3.545 \(\int (d+e x)^4 (a+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=307 \[ \frac{x \left (a+c x^2\right )^{5/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{480 c^2}+\frac{a x \left (a+c x^2\right )^{3/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{384 c^2}+\frac{a^2 x \sqrt{a+c x^2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{256 c^2}+\frac{a^3 \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}+\frac{e \left (a+c x^2\right )^{7/2} \left (7 e x \left (116 c d^2-27 a e^2\right )+16 d \left (103 c d^2-40 a e^2\right )\right )}{5040 c^2}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)^3}{10 c}+\frac{13 d e \left (a+c x^2\right )^{7/2} (d+e x)^2}{90 c} \]

[Out]

(a^2*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*Sqrt[a + c*x^2])/(256*c^2) + (a*(80*c^2*d^4 - 60*a*c*d^2*e^2
+ 3*a^2*e^4)*x*(a + c*x^2)^(3/2))/(384*c^2) + ((80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*(a + c*x^2)^(5/2))/
(480*c^2) + (13*d*e*(d + e*x)^2*(a + c*x^2)^(7/2))/(90*c) + (e*(d + e*x)^3*(a + c*x^2)^(7/2))/(10*c) + (e*(16*
d*(103*c*d^2 - 40*a*e^2) + 7*e*(116*c*d^2 - 27*a*e^2)*x)*(a + c*x^2)^(7/2))/(5040*c^2) + (a^3*(80*c^2*d^4 - 60
*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(256*c^(5/2))

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Rubi [A]  time = 0.293323, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {743, 833, 780, 195, 217, 206} \[ \frac{x \left (a+c x^2\right )^{5/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{480 c^2}+\frac{a x \left (a+c x^2\right )^{3/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{384 c^2}+\frac{a^2 x \sqrt{a+c x^2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{256 c^2}+\frac{a^3 \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}+\frac{e \left (a+c x^2\right )^{7/2} \left (7 e x \left (116 c d^2-27 a e^2\right )+16 d \left (103 c d^2-40 a e^2\right )\right )}{5040 c^2}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)^3}{10 c}+\frac{13 d e \left (a+c x^2\right )^{7/2} (d+e x)^2}{90 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a + c*x^2)^(5/2),x]

[Out]

(a^2*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*Sqrt[a + c*x^2])/(256*c^2) + (a*(80*c^2*d^4 - 60*a*c*d^2*e^2
+ 3*a^2*e^4)*x*(a + c*x^2)^(3/2))/(384*c^2) + ((80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2*e^4)*x*(a + c*x^2)^(5/2))/
(480*c^2) + (13*d*e*(d + e*x)^2*(a + c*x^2)^(7/2))/(90*c) + (e*(d + e*x)^3*(a + c*x^2)^(7/2))/(10*c) + (e*(16*
d*(103*c*d^2 - 40*a*e^2) + 7*e*(116*c*d^2 - 27*a*e^2)*x)*(a + c*x^2)^(7/2))/(5040*c^2) + (a^3*(80*c^2*d^4 - 60
*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(256*c^(5/2))

