3.2355 \(\int (d+e x)^3 (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=400 \[ \frac{e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{2016 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{768 c^4}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{12288 c^5}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{32768 c^6}-\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]

[Out]

(5*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(b + 2*c*x)*Sqrt[a + b*x +
c*x^2])/(32768*c^6) - (5*(b^2 - 4*a*c)*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(b + 2*
c*x)*(a + b*x + c*x^2)^(3/2))/(12288*c^5) + ((2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(
b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(768*c^4) + (e*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/(9*c) + (e*(640*c^2*d^
2 + 99*b^2*e^2 - 2*c*e*(243*b*d + 32*a*e) + 154*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/2))/(2016*c^3) - (5*
(b^2 - 4*a*c)^3*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]
*Sqrt[a + b*x + c*x^2])])/(65536*c^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 0.43993, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {742, 779, 612, 621, 206} \[ \frac{e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{2016 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{768 c^4}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{12288 c^5}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{32768 c^6}-\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]

[Out]

(5*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(b + 2*c*x)*Sqrt[a + b*x +
c*x^2])/(32768*c^6) - (5*(b^2 - 4*a*c)*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(b + 2*
c*x)*(a + b*x + c*x^2)^(3/2))/(12288*c^5) + ((2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*(
b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(768*c^4) + (e*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/(9*c) + (e*(640*c^2*d^
2 + 99*b^2*e^2 - 2*c*e*(243*b*d + 32*a*e) + 154*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/2))/(2016*c^3) - (5*
(b^2 - 4*a*c)^3*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]
*Sqrt[a + b*x + c*x^2])])/(65536*c^(13/2))

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 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 621

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

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)^3 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{\int (d+e x) \left (\frac{1}{2} \left (18 c d^2-e (7 b d+4 a e)\right )+\frac{11}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{\left ((2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{64 c^3}\\ &=\frac{(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{1536 c^4}\\ &=-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{8192 c^5}\\ &=\frac{5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{32768 c^6}-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{65536 c^6}\\ &=\frac{5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{32768 c^6}-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{32768 c^6}\\ &=\frac{5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{32768 c^6}-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac{(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.821426, size = 280, normalized size = 0.7 \[ \frac{\frac{3 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{65536 c^{11/2}}+\frac{e (a+x (b+c x))^{7/2} \left (-2 c e (32 a e+243 b d+77 b e x)+99 b^2 e^2+4 c^2 d (160 d+77 e x)\right )}{224 c^2}+e (d+e x)^2 (a+x (b+c x))^{7/2}}{9 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]

[Out]

(e*(d + e*x)^2*(a + x*(b + c*x))^(7/2) + (e*(a + x*(b + c*x))^(7/2)*(99*b^2*e^2 + 4*c^2*d*(160*d + 77*e*x) - 2
*c*e*(243*b*d + 32*a*e + 77*b*e*x)))/(224*c^2) + (3*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*
a*e))*(256*c^(5/2)*(b + 2*c*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c
*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))))/(65536*c^(11/2)))/(9*c)

________________________________________________________________________________________

Maple [B]  time = 0.056, size = 2294, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x)

[Out]

