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

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 = 0.190785, 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, Rules used = {779, 612, 620, 206} \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{192 c^3}+\frac{b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{512 c^{9/2}}-\frac{\left (b x+c x^2\right )^{5/2} (-12 c (A e+B d)+7 b B e-10 B c e x)}{60 c^2} \]

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))

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int (A+B x) (d+e x) \left (b x+c x^2\right )^{3/2} \, dx &=-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{\left (\frac{7}{2} b^2 B e+6 c (2 A c d-b (B d+A e))\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac{\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}-\frac{\left (b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right )\right ) \int \sqrt{b x+c x^2} \, dx}{128 c^3}\\ &=-\frac{b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{\left (b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac{b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{\left (b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{512 c^4}\\ &=-\frac{b^2 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \sqrt{b x+c x^2}}{512 c^4}+\frac{\left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac{(7 b B e-12 c (B d+A e)-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac{b^4 \left (24 A c^2 d+7 b^2 B e-12 b c (B d+A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.630616, size = 245, normalized size = 1.17 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-8 b^3 c^2 (15 A (3 d+e x)+B x (15 d+7 e x))+48 b^2 c^3 x (A (5 d+2 e x)+B x (2 d+e x))+10 b^4 c (18 A e+18 B d+7 B e x)+64 b c^4 x^2 (A (45 d+33 e x)+B x (33 d+26 e x))+128 c^5 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))-105 b^5 B e\right )+\frac{15 b^{7/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right ) \left (-12 b c (A e+B d)+24 A c^2 d+7 b^2 B e\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{7680 c^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-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*b^(7/2)*(24*A*c^2*d + 7*b^
2*B*e - 12*b*c*(B*d + A*e))*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[x]*Sqrt[1 + (c*x)/b])))/(7680*c^(9/2))

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Maple [B]  time = 0.007, size = 544, normalized size = 2.6 \begin{align*}{\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 ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{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{A{b}^{2}e}{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\,{b}^{3}xAe}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{3}Bxd}{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 ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}-{\frac{3\,B{b}^{5}d}{256}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\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\,dA{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,dA{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \end{align*}

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]

1/6*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))+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+1/4*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))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.05214, size = 1355, normalized size = 6.48 \begin{align*} \left [\frac{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 )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (1280 \, B c^{6} e x^{5} + 128 \,{\left (12 \, B c^{6} d +{\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} e\right )} x^{4} + 48 \,{\left (4 \,{\left (11 \, B b c^{5} + 10 \, A c^{6}\right )} d +{\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} e\right )} x^{3} + 8 \,{\left (12 \,{\left (B b^{2} c^{4} + 30 \, A b c^{5}\right )} d -{\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} e\right )} x^{2} + 180 \,{\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d - 15 \,{\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )} e - 10 \,{\left (12 \,{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d -{\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x}}{15360 \, c^{5}}, \frac{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 )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (1280 \, B c^{6} e x^{5} + 128 \,{\left (12 \, B c^{6} d +{\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} e\right )} x^{4} + 48 \,{\left (4 \,{\left (11 \, B b c^{5} + 10 \, A c^{6}\right )} d +{\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} e\right )} x^{3} + 8 \,{\left (12 \,{\left (B b^{2} c^{4} + 30 \, A b c^{5}\right )} d -{\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} e\right )} x^{2} + 180 \,{\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d - 15 \,{\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )} e - 10 \,{\left (12 \,{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d -{\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x}}{7680 \, c^{5}}\right ] \end{align*}

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, algorithm="fricas")

[Out]

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

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

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 [A]  time = 1.16881, size = 428, normalized size = 2.05 \begin{align*} \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 )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \end{align*}

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, 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 + 1
2*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) + 1/1024*(12*B*b^5*c*d - 24*A*b^4*c^2*d - 7*B
*b^6*e + 12*A*b^5*c*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9/2)