Rule 743

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

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 780

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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)^4 \left (a+c x^2\right )^{5/2} \, dx &=\frac{e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{\int (d+e x)^2 \left (10 c d^2-3 a e^2+13 c d e x\right ) \left (a+c x^2\right )^{5/2} \, dx}{10 c}\\ &=\frac{13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{\int (d+e x) \left (c d \left (90 c d^2-53 a e^2\right )+c e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{5/2} \, dx}{90 c^2}\\ &=\frac{13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{80 c^2}\\ &=\frac{\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac{13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{96 c^2}\\ &=\frac{a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{384 c^2}+\frac{\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac{13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (a^2 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \sqrt{a+c x^2} \, dx}{128 c^2}\\ &=\frac{a^2 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \sqrt{a+c x^2}}{256 c^2}+\frac{a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{384 c^2}+\frac{\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac{13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (a^3 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{256 c^2}\\ &=\frac{a^2 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \sqrt{a+c x^2}}{256 c^2}+\frac{a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{384 c^2}+\frac{\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac{13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{\left (a^3 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{256 c^2}\\ &=\frac{a^2 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \sqrt{a+c x^2}}{256 c^2}+\frac{a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{384 c^2}+\frac{\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac{13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac{e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac{e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac{a^3 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{256 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.251842, size = 283, normalized size = 0.92 \[ \frac{\sqrt{a+c x^2} \left (24 a^2 c^2 x \left (6195 d^2 e^2 x^2+5760 d^3 e x+2310 d^4+3200 d e^3 x^3+651 e^4 x^4\right )+10 a^3 c e \left (1890 d^2 e x+4608 d^3+512 d e^2 x^2+63 e^3 x^3\right )-5 a^4 e^3 (2048 d+189 e x)+16 a c^3 x^3 \left (10710 d^2 e^2 x^2+8640 d^3 e x+2730 d^4+6080 d e^3 x^3+1323 e^4 x^4\right )+64 c^4 x^5 \left (945 d^2 e^2 x^2+720 d^3 e x+210 d^4+560 d e^3 x^3+126 e^4 x^4\right )\right )}{80640 c^2}+\frac{a^3 \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{256 c^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(a + c*x^2)^(5/2),x]

[Out]

(Sqrt[a + c*x^2]*(-5*a^4*e^3*(2048*d + 189*e*x) + 10*a^3*c*e*(4608*d^3 + 1890*d^2*e*x + 512*d*e^2*x^2 + 63*e^3
*x^3) + 64*c^4*x^5*(210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4) + 24*a^2*c^2*x*(231
0*d^4 + 5760*d^3*e*x + 6195*d^2*e^2*x^2 + 3200*d*e^3*x^3 + 651*e^4*x^4) + 16*a*c^3*x^3*(2730*d^4 + 8640*d^3*e*
x + 10710*d^2*e^2*x^2 + 6080*d*e^3*x^3 + 1323*e^4*x^4)))/(80640*c^2) + (a^3*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a
^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(256*c^(5/2))

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Maple [A]  time = 0.054, size = 386, normalized size = 1.3 \begin{align*}{\frac{{e}^{4}{x}^{3}}{10\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{e}^{4}ax}{80\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}{e}^{4}x}{160\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{4}{a}^{3}x}{128\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{4}{a}^{4}x}{256\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,{e}^{4}{a}^{5}}{256}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{4\,d{e}^{3}{x}^{2}}{9\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,d{e}^{3}a}{63\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{d}^{2}{e}^{2}x}{4\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{d}^{2}{e}^{2}ax}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{d}^{2}{e}^{2}{a}^{2}x}{32\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{d}^{2}{e}^{2}{a}^{3}x}{64\,c}\sqrt{c{x}^{2}+a}}-{\frac{15\,{d}^{2}{e}^{2}{a}^{4}}{64}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{4\,{d}^{3}e}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{d}^{4}x}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{d}^{4}ax}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{4}{a}^{2}x}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,{d}^{4}{a}^{3}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(c*x^2+a)^(5/2),x)

[Out]