1/6*d^3*x*(c*x^2+b*x+a)^(5/2)-5/128*d*e^2*a^2/c^2*(c*x^2+b*x+a)^(3/2)*b-15/128*d*e^2*a^3/c*(c*x^2+b*x+a)^(1/2)
*x+45/128*d*e^2*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+125/4096*e^3*b^5/c^4*(c*x^2+b*x+a)
^(1/2)*x*a+5/128*e^3*b/c^2*a^2*(c*x^2+b*x+a)^(3/2)*x+9/64*d*e^2*b^2/c^2*x*(c*x^2+b*x+a)^(5/2)-45/1024*d*e^2*b^
4/c^3*(c*x^2+b*x+a)^(3/2)*x+25/256*d*e^2*b^3/c^3*(c*x^2+b*x+a)^(3/2)*a+135/8192*d*e^2*b^6/c^4*(c*x^2+b*x+a)^(1
/2)*x+165/1024*d*e^2*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a^2-285/4096*d*e^2*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a+55/65536*e^3
*b^9/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-2/63*e^3*a/c^2*(c*x^2+b*x+a)^(7/2)+55/12288*e^3*b^6/
c^5*(c*x^2+b*x+a)^(3/2)-55/32768*e^3*b^8/c^6*(c*x^2+b*x+a)^(1/2)+11/224*e^3*b^2/c^3*(c*x^2+b*x+a)^(7/2)-11/768
*e^3*b^4/c^4*(c*x^2+b*x+a)^(5/2)+1/12*d^3/c*(c*x^2+b*x+a)^(5/2)*b+5/24*d^3*(c*x^2+b*x+a)^(3/2)*x*a-5/192*d^3/c
^2*(c*x^2+b*x+a)^(3/2)*b^3+5/16*d^3*(c*x^2+b*x+a)^(1/2)*x*a^2+5/512*d^3/c^3*(c*x^2+b*x+a)^(1/2)*b^5+5/16*d^3/c
^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-5/1024*d^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*b^6+3/7*d^2*e*(c*x^2+b*x+a)^(7/2)/c-11/144*e^3*b/c^2*x*(c*x^2+b*x+a)^(7/2)-11/384*e^3*b^3/c^3*x*(c*x^2
+b*x+a)^(5/2)+1/9*e^3*x^2*(c*x^2+b*x+a)^(7/2)/c-285/2048*d*e^2*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*a+25/128*d*e^2*b^
2/c^2*(c*x^2+b*x+a)^(3/2)*x*a+15/64*d^2*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a+165/512*d*e^2*b^2/c^2*(c*x^2+b*x+a)^
(1/2)*x*a^2-5/16*d^2*e*b/c*(c*x^2+b*x+a)^(3/2)*x*a-15/32*d^2*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a^2+15/256*d^3/c^(5/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^4*a-135/32768*d*e^2*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))-15/128*d*e^2*a^4/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/8*d*e^2*x*(c*x^2+b*x+a)^
(7/2)/c-45/2048*d*e^2*b^5/c^4*(c*x^2+b*x+a)^(3/2)+135/16384*d*e^2*b^7/c^5*(c*x^2+b*x+a)^(1/2)+125/8192*e^3*b^6
/c^5*(c*x^2+b*x+a)^(1/2)*a+1/64*e^3*b^2/c^3*a*(c*x^2+b*x+a)^(5/2)+5/256*e^3*b^2/c^3*a^2*(c*x^2+b*x+a)^(3/2)+15
/512*e^3*b^2/c^3*a^3*(c*x^2+b*x+a)^(1/2)-1/8*d^2*e*b^2/c^2*(c*x^2+b*x+a)^(5/2)+5/128*d^2*e*b^4/c^3*(c*x^2+b*x+
a)^(3/2)-15/1024*d^2*e*b^6/c^4*(c*x^2+b*x+a)^(1/2)+15/2048*d^2*e*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))-27/112*d*e^2*b/c^2*(c*x^2+b*x+a)^(7/2)+9/128*d*e^2*b^3/c^3*(c*x^2+b*x+a)^(5/2)-5/96*d^3/c*(c*x^2+b*
x+a)^(3/2)*x*b^2+5/48*d^3/c*(c*x^2+b*x+a)^(3/2)*b*a+5/256*d^3/c^2*(c*x^2+b*x+a)^(1/2)*x*b^4+5/32*d^3/c*(c*x^2+
b*x+a)^(1/2)*b*a^2-5/64*d^3/c^2*(c*x^2+b*x+a)^(1/2)*b^3*a-15/64*d^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*b^2*a^2+55/6144*e^3*b^5/c^4*(c*x^2+b*x+a)^(3/2)*x+105/2048*e^3*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*a^2-45/4096*e^3*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-25/256*e^3*b^3/c
^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+15/256*e^3*b/c^(5/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))-35/1536*e^3*b^4/c^4*(c*x^2+b*x+a)^(3/2)*a-55/16384*e^3*b^7/c^5*(c*x^2+b*x+a)^(1/2)*x-85/2048*e^3
*b^4/c^4*(c*x^2+b*x+a)^(1/2)*a^2-15/256*d*e^2*a^3/c^2*(c*x^2+b*x+a)^(1/2)*b-1/16*d*e^2*a/c*x*(c*x^2+b*x+a)^(5/
2)-1/32*d*e^2*a/c^2*(c*x^2+b*x+a)^(5/2)*b-5/64*d*e^2*a^2/c*(c*x^2+b*x+a)^(3/2)*x-5/32*d^3/c*(c*x^2+b*x+a)^(1/2
)*x*a*b^2-85/1024*e^3*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x*a^2-35/768*e^3*b^3/c^3*(c*x^2+b*x+a)^(3/2)*x*a-15/512*d^2*
e*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x-15/64*d^2*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a^2+15/128*d^2*e*b^4/c^3*(c*x^2+b*x+a)
^(1/2)*a-15/32*d^2*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+45/128*d^2*e*b^3/c^(5/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-45/512*d^2*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*a+15/256*e^3*b/c^2*a^3*(c*x^2+b*x+a)^(1/2)*x-1/4*d^2*e*b/c*x*(c*x^2+b*x+a)^(5/2)+5/64*d^2*e*b^3/c^2*(c*x^2+b*
x+a)^(3/2)*x-5/32*d^2*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)*a+1/32*e^3*b/c^2*a*x*(c*x^2+b*x+a)^(5/2)-225/1024*d*e^2*b^
4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+105/2048*d*e^2*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*a

