3.1173 \(\int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=209 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{192 c^3}+\frac{b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2} \]

[Out]

-(b^2*(24*A*c^2*d + 7*b^2*B*e - 12*b*c*(B*d + A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2
])/(512*c^4) + ((24*A*c^2*d + 7*b^2*B*e - 12*b*c*(B*d + A*e))*(b + 2*c*x)*(b*x +
c*x^2)^(3/2))/(192*c^3) - ((7*b*B*e - 12*c*(B*d + A*e) - 10*B*c*e*x)*(b*x + c*x
^2)^(5/2))/(60*c^2) + (b^4*(24*A*c^2*d + 7*b^2*B*e - 12*b*c*(B*d + A*e))*ArcTanh
[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(9/2))

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Rubi [A]  time = 0.40609, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{192 c^3}+\frac{b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2} \]

Antiderivative was successfully verified.

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

[Out]

-(b^2*(24*A*c^2*d + 7*b^2*B*e - 12*b*c*(B*d + A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2
])/(512*c^4) + ((24*A*c^2*d + 7*b^2*B*e - 12*b*c*(B*d + A*e))*(b + 2*c*x)*(b*x +
c*x^2)^(3/2))/(192*c^3) - ((7*b*B*e - 12*c*(B*d + A*e) - 10*B*c*e*x)*(b*x + c*x
^2)^(5/2))/(60*c^2) + (b^4*(24*A*c^2*d + 7*b^2*B*e - 12*b*c*(B*d + A*e))*ArcTanh
[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(512*c^(9/2))

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Rubi in Sympy [A]  time = 24.1028, size = 206, normalized size = 0.99 \[ - \frac{b^{4} \left (- 7 B b^{2} e + 12 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{512 c^{\frac{9}{2}}} + \frac{b^{2} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (- 7 B b^{2} e + 12 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right )}{512 c^{4}} + \frac{\left (b x + c x^{2}\right )^{\frac{5}{2}} \left (- \frac{7 B b e}{2} + 5 B c e x + 6 c \left (A e + B d\right )\right )}{30 c^{2}} - \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (- 7 B b^{2} e + 12 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right )}{192 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**4*(-7*B*b**2*e + 12*c*(-2*A*c*d + b*(A*e + B*d)))*atanh(sqrt(c)*x/sqrt(b*x +
c*x**2))/(512*c**(9/2)) + b**2*(b + 2*c*x)*sqrt(b*x + c*x**2)*(-7*B*b**2*e + 12
*c*(-2*A*c*d + b*(A*e + B*d)))/(512*c**4) + (b*x + c*x**2)**(5/2)*(-7*B*b*e/2 +
5*B*c*e*x + 6*c*(A*e + B*d))/(30*c**2) - (b + 2*c*x)*(b*x + c*x**2)**(3/2)*(-7*B
*b**2*e + 12*c*(-2*A*c*d + b*(A*e + B*d)))/(192*c**3)

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Mathematica [A]  time = 0.545968, size = 259, normalized size = 1.24 \[ \frac{(x (b+c x))^{3/2} \left (\frac{b^4 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{c^{9/2} (b+c x)^{3/2}}+\frac{\sqrt{x} \left (10 b^4 c (18 A e+18 B d+7 B e x)-8 b^3 c^2 (15 A (3 d+e x)+B x (15 d+7 e x))+48 b^2 c^3 x (A (5 d+2 e x)+B x (2 d+e x))+64 b c^4 x^2 (A (45 d+33 e x)+B x (33 d+26 e x))+128 c^5 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))-105 b^5 B e\right )}{15 c^4 (b+c x)}\right )}{512 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((x*(b + c*x))^(3/2)*((Sqrt[x]*(-105*b^5*B*e + 10*b^4*c*(18*B*d + 18*A*e + 7*B*e
*x) + 48*b^2*c^3*x*(B*x*(2*d + e*x) + A*(5*d + 2*e*x)) + 128*c^5*x^3*(3*A*(5*d +
4*e*x) + 2*B*x*(6*d + 5*e*x)) - 8*b^3*c^2*(15*A*(3*d + e*x) + B*x*(15*d + 7*e*x
)) + 64*b*c^4*x^2*(B*x*(33*d + 26*e*x) + A*(45*d + 33*e*x))))/(15*c^4*(b + c*x))
+ (b^4*(24*A*c^2*d + 7*b^2*B*e - 12*b*c*(B*d + A*e))*Log[c*Sqrt[x] + Sqrt[c]*Sq
rt[b + c*x]])/(c^(9/2)*(b + c*x)^(3/2))))/(512*x^(3/2))