1/10*e^4*x^3*(c*x^2+a)^(7/2)/c-3/80*e^4*a/c^2*x*(c*x^2+a)^(7/2)+1/160*e^4*a^2/c^2*x*(c*x^2+a)^(5/2)+1/128*e^4*
a^3/c^2*x*(c*x^2+a)^(3/2)+3/256*e^4*a^4/c^2*x*(c*x^2+a)^(1/2)+3/256*e^4*a^5/c^(5/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/
2))+4/9*d*e^3*x^2*(c*x^2+a)^(7/2)/c-8/63*d*e^3*a/c^2*(c*x^2+a)^(7/2)+3/4*d^2*e^2*x*(c*x^2+a)^(7/2)/c-1/8*d^2*e
^2*a/c*x*(c*x^2+a)^(5/2)-5/32*d^2*e^2*a^2/c*x*(c*x^2+a)^(3/2)-15/64*d^2*e^2*a^3/c*x*(c*x^2+a)^(1/2)-15/64*d^2*
e^2*a^4/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+4/7*d^3*e*(c*x^2+a)^(7/2)/c+1/6*d^4*x*(c*x^2+a)^(5/2)+5/24*d^4*a
*x*(c*x^2+a)^(3/2)+5/16*d^4*a^2*x*(c*x^2+a)^(1/2)+5/16*d^4*a^3/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(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)^4*(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.97832, size = 1577, normalized size = 5.14 \begin{align*} \left [\frac{315 \,{\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (8064 \, c^{5} e^{4} x^{9} + 35840 \, c^{5} d e^{3} x^{8} + 46080 \, a^{3} c^{2} d^{3} e - 10240 \, a^{4} c d e^{3} + 3024 \,{\left (20 \, c^{5} d^{2} e^{2} + 7 \, a c^{4} e^{4}\right )} x^{7} + 5120 \,{\left (9 \, c^{5} d^{3} e + 19 \, a c^{4} d e^{3}\right )} x^{6} + 168 \,{\left (80 \, c^{5} d^{4} + 1020 \, a c^{4} d^{2} e^{2} + 93 \, a^{2} c^{3} e^{4}\right )} x^{5} + 15360 \,{\left (9 \, a c^{4} d^{3} e + 5 \, a^{2} c^{3} d e^{3}\right )} x^{4} + 210 \,{\left (208 \, a c^{4} d^{4} + 708 \, a^{2} c^{3} d^{2} e^{2} + 3 \, a^{3} c^{2} e^{4}\right )} x^{3} + 5120 \,{\left (27 \, a^{2} c^{3} d^{3} e + a^{3} c^{2} d e^{3}\right )} x^{2} + 315 \,{\left (176 \, a^{2} c^{3} d^{4} + 60 \, a^{3} c^{2} d^{2} e^{2} - 3 \, a^{4} c e^{4}\right )} x\right )} \sqrt{c x^{2} + a}}{161280 \, c^{3}}, -\frac{315 \,{\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (8064 \, c^{5} e^{4} x^{9} + 35840 \, c^{5} d e^{3} x^{8} + 46080 \, a^{3} c^{2} d^{3} e - 10240 \, a^{4} c d e^{3} + 3024 \,{\left (20 \, c^{5} d^{2} e^{2} + 7 \, a c^{4} e^{4}\right )} x^{7} + 5120 \,{\left (9 \, c^{5} d^{3} e + 19 \, a c^{4} d e^{3}\right )} x^{6} + 168 \,{\left (80 \, c^{5} d^{4} + 1020 \, a c^{4} d^{2} e^{2} + 93 \, a^{2} c^{3} e^{4}\right )} x^{5} + 15360 \,{\left (9 \, a c^{4} d^{3} e + 5 \, a^{2} c^{3} d e^{3}\right )} x^{4} + 210 \,{\left (208 \, a c^{4} d^{4} + 708 \, a^{2} c^{3} d^{2} e^{2} + 3 \, a^{3} c^{2} e^{4}\right )} x^{3} + 5120 \,{\left (27 \, a^{2} c^{3} d^{3} e + a^{3} c^{2} d e^{3}\right )} x^{2} + 315 \,{\left (176 \, a^{2} c^{3} d^{4} + 60 \, a^{3} c^{2} d^{2} e^{2} - 3 \, a^{4} c e^{4}\right )} x\right )} \sqrt{c x^{2} + a}}{80640 \, c^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