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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)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 6.85473, size = 4959, normalized size = 12.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/8257536*(315*(64*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3 - 96*(b^7*c^2 - 12*a*b^5*c^3 +
48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*e + 6*(9*b^8*c - 112*a*b^6*c^2 + 480*a^2*b^4*c^3 - 768*a^3*b^2*c^4 + 256*a
^4*c^5)*d*e^2 - (11*b^9 - 144*a*b^7*c + 672*a^2*b^5*c^2 - 1280*a^3*b^3*c^3 + 768*a^4*b*c^4)*e^3)*sqrt(c)*log(-
8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(229376*c^9*e^3*x^8 + 143
36*(54*c^9*d*e^2 + 37*b*c^8*e^3)*x^7 + 1024*(864*c^9*d^2*e + 1782*b*c^8*d*e^2 + (309*b^2*c^7 + 608*a*c^8)*e^3)
*x^6 + 256*(1344*c^9*d^3 + 8352*b*c^8*d^2*e + 18*(243*b^2*c^7 + 476*a*c^8)*d*e^2 + (5*b^3*c^6 + 3012*a*b*c^7)*
e^3)*x^5 + 128*(6720*b*c^8*d^3 + 288*(37*b^2*c^7 + 72*a*c^8)*d^2*e + 18*(3*b^3*c^6 + 1228*a*b*c^7)*d*e^2 - (11
*b^4*c^5 - 84*a*b^2*c^6 - 3840*a^2*c^7)*e^3)*x^4 + 1344*(15*b^5*c^4 - 160*a*b^3*c^5 + 528*a^2*b*c^6)*d^3 - 288
*(105*b^6*c^3 - 1120*a*b^4*c^4 + 3696*a^2*b^2*c^5 - 3072*a^3*c^6)*d^2*e + 18*(945*b^7*c^2 - 10500*a*b^5*c^3 +
37744*a^2*b^3*c^4 - 42432*a^3*b*c^5)*d*e^2 - (3465*b^8*c - 40740*a*b^6*c^2 + 162288*a^2*b^4*c^3 - 234432*a^3*b
^2*c^4 + 65536*a^4*c^5)*e^3 + 16*(1344*(27*b^2*c^7 + 52*a*c^8)*d^3 + 288*(3*b^3*c^6 + 788*a*b*c^7)*d^2*e - 18*
(27*b^4*c^5 - 216*a*b^2*c^6 - 6608*a^2*c^7)*d*e^2 + (99*b^5*c^4 - 856*a*b^3*c^5 + 1968*a^2*b*c^6)*e^3)*x^3 + 8
*(1344*(b^3*c^6 + 156*a*b*c^7)*d^3 - 288*(7*b^4*c^5 - 60*a*b^2*c^6 - 1152*a^2*c^7)*d^2*e + 18*(63*b^5*c^4 - 56
8*a*b^3*c^5 + 1392*a^2*b*c^6)*d*e^2 - (231*b^6*c^3 - 2232*a*b^4*c^4 + 6384*a^2*b^2*c^5 - 4096*a^3*c^6)*e^3)*x^
2 - 2*(1344*(5*b^4*c^5 - 48*a*b^2*c^6 - 528*a^2*c^7)*d^3 - 288*(35*b^5*c^4 - 336*a*b^3*c^5 + 912*a^2*b*c^6)*d^
2*e + 18*(315*b^6*c^3 - 3164*a*b^4*c^4 + 9552*a^2*b^2*c^5 - 6720*a^3*c^6)*d*e^2 - (1155*b^7*c^2 - 12348*a*b^5*
