3.1164 \(\int (A+B x) (d+e x) \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=154 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{64 c^{7/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{64 c^3}-\frac{\left (b x+c x^2\right )^{3/2} (-8 c (A e+B d)+5 b B e-6 B c e x)}{24 c^2} \]

[Out]

((16*A*c^2*d + 5*b^2*B*e - 8*b*c*(B*d + A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64
*c^3) - ((5*b*B*e - 8*c*(B*d + A*e) - 6*B*c*e*x)*(b*x + c*x^2)^(3/2))/(24*c^2) -
(b^2*(16*A*c^2*d + 5*b^2*B*e - 8*b*c*(B*d + A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/(64*c^(7/2))

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Rubi [A]  time = 0.306905, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{64 c^{7/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{64 c^3}-\frac{\left (b x+c x^2\right )^{3/2} (-8 c (A e+B d)+5 b B e-6 B c e x)}{24 c^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)*Sqrt[b*x + c*x^2],x]

[Out]

((16*A*c^2*d + 5*b^2*B*e - 8*b*c*(B*d + A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(64
*c^3) - ((5*b*B*e - 8*c*(B*d + A*e) - 6*B*c*e*x)*(b*x + c*x^2)^(3/2))/(24*c^2) -
(b^2*(16*A*c^2*d + 5*b^2*B*e - 8*b*c*(B*d + A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x
+ c*x^2]])/(64*c^(7/2))

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Rubi in Sympy [A]  time = 20.3593, size = 151, normalized size = 0.98 \[ \frac{b^{2} \left (- 5 B b^{2} e + 8 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{7}{2}}} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}} \left (- \frac{5 B b e}{2} + 3 B c e x + 4 c \left (A e + B d\right )\right )}{12 c^{2}} - \frac{\left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (- 5 B b^{2} e + 8 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right )}{64 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**(1/2),x)

[Out]

b**2*(-5*B*b**2*e + 8*c*(-2*A*c*d + b*(A*e + B*d)))*atanh(sqrt(c)*x/sqrt(b*x + c
*x**2))/(64*c**(7/2)) + (b*x + c*x**2)**(3/2)*(-5*B*b*e/2 + 3*B*c*e*x + 4*c*(A*e
+ B*d))/(12*c**2) - (b + 2*c*x)*sqrt(b*x + c*x**2)*(-5*B*b**2*e + 8*c*(-2*A*c*d
+ b*(A*e + B*d)))/(64*c**3)

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Mathematica [A]  time = 0.360254, size = 179, normalized size = 1.16 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-2 b^2 c (12 A e+12 B d+5 B e x)+8 b c^2 (2 A (3 d+e x)+B x (2 d+e x))+16 c^3 x (A (6 d+4 e x)+B x (4 d+3 e x))+15 b^3 B e\right )-\frac{3 b^2 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{192 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)*Sqrt[b*x + c*x^2],x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(15*b^3*B*e - 2*b^2*c*(12*B*d + 12*A*e + 5*B*e*x) +
8*b*c^2*(B*x*(2*d + e*x) + 2*A*(3*d + e*x)) + 16*c^3*x*(B*x*(4*d + 3*e*x) + A*(6
*d + 4*e*x))) - (3*b^2*(16*A*c^2*d + 5*b^2*B*e - 8*b*c*(B*d + A*e))*Log[c*Sqrt[x
] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(192*c^(7/2))

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Maple [B]  time = 0.01, size = 372, normalized size = 2.4 \[{\frac{Adx}{2}\sqrt{c{x}^{2}+bx}}+{\frac{Abd}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{Ad{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{\frac{Ae}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{Bd}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{Abex}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{Bbdx}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}Ae}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}Bd}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{A{b}^{3}e}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{B{b}^{3}d}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{Bex}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,bBe}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}Bex}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{3}Be}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,Be{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)*(c*x^2+b*x)^(1/2),x)

[Out]