[1/161280*(315*(80*a^3*c^2*d^4 - 60*a^4*c*d^2*e^2 + 3*a^5*e^4)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c
)*x - a) + 2*(8064*c^5*e^4*x^9 + 35840*c^5*d*e^3*x^8 + 46080*a^3*c^2*d^3*e - 10240*a^4*c*d*e^3 + 3024*(20*c^5*
d^2*e^2 + 7*a*c^4*e^4)*x^7 + 5120*(9*c^5*d^3*e + 19*a*c^4*d*e^3)*x^6 + 168*(80*c^5*d^4 + 1020*a*c^4*d^2*e^2 +
93*a^2*c^3*e^4)*x^5 + 15360*(9*a*c^4*d^3*e + 5*a^2*c^3*d*e^3)*x^4 + 210*(208*a*c^4*d^4 + 708*a^2*c^3*d^2*e^2 +
3*a^3*c^2*e^4)*x^3 + 5120*(27*a^2*c^3*d^3*e + a^3*c^2*d*e^3)*x^2 + 315*(176*a^2*c^3*d^4 + 60*a^3*c^2*d^2*e^2
- 3*a^4*c*e^4)*x)*sqrt(c*x^2 + a))/c^3, -1/80640*(315*(80*a^3*c^2*d^4 - 60*a^4*c*d^2*e^2 + 3*a^5*e^4)*sqrt(-c)
*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (8064*c^5*e^4*x^9 + 35840*c^5*d*e^3*x^8 + 46080*a^3*c^2*d^3*e - 10240*a^
4*c*d*e^3 + 3024*(20*c^5*d^2*e^2 + 7*a*c^4*e^4)*x^7 + 5120*(9*c^5*d^3*e + 19*a*c^4*d*e^3)*x^6 + 168*(80*c^5*d^
4 + 1020*a*c^4*d^2*e^2 + 93*a^2*c^3*e^4)*x^5 + 15360*(9*a*c^4*d^3*e + 5*a^2*c^3*d*e^3)*x^4 + 210*(208*a*c^4*d^
4 + 708*a^2*c^3*d^2*e^2 + 3*a^3*c^2*e^4)*x^3 + 5120*(27*a^2*c^3*d^3*e + a^3*c^2*d*e^3)*x^2 + 315*(176*a^2*c^3*
d^4 + 60*a^3*c^2*d^2*e^2 - 3*a^4*c*e^4)*x)*sqrt(c*x^2 + a))/c^3]

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Sympy [A]  time = 75.149, size = 1062, normalized size = 3.46 \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)**4*(c*x**2+a)**(5/2),x)

[Out]

-3*a**(9/2)*e**4*x/(256*c**2*sqrt(1 + c*x**2/a)) + 15*a**(7/2)*d**2*e**2*x/(64*c*sqrt(1 + c*x**2/a)) - a**(7/2
)*e**4*x**3/(256*c*sqrt(1 + c*x**2/a)) + a**(5/2)*d**4*x*sqrt(1 + c*x**2/a)/2 + 3*a**(5/2)*d**4*x/(16*sqrt(1 +
c*x**2/a)) + 133*a**(5/2)*d**2*e**2*x**3/(64*sqrt(1 + c*x**2/a)) + 129*a**(5/2)*e**4*x**5/(640*sqrt(1 + c*x**
2/a)) + 35*a**(3/2)*c*d**4*x**3/(48*sqrt(1 + c*x**2/a)) + 127*a**(3/2)*c*d**2*e**2*x**5/(32*sqrt(1 + c*x**2/a)
) + 73*a**(3/2)*c*e**4*x**7/(160*sqrt(1 + c*x**2/a)) + 17*sqrt(a)*c**2*d**4*x**5/(24*sqrt(1 + c*x**2/a)) + 23*
sqrt(a)*c**2*d**2*e**2*x**7/(8*sqrt(1 + c*x**2/a)) + 29*sqrt(a)*c**2*e**4*x**9/(80*sqrt(1 + c*x**2/a)) + 3*a**
5*e**4*asinh(sqrt(c)*x/sqrt(a))/(256*c**(5/2)) - 15*a**4*d**2*e**2*asinh(sqrt(c)*x/sqrt(a))/(64*c**(3/2)) + 5*
a**3*d**4*asinh(sqrt(c)*x/sqrt(a))/(16*sqrt(c)) + 4*a**2*d**3*e*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*
x**2)**(3/2)/(3*c), True)) + 4*a**2*d*e**3*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x
**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + 8*a*c*d**3*e*Piecewise((-2*a**2*sq
rt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/
4, True)) + 8*a*c*d*e**3*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**
2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, True)) + 4*c**2*d**
3*e*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*sqrt(a +
c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, True)) + 4*c**2*d*e**3*Piecewise((-16*a*
*4*sqrt(a + c*x**2)/(315*c**4) + 8*a**3*x**2*sqrt(a + c*x**2)/(315*c**3) - 2*a**2*x**4*sqrt(a + c*x**2)/(105*c
**2) + a*x**6*sqrt(a + c*x**2)/(63*c) + x**8*sqrt(a + c*x**2)/9, Ne(c, 0)), (sqrt(a)*x**8/8, True)) + c**3*d**
4*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + 3*c**3*d**2*e**2*x**9/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c**3*e**4*x**11
/(10*sqrt(a)*sqrt(1 + c*x**2/a))

