3} + 3 \, a d^{2} e\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^3,x, algorithm="fricas")

[Out]

integral((c*e^3*x^5 + (3*c*d*e^2 + b*e^3)*x^4 + a*d^3 + (3*c*d^2*e + 3*b*d*e^2 +
a*e^3)*x^3 + (c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*x^2 + (b*d^3 + 3*a*d^2*e)*x)*(c*x^
2 + b*x + a)^(1/4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{5}{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(c*x**2+b*x+a)**(5/4),x)

[Out]

Integral((d + e*x)**3*(a + b*x + c*x**2)**(5/4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}{\left (e x + d\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^3,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/4)*(e*x + d)^3, x)