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Maple [B]  time = 0.012, size = 544, normalized size = 2.6 \[{\frac{dAx}{4} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{Abd}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,dA{b}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,A{b}^{3}d}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,dA{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{Ae}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{Bd}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{Abex}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{Bbdx}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}Ae}{16\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}Bd}{16\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,A{b}^{3}xe}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{3}xBd}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,A{b}^{4}e}{128\,{c}^{3}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}Bd}{128\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,A{b}^{5}e}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}-{\frac{3\,{b}^{5}Bd}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{Bex}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,bBe}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}Bex}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{3}Be}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,Be{b}^{4}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,Be{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,Be{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*d*A*(c*x^2+b*x)^(3/2)*x+1/8*d*A/c*(c*x^2+b*x)^(3/2)*b-3/32*d*A*b^2/c*(c*x^2+
b*x)^(1/2)*x-3/64*d*A*b^3/c^2*(c*x^2+b*x)^(1/2)+3/128*d*A*b^4/c^(5/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/5*(c*x^2+b*x)^(5/2)/c*A*e+1/5*(c*x^2+b*x)^(5/2
)/c*B*d-1/8*b/c*(c*x^2+b*x)^(3/2)*x*A*e-1/8*b/c*(c*x^2+b*x)^(3/2)*x*B*d-1/16*b^2
/c^2*(c*x^2+b*x)^(3/2)*A*e-1/16*b^2/c^2*(c*x^2+b*x)^(3/2)*B*d+3/64*b^3/c^2*(c*x^
2+b*x)^(1/2)*x*A*e+3/64*b^3/c^2*(c*x^2+b*x)^(1/2)*x*B*d+3/128*b^4/c^3*(c*x^2+b*x
)^(1/2)*A*e+3/128*b^4/c^3*(c*x^2+b*x)^(1/2)*B*d-3/256*b^5/c^(7/2)*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x)^(1/2))*A*e-3/256*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x)^(1/2))*B*d+1/6*B*e*x*(c*x^2+b*x)^(5/2)/c-7/60*B*e*b/c^2*(c*x^2+b*x)^(5/2)+7
/96*B*e*b^2/c^2*(c*x^2+b*x)^(3/2)*x+7/192*B*e*b^3/c^3*(c*x^2+b*x)^(3/2)-7/256*B*
e*b^4/c^3*(c*x^2+b*x)^(1/2)*x-7/512*B*e*b^5/c^4*(c*x^2+b*x)^(1/2)+7/1024*B*e*b^6
/c^(9/2)*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((c*x^2 + b*x)^(3/2)*(B*x + A)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310073, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, B c^{5} e x^{5} + 128 \,{\left (12 \, B c^{5} d +{\left (13 \, B b c^{4} + 12 \, A c^{5}\right )} e\right )} x^{4} + 48 \,{\left (4 \,{\left (11 \, B b c^{4} + 10 \, A c^{5}\right )} d +{\left (B b^{2} c^{3} + 44 \, A b c^{4}\right )} e\right )} x^{3} + 8 \,{\left (12 \,{\left (B b^{2} c^{3} + 30 \, A b c^{4}\right )} d -{\left (7 \, B b^{3} c^{2} - 12 \, A b^{2} c^{3}\right )} e\right )} x^{2} + 180 \,{\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d - 15 \,{\left (7 \, B b^{5} - 12 \, A b^{4} c\right )} e - 10 \,{\left (12 \,{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d -{\left (7 \, B b^{4} c - 12 \, A b^{3} c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} + 15 \,{\left (12 \,{\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} d -{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{15360 \, c^{\frac{9}{2}}}, \frac{{\left (1280 \, B c^{5} e x^{5} + 128 \,{\left (12 \, B c^{5} d +{\left (13 \, B b c^{4} + 12 \, A c^{5}\right )} e\right )} x^{4} + 48 \,{\left (4 \,{\left (11 \, B b c^{4} + 10 \, A c^{5}\right )} d +{\left (B b^{2} c^{3} + 44 \, A b c^{4}\right )} e\right )} x^{3} + 8 \,{\left (12 \,{\left (B b^{2} c^{3} + 30 \, A b c^{4}\right )} d -{\left (7 \, B b^{3} c^{2} - 12 \, A b^{2} c^{3}\right )} e\right )} x^{2} + 180 \,{\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d - 15 \,{\left (7 \, B b^{5} - 12 \, A b^{4} c\right )} e - 10 \,{\left (12 \,{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d -{\left (7 \, B b^{4} c - 12 \, A b^{3} c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 15 \,{\left (12 \,{\left (B b^{5} c - 2 \, A b^{4} c^{2}\right )} d -{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{7680 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15360*(2*(1280*B*c^5*e*x^5 + 128*(12*B*c^5*d + (13*B*b*c^4 + 12*A*c^5)*e)*x^4
+ 48*(4*(11*B*b*c^4 + 10*A*c^5)*d + (B*b^2*c^3 + 44*A*b*c^4)*e)*x^3 + 8*(12*(B*
b^2*c^3 + 30*A*b*c^4)*d - (7*B*b^3*c^2 - 12*A*b^2*c^3)*e)*x^2 + 180*(B*b^4*c - 2
*A*b^3*c^2)*d - 15*(7*B*b^5 - 12*A*b^4*c)*e - 10*(12*(B*b^3*c^2 - 2*A*b^2*c^3)*d
- (7*B*b^4*c - 12*A*b^3*c^2)*e)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 15*(12*(B*b^5*c
- 2*A*b^4*c^2)*d - (7*B*b^6 - 12*A*b^5*c)*e)*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*
x^2 + b*x)*c))/c^(9/2), 1/7680*((1280*B*c^5*e*x^5 + 128*(12*B*c^5*d + (13*B*b*c^
4 + 12*A*c^5)*e)*x^4 + 48*(4*(11*B*b*c^4 + 10*A*c^5)*d + (B*b^2*c^3 + 44*A*b*c^4
)*e)*x^3 + 8*(12*(B*b^2*c^3 + 30*A*b*c^4)*d - (7*B*b^3*c^2 - 12*A*b^2*c^3)*e)*x^
2 + 180*(B*b^4*c - 2*A*b^3*c^2)*d - 15*(7*B*b^5 - 12*A*b^4*c)*e - 10*(12*(B*b^3*
c^2 - 2*A*b^2*c^3)*d - (7*B*b^4*c - 12*A*b^3*c^2)*e)*x)*sqrt(c*x^2 + b*x)*sqrt(-
c) - 15*(12*(B*b^5*c - 2*A*b^4*c^2)*d - (7*B*b^6 - 12*A*b^5*c)*e)*arctan(sqrt(c*
x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.288456, size = 428, normalized size = 2.05 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B c x e + \frac{12 \, B c^{6} d + 13 \, B b c^{5} e + 12 \, A c^{6} e}{c^{5}}\right )} x + \frac{3 \,{\left (44 \, B b c^{5} d + 40 \, A c^{6} d + B b^{2} c^{4} e + 44 \, A b c^{5} e\right )}}{c^{5}}\right )} x + \frac{12 \, B b^{2} c^{4} d + 360 \, A b c^{5} d - 7 \, B b^{3} c^{3} e + 12 \, A b^{2} c^{4} e}{c^{5}}\right )} x - \frac{5 \,{\left (12 \, B b^{3} c^{3} d - 24 \, A b^{2} c^{4} d - 7 \, B b^{4} c^{2} e + 12 \, A b^{3} c^{3} e\right )}}{c^{5}}\right )} x + \frac{15 \,{\left (12 \, B b^{4} c^{2} d - 24 \, A b^{3} c^{3} d - 7 \, B b^{5} c e + 12 \, A b^{4} c^{2} e\right )}}{c^{5}}\right )} + \frac{{\left (12 \, B b^{5} c d - 24 \, A b^{4} c^{2} d - 7 \, B b^{6} e + 12 \, A b^{5} c e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*B*c*x*e + (12*B*c^6*d + 13*B*b*c^5*e +
12*A*c^6*e)/c^5)*x + 3*(44*B*b*c^5*d + 40*A*c^6*d + B*b^2*c^4*e + 44*A*b*c^5*e)/
c^5)*x + (12*B*b^2*c^4*d + 360*A*b*c^5*d - 7*B*b^3*c^3*e + 12*A*b^2*c^4*e)/c^5)*
x - 5*(12*B*b^3*c^3*d - 24*A*b^2*c^4*d - 7*B*b^4*c^2*e + 12*A*b^3*c^3*e)/c^5)*x
+ 15*(12*B*b^4*c^2*d - 24*A*b^3*c^3*d - 7*B*b^5*c*e + 12*A*b^4*c^2*e)/c^5) + 1/1
024*(12*B*b^5*c*d - 24*A*b^4*c^2*d - 7*B*b^6*e + 12*A*b^5*c*e)*ln(abs(-2*(sqrt(c
)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2)