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Giac [A]  time = 1.38486, size = 486, normalized size = 1.58 \begin{align*} \frac{1}{80640} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (2 \,{\left (7 \,{\left (8 \,{\left (9 \, c^{2} x e^{4} + 40 \, c^{2} d e^{3}\right )} x + \frac{27 \,{\left (20 \, c^{10} d^{2} e^{2} + 7 \, a c^{9} e^{4}\right )}}{c^{8}}\right )} x + \frac{320 \,{\left (9 \, c^{10} d^{3} e + 19 \, a c^{9} d e^{3}\right )}}{c^{8}}\right )} x + \frac{21 \,{\left (80 \, c^{10} d^{4} + 1020 \, a c^{9} d^{2} e^{2} + 93 \, a^{2} c^{8} e^{4}\right )}}{c^{8}}\right )} x + \frac{1920 \,{\left (9 \, a c^{9} d^{3} e + 5 \, a^{2} c^{8} d e^{3}\right )}}{c^{8}}\right )} x + \frac{105 \,{\left (208 \, a c^{9} d^{4} + 708 \, a^{2} c^{8} d^{2} e^{2} + 3 \, a^{3} c^{7} e^{4}\right )}}{c^{8}}\right )} x + \frac{2560 \,{\left (27 \, a^{2} c^{8} d^{3} e + a^{3} c^{7} d e^{3}\right )}}{c^{8}}\right )} x + \frac{315 \,{\left (176 \, a^{2} c^{8} d^{4} + 60 \, a^{3} c^{7} d^{2} e^{2} - 3 \, a^{4} c^{6} e^{4}\right )}}{c^{8}}\right )} x + \frac{5120 \,{\left (9 \, a^{3} c^{7} d^{3} e - 2 \, a^{4} c^{6} d e^{3}\right )}}{c^{8}}\right )} - \frac{{\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{256 \, c^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/80640*sqrt(c*x^2 + a)*((2*((4*((2*(7*(8*(9*c^2*x*e^4 + 40*c^2*d*e^3)*x + 27*(20*c^10*d^2*e^2 + 7*a*c^9*e^4)/
c^8)*x + 320*(9*c^10*d^3*e + 19*a*c^9*d*e^3)/c^8)*x + 21*(80*c^10*d^4 + 1020*a*c^9*d^2*e^2 + 93*a^2*c^8*e^4)/c
^8)*x + 1920*(9*a*c^9*d^3*e + 5*a^2*c^8*d*e^3)/c^8)*x + 105*(208*a*c^9*d^4 + 708*a^2*c^8*d^2*e^2 + 3*a^3*c^7*e
^4)/c^8)*x + 2560*(27*a^2*c^8*d^3*e + a^3*c^7*d*e^3)/c^8)*x + 315*(176*a^2*c^8*d^4 + 60*a^3*c^7*d^2*e^2 - 3*a^
4*c^6*e^4)/c^8)*x + 5120*(9*a^3*c^7*d^3*e - 2*a^4*c^6*d*e^3)/c^8) - 1/256*(80*a^3*c^2*d^4 - 60*a^4*c*d^2*e^2 +
3*a^5*e^4)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)