c^3 + 42192*a^2*b^3*c^4 - 44096*a^3*b*c^5)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^7, 1/4128768*(315*(64*(b^6*c^3 - 1
2*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^3 - 96*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d
^2*e + 6*(9*b^8*c - 112*a*b^6*c^2 + 480*a^2*b^4*c^3 - 768*a^3*b^2*c^4 + 256*a^4*c^5)*d*e^2 - (11*b^9 - 144*a*b
^7*c + 672*a^2*b^5*c^2 - 1280*a^3*b^3*c^3 + 768*a^4*b*c^4)*e^3)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c
*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(229376*c^9*e^3*x^8 + 14336*(54*c^9*d*e^2 + 37*b*c^8*e^3)*x^7 +
1024*(864*c^9*d^2*e + 1782*b*c^8*d*e^2 + (309*b^2*c^7 + 608*a*c^8)*e^3)*x^6 + 256*(1344*c^9*d^3 + 8352*b*c^8*d
^2*e + 18*(243*b^2*c^7 + 476*a*c^8)*d*e^2 + (5*b^3*c^6 + 3012*a*b*c^7)*e^3)*x^5 + 128*(6720*b*c^8*d^3 + 288*(3
7*b^2*c^7 + 72*a*c^8)*d^2*e + 18*(3*b^3*c^6 + 1228*a*b*c^7)*d*e^2 - (11*b^4*c^5 - 84*a*b^2*c^6 - 3840*a^2*c^7)
*e^3)*x^4 + 1344*(15*b^5*c^4 - 160*a*b^3*c^5 + 528*a^2*b*c^6)*d^3 - 288*(105*b^6*c^3 - 1120*a*b^4*c^4 + 3696*a
^2*b^2*c^5 - 3072*a^3*c^6)*d^2*e + 18*(945*b^7*c^2 - 10500*a*b^5*c^3 + 37744*a^2*b^3*c^4 - 42432*a^3*b*c^5)*d*
e^2 - (3465*b^8*c - 40740*a*b^6*c^2 + 162288*a^2*b^4*c^3 - 234432*a^3*b^2*c^4 + 65536*a^4*c^5)*e^3 + 16*(1344*
(27*b^2*c^7 + 52*a*c^8)*d^3 + 288*(3*b^3*c^6 + 788*a*b*c^7)*d^2*e - 18*(27*b^4*c^5 - 216*a*b^2*c^6 - 6608*a^2*
c^7)*d*e^2 + (99*b^5*c^4 - 856*a*b^3*c^5 + 1968*a^2*b*c^6)*e^3)*x^3 + 8*(1344*(b^3*c^6 + 156*a*b*c^7)*d^3 - 28
8*(7*b^4*c^5 - 60*a*b^2*c^6 - 1152*a^2*c^7)*d^2*e + 18*(63*b^5*c^4 - 568*a*b^3*c^5 + 1392*a^2*b*c^6)*d*e^2 - (
231*b^6*c^3 - 2232*a*b^4*c^4 + 6384*a^2*b^2*c^5 - 4096*a^3*c^6)*e^3)*x^2 - 2*(1344*(5*b^4*c^5 - 48*a*b^2*c^6 -
528*a^2*c^7)*d^3 - 288*(35*b^5*c^4 - 336*a*b^3*c^5 + 912*a^2*b*c^6)*d^2*e + 18*(315*b^6*c^3 - 3164*a*b^4*c^4
+ 9552*a^2*b^2*c^5 - 6720*a^3*c^6)*d*e^2 - (1155*b^7*c^2 - 12348*a*b^5*c^3 + 42192*a^2*b^3*c^4 - 44096*a^3*b*c
^5)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^7]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral((d + e*x)**3*(a + b*x + c*x**2)**(5/2), x)

