3.303 \(\int (d+e x)^2 (b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=266 \[ \frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{16384 c^5}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{6144 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{384 c^3}-\frac{5 b^6 \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{9 e \left (b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac{e \left (b x+c x^2\right )^{7/2} (d+e x)}{8 c} \]

[Out]

(5*b^4*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^5) - (5*b^2*(32*c^2*d^2 -
32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*
(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) + (9*e*(2*c*d - b*e)*(b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*
(b*x + c*x^2)^(7/2))/(8*c) - (5*b^6*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2
]])/(16384*c^(11/2))

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Rubi [A]  time = 0.220245, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {742, 640, 612, 620, 206} \[ \frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{16384 c^5}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{6144 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{384 c^3}-\frac{5 b^6 \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{9 e \left (b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac{e \left (b x+c x^2\right )^{7/2} (d+e x)}{8 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^4*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^5) - (5*b^2*(32*c^2*d^2 -
32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*
(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) + (9*e*(2*c*d - b*e)*(b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*
(b*x + c*x^2)^(7/2))/(8*c) - (5*b^6*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2
]])/(16384*c^(11/2))

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[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 (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx &=\frac{e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}+\frac{\int \left (\frac{1}{2} d (16 c d-7 b e)+\frac{9}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac{9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}+\frac{\left (c d (16 c d-7 b e)-\frac{9}{2} b e (2 c d-b e)\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac{\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac{\left (5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac{5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \int \sqrt{b x+c x^2} \, dx}{4096 c^4}\\ &=\frac{5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{16384 c^5}-\frac{5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac{\left (5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac{5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{16384 c^5}-\frac{5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac{\left (5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{16384 c^5}\\ &=\frac{5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{16384 c^5}-\frac{5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac{\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac{9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac{5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.591327, size = 219, normalized size = 0.82 \[ \frac{(x (b+c x))^{5/2} \left (\frac{\left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \left (\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \left (8 b^3 c^2 x^2+432 b^2 c^3 x^3-10 b^4 c x+15 b^5+640 b c^4 x^4+256 c^5 x^5\right )-15 b^{11/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )}{6144 c^{9/2} (b+c x)^2 \sqrt{\frac{c x}{b}+1}}+\frac{9 e x^{7/2} (b+c x) (2 c d-b e)}{14 c}+e x^{7/2} (b+c x) (d+e x)\right )}{8 c x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(b*x + c*x^2)^(5/2),x]

[Out]

((x*(b + c*x))^(5/2)*((9*e*(2*c*d - b*e)*x^(7/2)*(b + c*x))/(14*c) + e*x^(7/2)*(b + c*x)*(d + e*x) + ((32*c^2*
d^2 - 32*b*c*d*e + 9*b^2*e^2)*(Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*(15*b^5 - 10*b^4*c*x + 8*b^3*c^2*x^2 + 432*b^
2*c^3*x^3 + 640*b*c^4*x^4 + 256*c^5*x^5) - 15*b^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]))/(6144*c^(9/2)*(b +
c*x)^2*Sqrt[1 + (c*x)/b])))/(8*c*x^(5/2))

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Maple [B]  time = 0.052, size = 553, normalized size = 2.1 \begin{align*}{\frac{{e}^{2}x}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{9\,{e}^{2}b}{112\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{b}^{2}{e}^{2}x}{64\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{b}^{3}{e}^{2}}{128\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{15\,{e}^{2}{b}^{4}x}{1024\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{e}^{2}{b}^{5}}{2048\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{45\,{e}^{2}{b}^{6}x}{8192\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{45\,{e}^{2}{b}^{7}}{16384\,{c}^{5}}\sqrt{c{x}^{2}+bx}}-{\frac{45\,{e}^{2}{b}^{8}}{32768}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{11}{2}}}}+{\frac{2\,de}{7\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{bdex}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}de}{12\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{5\,de{b}^{3}x}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,de{b}^{4}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,de{b}^{5}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,de{b}^{6}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,de{b}^{7}}{1024}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{{d}^{2}x}{6} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{2}b}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}{d}^{2}x}{96\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{2}{b}^{3}}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{2}{b}^{4}x}{256\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{d}^{2}{b}^{5}}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{d}^{2}{b}^{6}}{1024}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(c*x^2+b*x)^(5/2),x)

[Out]

