3.1192 \(\int \frac{(A+B x) (d+e x)^2}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=189 \[ \frac{\sqrt{b x+c x^2} \left (2 c e x (6 A c e-5 b B e+4 B c d)+6 A c e (8 c d-3 b e)+B \left (15 b^2 e^2-36 b c d e+16 c^2 d^2\right )\right )}{24 c^3}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (6 b^2 c e (A e+2 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2-5 b^3 B e^2\right )}{8 c^{7/2}}+\frac{B \sqrt{b x+c x^2} (d+e x)^2}{3 c} \]

[Out]

(B*(d + e*x)^2*Sqrt[b*x + c*x^2])/(3*c) + ((6*A*c*e*(8*c*d - 3*b*e) + B*(16*c^2*
d^2 - 36*b*c*d*e + 15*b^2*e^2) + 2*c*e*(4*B*c*d - 5*b*B*e + 6*A*c*e)*x)*Sqrt[b*x
+ c*x^2])/(24*c^3) + ((16*A*c^3*d^2 - 5*b^3*B*e^2 + 6*b^2*c*e*(2*B*d + A*e) - 8
*b*c^2*d*(B*d + 2*A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(7/2))

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Rubi [A]  time = 0.47943, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{b x+c x^2} \left (2 c e x (6 A c e-5 b B e+4 B c d)+6 A c e (8 c d-3 b e)+B \left (15 b^2 e^2-36 b c d e+16 c^2 d^2\right )\right )}{24 c^3}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (6 b^2 c e (A e+2 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2-5 b^3 B e^2\right )}{8 c^{7/2}}+\frac{B \sqrt{b x+c x^2} (d+e x)^2}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(B*(d + e*x)^2*Sqrt[b*x + c*x^2])/(3*c) + ((6*A*c*e*(8*c*d - 3*b*e) + B*(16*c^2*
d^2 - 36*b*c*d*e + 15*b^2*e^2) + 2*c*e*(4*B*c*d - 5*b*B*e + 6*A*c*e)*x)*Sqrt[b*x
+ c*x^2])/(24*c^3) + ((16*A*c^3*d^2 - 5*b^3*B*e^2 + 6*b^2*c*e*(2*B*d + A*e) - 8
*b*c^2*d*(B*d + 2*A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(7/2))

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Rubi in Sympy [A]  time = 48.1125, size = 216, normalized size = 1.14 \[ \frac{B \left (d + e x\right )^{2} \sqrt{b x + c x^{2}}}{3 c} + \frac{\sqrt{b x + c x^{2}} \left (- \frac{9 A b c e^{2}}{2} + 12 A c^{2} d e + \frac{15 B b^{2} e^{2}}{4} - 9 B b c d e + 4 B c^{2} d^{2} + \frac{c e x \left (6 A c e - 5 B b e + 4 B c d\right )}{2}\right )}{6 c^{3}} + \frac{\left (6 A b^{2} c e^{2} - 16 A b c^{2} d e + 16 A c^{3} d^{2} - 5 B b^{3} e^{2} + 12 B b^{2} c d e - 8 B b c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(d + e*x)**2*sqrt(b*x + c*x**2)/(3*c) + sqrt(b*x + c*x**2)*(-9*A*b*c*e**2/2 +
12*A*c**2*d*e + 15*B*b**2*e**2/4 - 9*B*b*c*d*e + 4*B*c**2*d**2 + c*e*x*(6*A*c*e
- 5*B*b*e + 4*B*c*d)/2)/(6*c**3) + (6*A*b**2*c*e**2 - 16*A*b*c**2*d*e + 16*A*c**
3*d**2 - 5*B*b**3*e**2 + 12*B*b**2*c*d*e - 8*B*b*c**2*d**2)*atanh(sqrt(c)*x/sqrt
(b*x + c*x**2))/(8*c**(7/2))

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Mathematica [A]  time = 0.317376, size = 193, normalized size = 1.02 \[ \frac{\sqrt{x} \left (\frac{\sqrt{x} (b+c x) \left (6 A c e (-3 b e+8 c d+2 c e x)+B \left (15 b^2 e^2-2 b c e (18 d+5 e x)+8 c^2 \left (3 d^2+3 d e x+e^2 x^2\right )\right )\right )}{3 c^3}+\frac{\sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (6 b^2 c e (A e+2 B d)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2-5 b^3 B e^2\right )}{c^{7/2}}\right )}{8 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*((Sqrt[x]*(b + c*x)*(6*A*c*e*(8*c*d - 3*b*e + 2*c*e*x) + B*(15*b^2*e^2
- 2*b*c*e*(18*d + 5*e*x) + 8*c^2*(3*d^2 + 3*d*e*x + e^2*x^2))))/(3*c^3) + ((16*A
*c^3*d^2 - 5*b^3*B*e^2 + 6*b^2*c*e*(2*B*d + A*e) - 8*b*c^2*d*(B*d + 2*A*e))*Sqrt
[b + c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/c^(7/2)))/(8*Sqrt[x*(b + c*x)]
)

