3.2334 \(\int (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=161 \[ \frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{128 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{16 c^2}+\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 c} \]

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^3) + (
(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(16*c^2) + (e*(a + b*x + c*x^
2)^(5/2))/(5*c) + (3*(b^2 - 4*a*c)^2*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c
]*Sqrt[a + b*x + c*x^2])])/(256*c^(7/2))

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Rubi [A]  time = 0.180161, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{128 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{16 c^2}+\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^3) + (
(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(16*c^2) + (e*(a + b*x + c*x^
2)^(5/2))/(5*c) + (3*(b^2 - 4*a*c)^2*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c
]*Sqrt[a + b*x + c*x^2])])/(256*c^(7/2))

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Rubi in Sympy [A]  time = 19.6882, size = 151, normalized size = 0.94 \[ \frac{e \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{5 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{16 c^{2}} + \frac{3 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{128 c^{3}} - \frac{3 \left (- 4 a c + b^{2}\right )^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(a + b*x + c*x**2)**(5/2)/(5*c) - (b + 2*c*x)*(b*e - 2*c*d)*(a + b*x + c*x**2)
**(3/2)/(16*c**2) + 3*(b + 2*c*x)*(-4*a*c + b**2)*(b*e - 2*c*d)*sqrt(a + b*x + c
*x**2)/(128*c**3) - 3*(-4*a*c + b**2)**2*(b*e - 2*c*d)*atanh((b + 2*c*x)/(2*sqrt
(c)*sqrt(a + b*x + c*x**2)))/(256*c**(7/2))

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Mathematica [A]  time = 0.270355, size = 187, normalized size = 1.16 \[ \frac{\sqrt{a+x (b+c x)} \left (16 c^2 \left (8 a^2 e+a c x (25 d+16 e x)+2 c^2 x^3 (5 d+4 e x)\right )+4 b^2 c (c x (5 d+2 e x)-25 a e)+8 b c^2 \left (25 a d+7 a e x+30 c d x^2+22 c e x^3\right )+15 b^4 e-10 b^3 c (3 d+e x)\right )}{640 c^3}-\frac{3 \left (b^2-4 a c\right )^2 (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{256 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(15*b^4*e - 10*b^3*c*(3*d + e*x) + 8*b*c^2*(25*a*d + 7*a*
e*x + 30*c*d*x^2 + 22*c*e*x^3) + 4*b^2*c*(-25*a*e + c*x*(5*d + 2*e*x)) + 16*c^2*
(8*a^2*e + 2*c^2*x^3*(5*d + 4*e*x) + a*c*x*(25*d + 16*e*x))))/(640*c^3) - (3*(b^
2 - 4*a*c)^2*(-2*c*d + b*e)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(2
56*c^(7/2))