1/8*e^2*x*(c*x^2+b*x)^(7/2)/c-9/112*e^2*b/c^2*(c*x^2+b*x)^(7/2)+3/64*e^2*b^2/c^2*x*(c*x^2+b*x)^(5/2)+3/128*e^2
*b^3/c^3*(c*x^2+b*x)^(5/2)-15/1024*e^2*b^4/c^3*(c*x^2+b*x)^(3/2)*x-15/2048*e^2*b^5/c^4*(c*x^2+b*x)^(3/2)+45/81
92*e^2*b^6/c^4*(c*x^2+b*x)^(1/2)*x+45/16384*e^2*b^7/c^5*(c*x^2+b*x)^(1/2)-45/32768*e^2*b^8/c^(11/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+2/7*d*e*(c*x^2+b*x)^(7/2)/c-1/6*d*e*b/c*x*(c*x^2+b*x)^(5/2)-1/12*d*e*b^2/c^2*(
c*x^2+b*x)^(5/2)+5/96*d*e*b^3/c^2*(c*x^2+b*x)^(3/2)*x+5/192*d*e*b^4/c^3*(c*x^2+b*x)^(3/2)-5/256*d*e*b^5/c^3*(c
*x^2+b*x)^(1/2)*x-5/512*d*e*b^6/c^4*(c*x^2+b*x)^(1/2)+5/1024*d*e*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
)^(1/2))+1/6*d^2*x*(c*x^2+b*x)^(5/2)+1/12*d^2/c*(c*x^2+b*x)^(5/2)*b-5/96*d^2*b^2/c*(c*x^2+b*x)^(3/2)*x-5/192*d
^2*b^3/c^2*(c*x^2+b*x)^(3/2)+5/256*d^2*b^4/c^2*(c*x^2+b*x)^(1/2)*x+5/512*d^2*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*
d^2*b^6/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(-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((e*x+d)^2*(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.16372, size = 1481, normalized size = 5.57 \begin{align*} \left [\frac{105 \,{\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (43008 \, c^{8} e^{2} x^{7} + 3360 \, b^{5} c^{3} d^{2} - 3360 \, b^{6} c^{2} d e + 945 \, b^{7} c e^{2} + 3072 \,{\left (32 \, c^{8} d e + 33 \, b c^{7} e^{2}\right )} x^{6} + 256 \,{\left (224 \, c^{8} d^{2} + 928 \, b c^{7} d e + 243 \, b^{2} c^{6} e^{2}\right )} x^{5} + 128 \,{\left (1120 \, b c^{7} d^{2} + 1184 \, b^{2} c^{6} d e + 3 \, b^{3} c^{5} e^{2}\right )} x^{4} + 48 \,{\left (2016 \, b^{2} c^{6} d^{2} + 32 \, b^{3} c^{5} d e - 9 \, b^{4} c^{4} e^{2}\right )} x^{3} + 56 \,{\left (32 \, b^{3} c^{5} d^{2} - 32 \, b^{4} c^{4} d e + 9 \, b^{5} c^{3} e^{2}\right )} x^{2} - 70 \,{\left (32 \, b^{4} c^{4} d^{2} - 32 \, b^{5} c^{3} d e + 9 \, b^{6} c^{2} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{688128 \, c^{6}}, \frac{105 \,{\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (43008 \, c^{8} e^{2} x^{7} + 3360 \, b^{5} c^{3} d^{2} - 3360 \, b^{6} c^{2} d e + 945 \, b^{7} c e^{2} + 3072 \,{\left (32 \, c^{8} d e + 33 \, b c^{7} e^{2}\right )} x^{6} + 256 \,{\left (224 \, c^{8} d^{2} + 928 \, b c^{7} d e + 243 \, b^{2} c^{6} e^{2}\right )} x^{5} + 128 \,{\left (1120 \, b c^{7} d^{2} + 1184 \, b^{2} c^{6} d e + 3 \, b^{3} c^{5} e^{2}\right )} x^{4} + 48 \,{\left (2016 \, b^{2} c^{6} d^{2} + 32 \, b^{3} c^{5} d e - 9 \, b^{4} c^{4} e^{2}\right )} x^{3} + 56 \,{\left (32 \, b^{3} c^{5} d^{2} - 32 \, b^{4} c^{4} d e + 9 \, b^{5} c^{3} e^{2}\right )} x^{2} - 70 \,{\left (32 \, b^{4} c^{4} d^{2} - 32 \, b^{5} c^{3} d e + 9 \, b^{6} c^{2} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{344064 \, c^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[1/688128*(105*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)
) + 2*(43008*c^8*e^2*x^7 + 3360*b^5*c^3*d^2 - 3360*b^6*c^2*d*e + 945*b^7*c*e^2 + 3072*(32*c^8*d*e + 33*b*c^7*e
^2)*x^6 + 256*(224*c^8*d^2 + 928*b*c^7*d*e + 243*b^2*c^6*e^2)*x^5 + 128*(1120*b*c^7*d^2 + 1184*b^2*c^6*d*e + 3
*b^3*c^5*e^2)*x^4 + 48*(2016*b^2*c^6*d^2 + 32*b^3*c^5*d*e - 9*b^4*c^4*e^2)*x^3 + 56*(32*b^3*c^5*d^2 - 32*b^4*c
^4*d*e + 9*b^5*c^3*e^2)*x^2 - 70*(32*b^4*c^4*d^2 - 32*b^5*c^3*d*e + 9*b^6*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^6,
1/344064*(105*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) +
(43008*c^8*e^2*x^7 + 3360*b^5*c^3*d^2 - 3360*b^6*c^2*d*e + 945*b^7*c*e^2 + 3072*(32*c^8*d*e + 33*b*c^7*e^2)*x^
6 + 256*(224*c^8*d^2 + 928*b*c^7*d*e + 243*b^2*c^6*e^2)*x^5 + 128*(1120*b*c^7*d^2 + 1184*b^2*c^6*d*e + 3*b^3*c
^5*e^2)*x^4 + 48*(2016*b^2*c^6*d^2 + 32*b^3*c^5*d*e - 9*b^4*c^4*e^2)*x^3 + 56*(32*b^3*c^5*d^2 - 32*b^4*c^4*d*e
+ 9*b^5*c^3*e^2)*x^2 - 70*(32*b^4*c^4*d^2 - 32*b^5*c^3*d*e + 9*b^6*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^6]