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Maple [B]  time = 0.015, size = 395, normalized size = 2.1 \[{A{d}^{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}+{\frac{xA{e}^{2}}{2\,c}\sqrt{c{x}^{2}+bx}}+{\frac{Bxde}{c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,Ab{e}^{2}}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,Bbde}{2\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,A{b}^{2}{e}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{3\,{b}^{2}Bde}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{\sqrt{c{x}^{2}+bx}Ade}{c}}+{\frac{B{d}^{2}}{c}\sqrt{c{x}^{2}+bx}}-{Abde\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}-{\frac{Bb{d}^{2}}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{\frac{B{e}^{2}{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx}}-{\frac{5\,B{e}^{2}bx}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,B{e}^{2}{b}^{2}}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{3}B{e}^{2}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*d^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))/c^(1/2)+1/2*x/c*(c*x^2+b*x)^(1/2
)*A*e^2+x/c*(c*x^2+b*x)^(1/2)*B*d*e-3/4*b/c^2*(c*x^2+b*x)^(1/2)*A*e^2-3/2*b/c^2*
(c*x^2+b*x)^(1/2)*B*d*e+3/8*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2)
)*A*e^2+3/4*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d*e+2/c*(c*x
^2+b*x)^(1/2)*A*d*e+1/c*(c*x^2+b*x)^(1/2)*B*d^2-b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x)^(1/2))*A*d*e-1/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2)
)*B*d^2+1/3*B*e^2*x^2/c*(c*x^2+b*x)^(1/2)-5/12*B*e^2*b/c^2*x*(c*x^2+b*x)^(1/2)+5
/8*B*e^2*b^2/c^3*(c*x^2+b*x)^(1/2)-5/16*B*e^2*b^3/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((B*x + A)*(e*x + d)^2/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.341477, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B c^{2} e^{2} x^{2} + 24 \, B c^{2} d^{2} - 12 \,{\left (3 \, B b c - 4 \, A c^{2}\right )} d e + 3 \,{\left (5 \, B b^{2} - 6 \, A b c\right )} e^{2} + 2 \,{\left (12 \, B c^{2} d e -{\left (5 \, B b c - 6 \, A c^{2}\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 3 \,{\left (8 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \,{\left (3 \, B b^{2} c - 4 \, A b c^{2}\right )} d e +{\left (5 \, B b^{3} - 6 \, A b^{2} c\right )} e^{2}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{48 \, c^{\frac{7}{2}}}, \frac{{\left (8 \, B c^{2} e^{2} x^{2} + 24 \, B c^{2} d^{2} - 12 \,{\left (3 \, B b c - 4 \, A c^{2}\right )} d e + 3 \,{\left (5 \, B b^{2} - 6 \, A b c\right )} e^{2} + 2 \,{\left (12 \, B c^{2} d e -{\left (5 \, B b c - 6 \, A c^{2}\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 3 \,{\left (8 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \,{\left (3 \, B b^{2} c - 4 \, A b c^{2}\right )} d e +{\left (5 \, B b^{3} - 6 \, A b^{2} c\right )} e^{2}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{24 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(2*(8*B*c^2*e^2*x^2 + 24*B*c^2*d^2 - 12*(3*B*b*c - 4*A*c^2)*d*e + 3*(5*B*b
^2 - 6*A*b*c)*e^2 + 2*(12*B*c^2*d*e - (5*B*b*c - 6*A*c^2)*e^2)*x)*sqrt(c*x^2 + b
*x)*sqrt(c) - 3*(8*(B*b*c^2 - 2*A*c^3)*d^2 - 4*(3*B*b^2*c - 4*A*b*c^2)*d*e + (5*
B*b^3 - 6*A*b^2*c)*e^2)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/c^(7/2
), 1/24*((8*B*c^2*e^2*x^2 + 24*B*c^2*d^2 - 12*(3*B*b*c - 4*A*c^2)*d*e + 3*(5*B*b
^2 - 6*A*b*c)*e^2 + 2*(12*B*c^2*d*e - (5*B*b*c - 6*A*c^2)*e^2)*x)*sqrt(c*x^2 + b
*x)*sqrt(-c) - 3*(8*(B*b*c^2 - 2*A*c^3)*d^2 - 4*(3*B*b^2*c - 4*A*b*c^2)*d*e + (5
*B*b^3 - 6*A*b^2*c)*e^2)*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. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{2}}{\sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*(d + e*x)**2/sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.323286, size = 265, normalized size = 1.4 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (\frac{4 \, B x e^{2}}{c} + \frac{12 \, B c^{2} d e - 5 \, B b c e^{2} + 6 \, A c^{2} e^{2}}{c^{3}}\right )} x + \frac{3 \,{\left (8 \, B c^{2} d^{2} - 12 \, B b c d e + 16 \, A c^{2} d e + 5 \, B b^{2} e^{2} - 6 \, A b c e^{2}\right )}}{c^{3}}\right )} + \frac{{\left (8 \, B b c^{2} d^{2} - 16 \, A c^{3} d^{2} - 12 \, B b^{2} c d e + 16 \, A b c^{2} d e + 5 \, B b^{3} e^{2} - 6 \, A b^{2} c e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(c*x^2 + b*x)*(2*(4*B*x*e^2/c + (12*B*c^2*d*e - 5*B*b*c*e^2 + 6*A*c^2*e
^2)/c^3)*x + 3*(8*B*c^2*d^2 - 12*B*b*c*d*e + 16*A*c^2*d*e + 5*B*b^2*e^2 - 6*A*b*
c*e^2)/c^3) + 1/16*(8*B*b*c^2*d^2 - 16*A*c^3*d^2 - 12*B*b^2*c*d*e + 16*A*b*c^2*d
*e + 5*B*b^3*e^2 - 6*A*b^2*c*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt
(c) - b))/c^(7/2)