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Maple [B]  time = 0.009, size = 469, normalized size = 2.9 \[{\frac{dx}{4} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{bd}{8\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adx}{8}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,dx{b}^{2}}{32\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,bda}{16\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,d{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{a}^{2}d}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{3\,a{b}^{2}d}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,d{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{e}{5\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}}-{\frac{bex}{8\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}e}{16\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,bexa}{16\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,e{b}^{3}x}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,ae{b}^{2}}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,e{b}^{4}}{128\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{a}^{2}be}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,a{b}^{3}e}{32}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{3\,e{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*d*(c*x^2+b*x+a)^(3/2)*x+1/8*d/c*(c*x^2+b*x+a)^(3/2)*b+3/8*d*(c*x^2+b*x+a)^(1
/2)*x*a-3/32*d/c*(c*x^2+b*x+a)^(1/2)*x*b^2+3/16*d/c*(c*x^2+b*x+a)^(1/2)*b*a-3/64
*d/c^2*(c*x^2+b*x+a)^(1/2)*b^3+3/8*d/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a
)^(1/2))*a^2-3/16*d/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a+3/
128*d/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4+1/5*e*(c*x^2+b*x+a
)^(5/2)/c-1/8*e*b/c*(c*x^2+b*x+a)^(3/2)*x-1/16*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)-3/1
6*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a+3/64*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x-3/32*e*b^2/
c^2*(c*x^2+b*x+a)^(1/2)*a+3/128*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)-3/16*e*b/c^(3/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/32*e*b^3/c^(5/2)*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-3/256*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265323, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (128 \, c^{4} e x^{4} + 16 \,{\left (10 \, c^{4} d + 11 \, b c^{3} e\right )} x^{3} + 8 \,{\left (30 \, b c^{3} d +{\left (b^{2} c^{2} + 32 \, a c^{3}\right )} e\right )} x^{2} - 10 \,{\left (3 \, b^{3} c - 20 \, a b c^{2}\right )} d +{\left (15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2}\right )} e + 2 \,{\left (10 \,{\left (b^{2} c^{2} + 20 \, a c^{3}\right )} d -{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 15 \,{\left (2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d -{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{2560 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (128 \, c^{4} e x^{4} + 16 \,{\left (10 \, c^{4} d + 11 \, b c^{3} e\right )} x^{3} + 8 \,{\left (30 \, b c^{3} d +{\left (b^{2} c^{2} + 32 \, a c^{3}\right )} e\right )} x^{2} - 10 \,{\left (3 \, b^{3} c - 20 \, a b c^{2}\right )} d +{\left (15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2}\right )} e + 2 \,{\left (10 \,{\left (b^{2} c^{2} + 20 \, a c^{3}\right )} d -{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 15 \,{\left (2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d -{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{1280 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2560*(4*(128*c^4*e*x^4 + 16*(10*c^4*d + 11*b*c^3*e)*x^3 + 8*(30*b*c^3*d + (b^
2*c^2 + 32*a*c^3)*e)*x^2 - 10*(3*b^3*c - 20*a*b*c^2)*d + (15*b^4 - 100*a*b^2*c +
128*a^2*c^2)*e + 2*(10*(b^2*c^2 + 20*a*c^3)*d - (5*b^3*c - 28*a*b*c^2)*e)*x)*sq
rt(c*x^2 + b*x + a)*sqrt(c) - 15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d - (b^5
- 8*a*b^3*c + 16*a^2*b*c^2)*e)*log(4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*
c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/c^(7/2), 1/1280*(2*(128*c^4*e*x^4 + 1
6*(10*c^4*d + 11*b*c^3*e)*x^3 + 8*(30*b*c^3*d + (b^2*c^2 + 32*a*c^3)*e)*x^2 - 10
*(3*b^3*c - 20*a*b*c^2)*d + (15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*e + 2*(10*(b^2*
c^2 + 20*a*c^3)*d - (5*b^3*c - 28*a*b*c^2)*e)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c)
+ 15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*
e)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21785, size = 356, normalized size = 2.21 \[ \frac{1}{640} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x e + \frac{10 \, c^{5} d + 11 \, b c^{4} e}{c^{4}}\right )} x + \frac{30 \, b c^{4} d + b^{2} c^{3} e + 32 \, a c^{4} e}{c^{4}}\right )} x + \frac{10 \, b^{2} c^{3} d + 200 \, a c^{4} d - 5 \, b^{3} c^{2} e + 28 \, a b c^{3} e}{c^{4}}\right )} x - \frac{30 \, b^{3} c^{2} d - 200 \, a b c^{3} d - 15 \, b^{4} c e + 100 \, a b^{2} c^{2} e - 128 \, a^{2} c^{3} e}{c^{4}}\right )} - \frac{3 \,{\left (2 \, b^{4} c d - 16 \, a b^{2} c^{2} d + 32 \, a^{2} c^{3} d - b^{5} e + 8 \, a b^{3} c e - 16 \, a^{2} b c^{2} e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/640*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*c*x*e + (10*c^5*d + 11*b*c^4*e)/c^4)*x +
(30*b*c^4*d + b^2*c^3*e + 32*a*c^4*e)/c^4)*x + (10*b^2*c^3*d + 200*a*c^4*d - 5*
b^3*c^2*e + 28*a*b*c^3*e)/c^4)*x - (30*b^3*c^2*d - 200*a*b*c^3*d - 15*b^4*c*e +
100*a*b^2*c^2*e - 128*a^2*c^3*e)/c^4) - 3/256*(2*b^4*c*d - 16*a*b^2*c^2*d + 32*a
^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*c^2*e)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*sqrt(c) - b))/c^(7/2)