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Giac [B]  time = 1.17846, size = 1566, normalized size = 3.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/2064384*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*c^2*x*e^3 + (54*c^10*d*e^2 + 37*b*c^9*e^3)/c^8)*x +
(864*c^10*d^2*e + 1782*b*c^9*d*e^2 + 309*b^2*c^8*e^3 + 608*a*c^9*e^3)/c^8)*x + (1344*c^10*d^3 + 8352*b*c^9*d^2
*e + 4374*b^2*c^8*d*e^2 + 8568*a*c^9*d*e^2 + 5*b^3*c^7*e^3 + 3012*a*b*c^8*e^3)/c^8)*x + (6720*b*c^9*d^3 + 1065
6*b^2*c^8*d^2*e + 20736*a*c^9*d^2*e + 54*b^3*c^7*d*e^2 + 22104*a*b*c^8*d*e^2 - 11*b^4*c^6*e^3 + 84*a*b^2*c^7*e
^3 + 3840*a^2*c^8*e^3)/c^8)*x + (36288*b^2*c^8*d^3 + 69888*a*c^9*d^3 + 864*b^3*c^7*d^2*e + 226944*a*b*c^8*d^2*
e - 486*b^4*c^6*d*e^2 + 3888*a*b^2*c^7*d*e^2 + 118944*a^2*c^8*d*e^2 + 99*b^5*c^5*e^3 - 856*a*b^3*c^6*e^3 + 196
8*a^2*b*c^7*e^3)/c^8)*x + (1344*b^3*c^7*d^3 + 209664*a*b*c^8*d^3 - 2016*b^4*c^6*d^2*e + 17280*a*b^2*c^7*d^2*e
+ 331776*a^2*c^8*d^2*e + 1134*b^5*c^5*d*e^2 - 10224*a*b^3*c^6*d*e^2 + 25056*a^2*b*c^7*d*e^2 - 231*b^6*c^4*e^3
+ 2232*a*b^4*c^5*e^3 - 6384*a^2*b^2*c^6*e^3 + 4096*a^3*c^7*e^3)/c^8)*x - (6720*b^4*c^6*d^3 - 64512*a*b^2*c^7*d
^3 - 709632*a^2*c^8*d^3 - 10080*b^5*c^5*d^2*e + 96768*a*b^3*c^6*d^2*e - 262656*a^2*b*c^7*d^2*e + 5670*b^6*c^4*
d*e^2 - 56952*a*b^4*c^5*d*e^2 + 171936*a^2*b^2*c^6*d*e^2 - 120960*a^3*c^7*d*e^2 - 1155*b^7*c^3*e^3 + 12348*a*b
^5*c^4*e^3 - 42192*a^2*b^3*c^5*e^3 + 44096*a^3*b*c^6*e^3)/c^8)*x + (20160*b^5*c^5*d^3 - 215040*a*b^3*c^6*d^3 +
709632*a^2*b*c^7*d^3 - 30240*b^6*c^4*d^2*e + 322560*a*b^4*c^5*d^2*e - 1064448*a^2*b^2*c^6*d^2*e + 884736*a^3*
c^7*d^2*e + 17010*b^7*c^3*d*e^2 - 189000*a*b^5*c^4*d*e^2 + 679392*a^2*b^3*c^5*d*e^2 - 763776*a^3*b*c^6*d*e^2 -
3465*b^8*c^2*e^3 + 40740*a*b^6*c^3*e^3 - 162288*a^2*b^4*c^4*e^3 + 234432*a^3*b^2*c^5*e^3 - 65536*a^4*c^6*e^3)
/c^8) + 5/65536*(64*b^6*c^3*d^3 - 768*a*b^4*c^4*d^3 + 3072*a^2*b^2*c^5*d^3 - 4096*a^3*c^6*d^3 - 96*b^7*c^2*d^2
*e + 1152*a*b^5*c^3*d^2*e - 4608*a^2*b^3*c^4*d^2*e + 6144*a^3*b*c^5*d^2*e + 54*b^8*c*d*e^2 - 672*a*b^6*c^2*d*e
^2 + 2880*a^2*b^4*c^3*d*e^2 - 4608*a^3*b^2*c^4*d*e^2 + 1536*a^4*c^5*d*e^2 - 11*b^9*e^3 + 144*a*b^7*c*e^3 - 672
*a^2*b^5*c^2*e^3 + 1280*a^3*b^3*c^3*e^3 - 768*a^4*b*c^4*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sq
rt(c) - b))/c^(13/2)