1/2*A*d*(c*x^2+b*x)^(1/2)*x+1/4*A*d/c*(c*x^2+b*x)^(1/2)*b-1/8*A*d*b^2/c^(3/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/3*(c*x^2+b*x)^(3/2)/c*A*e+1/3*(c*x^2+b
*x)^(3/2)/c*B*d-1/4*b/c*(c*x^2+b*x)^(1/2)*x*A*e-1/4*b/c*(c*x^2+b*x)^(1/2)*x*B*d-
1/8*b^2/c^2*(c*x^2+b*x)^(1/2)*A*e-1/8*b^2/c^2*(c*x^2+b*x)^(1/2)*B*d+1/16*b^3/c^(
5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*e+1/16*b^3/c^(5/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d+1/4*B*e*x*(c*x^2+b*x)^(3/2)/c-5/24*B*e*b/c^2*(
c*x^2+b*x)^(3/2)+5/32*B*e*b^2/c^2*(c*x^2+b*x)^(1/2)*x+5/64*B*e*b^3/c^3*(c*x^2+b*
x)^(1/2)-5/128*B*e*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(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(sqrt(c*x^2 + b*x)*(B*x + A)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.328271, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B c^{3} e x^{3} + 8 \,{\left (8 \, B c^{3} d +{\left (B b c^{2} + 8 \, A c^{3}\right )} e\right )} x^{2} - 24 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} d + 3 \,{\left (5 \, B b^{3} - 8 \, A b^{2} c\right )} e + 2 \,{\left (8 \,{\left (B b c^{2} + 6 \, A c^{3}\right )} d -{\left (5 \, B b^{2} c - 8 \, A b c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} + 3 \,{\left (8 \,{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d -{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{384 \, c^{\frac{7}{2}}}, \frac{{\left (48 \, B c^{3} e x^{3} + 8 \,{\left (8 \, B c^{3} d +{\left (B b c^{2} + 8 \, A c^{3}\right )} e\right )} x^{2} - 24 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} d + 3 \,{\left (5 \, B b^{3} - 8 \, A b^{2} c\right )} e + 2 \,{\left (8 \,{\left (B b c^{2} + 6 \, A c^{3}\right )} d -{\left (5 \, B b^{2} c - 8 \, A b c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 3 \,{\left (8 \,{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d -{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{192 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)*(e*x + d),x, algorithm="fricas")

[Out]

[1/384*(2*(48*B*c^3*e*x^3 + 8*(8*B*c^3*d + (B*b*c^2 + 8*A*c^3)*e)*x^2 - 24*(B*b^
2*c - 2*A*b*c^2)*d + 3*(5*B*b^3 - 8*A*b^2*c)*e + 2*(8*(B*b*c^2 + 6*A*c^3)*d - (5
*B*b^2*c - 8*A*b*c^2)*e)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 3*(8*(B*b^3*c - 2*A*b^2*
c^2)*d - (5*B*b^4 - 8*A*b^3*c)*e)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*
c))/c^(7/2), 1/192*((48*B*c^3*e*x^3 + 8*(8*B*c^3*d + (B*b*c^2 + 8*A*c^3)*e)*x^2
- 24*(B*b^2*c - 2*A*b*c^2)*d + 3*(5*B*b^3 - 8*A*b^2*c)*e + 2*(8*(B*b*c^2 + 6*A*c
^3)*d - (5*B*b^2*c - 8*A*b*c^2)*e)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) + 3*(8*(B*b^3*c
- 2*A*b^2*c^2)*d - (5*B*b^4 - 8*A*b^3*c)*e)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(
c*x)))/(sqrt(-c)*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x \left (b + c x\right )} \left (A + B x\right ) \left (d + e x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**(1/2),x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)*(d + e*x), x)

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GIAC/XCAS [A]  time = 0.292745, size = 277, normalized size = 1.8 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, B x e + \frac{8 \, B c^{3} d + B b c^{2} e + 8 \, A c^{3} e}{c^{3}}\right )} x + \frac{8 \, B b c^{2} d + 48 \, A c^{3} d - 5 \, B b^{2} c e + 8 \, A b c^{2} e}{c^{3}}\right )} x - \frac{3 \,{\left (8 \, B b^{2} c d - 16 \, A b c^{2} d - 5 \, B b^{3} e + 8 \, A b^{2} c e\right )}}{c^{3}}\right )} - \frac{{\left (8 \, B b^{3} c d - 16 \, A b^{2} c^{2} d - 5 \, B b^{4} e + 8 \, A b^{3} c e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*(B*x + A)*(e*x + d),x, algorithm="giac")

[Out]

1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*B*x*e + (8*B*c^3*d + B*b*c^2*e + 8*A*c^3*e)/c^3
)*x + (8*B*b*c^2*d + 48*A*c^3*d - 5*B*b^2*c*e + 8*A*b*c^2*e)/c^3)*x - 3*(8*B*b^2
*c*d - 16*A*b*c^2*d - 5*B*b^3*e + 8*A*b^2*c*e)/c^3) - 1/128*(8*B*b^3*c*d - 16*A*
b^2*c^2*d - 5*B*b^4*e + 8*A*b^3*c*e)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*s
qrt(c) - b))/c^(7/2)