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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{5}{2}} \left (d + e x\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(c*x**2+b*x)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*(d + e*x)**2, x)

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Giac [A]  time = 1.89956, size = 473, normalized size = 1.78 \begin{align*} \frac{1}{344064} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (14 \, c^{2} x e^{2} + \frac{32 \, c^{9} d e + 33 \, b c^{8} e^{2}}{c^{7}}\right )} x + \frac{224 \, c^{9} d^{2} + 928 \, b c^{8} d e + 243 \, b^{2} c^{7} e^{2}}{c^{7}}\right )} x + \frac{1120 \, b c^{8} d^{2} + 1184 \, b^{2} c^{7} d e + 3 \, b^{3} c^{6} e^{2}}{c^{7}}\right )} x + \frac{3 \,{\left (2016 \, b^{2} c^{7} d^{2} + 32 \, b^{3} c^{6} d e - 9 \, b^{4} c^{5} e^{2}\right )}}{c^{7}}\right )} x + \frac{7 \,{\left (32 \, b^{3} c^{6} d^{2} - 32 \, b^{4} c^{5} d e + 9 \, b^{5} c^{4} e^{2}\right )}}{c^{7}}\right )} x - \frac{35 \,{\left (32 \, b^{4} c^{5} d^{2} - 32 \, b^{5} c^{4} d e + 9 \, b^{6} c^{3} e^{2}\right )}}{c^{7}}\right )} x + \frac{105 \,{\left (32 \, b^{5} c^{4} d^{2} - 32 \, b^{6} c^{3} d e + 9 \, b^{7} c^{2} e^{2}\right )}}{c^{7}}\right )} + \frac{5 \,{\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

1/344064*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*(14*c^2*x*e^2 + (32*c^9*d*e + 33*b*c^8*e^2)/c^7)*x + (224*c^9*d^
2 + 928*b*c^8*d*e + 243*b^2*c^7*e^2)/c^7)*x + (1120*b*c^8*d^2 + 1184*b^2*c^7*d*e + 3*b^3*c^6*e^2)/c^7)*x + 3*(
2016*b^2*c^7*d^2 + 32*b^3*c^6*d*e - 9*b^4*c^5*e^2)/c^7)*x + 7*(32*b^3*c^6*d^2 - 32*b^4*c^5*d*e + 9*b^5*c^4*e^2
)/c^7)*x - 35*(32*b^4*c^5*d^2 - 32*b^5*c^4*d*e + 9*b^6*c^3*e^2)/c^7)*x + 105*(32*b^5*c^4*d^2 - 32*b^6*c^3*d*e
+ 9*b^7*c^2*e^2)/c^7) + 5/32768*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*sqrt(c) - b))/c^(11/2)