\(\int (d+e x)^4 (a+c x^2)^{5/2} \, dx\) [545]

Optimal result

Integrand size = 19, antiderivative size = 307 \[ \int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx=\frac {a^2 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \sqrt {a+c x^2}}{256 c^2}+\frac {a \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{3/2}}{384 c^2}+\frac {\left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) x \left (a+c x^2\right )^{5/2}}{480 c^2}+\frac {13 d e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{90 c}+\frac {e (d+e x)^3 \left (a+c x^2\right )^{7/2}}{10 c}+\frac {e \left (16 d \left (103 c d^2-40 a e^2\right )+7 e \left (116 c d^2-27 a e^2\right ) x\right ) \left (a+c x^2\right )^{7/2}}{5040 c^2}+\frac {a^3 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{256 c^{5/2}} \]

[Out]

1/384*a*(3*a^2*e^4-60*a*c*d^2*e^2+80*c^2*d^4)*x*(c*x^2+a)^(3/2)/c^2+1/480*(3*a^2*e^4-60*a*c*d^2*e^2+80*c^2*d^4
)*x*(c*x^2+a)^(5/2)/c^2+13/90*d*e*(e*x+d)^2*(c*x^2+a)^(7/2)/c+1/10*e*(e*x+d)^3*(c*x^2+a)^(7/2)/c+1/5040*e*(16*
d*(-40*a*e^2+103*c*d^2)+7*e*(-27*a*e^2+116*c*d^2)*x)*(c*x^2+a)^(7/2)/c^2+1/256*a^3*(3*a^2*e^4-60*a*c*d^2*e^2+8
0*c^2*d^4)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(5/2)+1/256*a^2*(3*a^2*e^4-60*a*c*d^2*e^2+80*c^2*d^4)*x*(c*x^2
+a)^(1/2)/c^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {757, 847, 794, 201, 223, 212} \[ \int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx=\frac {x \left (a+c x^2\right )^{5/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{480 c^2}+\frac {a x \left (a+c x^2\right )^{3/2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{384 c^2}+\frac {a^2 x \sqrt {a+c x^2} \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{256 c^2}+\frac {a^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 e^4-60 a c d^2 e^2+80 c^2 d^4\right )}{256 c^{5/2}}+\frac {e \left (a+c x^2\right )^{7/2} \left (7 e x \left (116 c d^2-27 a e^2\right )+16 d \left (103 c d^2-40 a e^2\right )\right )}{5040 c^2}+\frac {e \left (a+c x^2\right )^{7/2} (d+e x)^3}{10 c}+\frac {13 d e \left (a+c x^2\right )^{7/2} (d+e x)^2}{90 c} \]

[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 201

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 212

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

Rule 223

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 757

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 794

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 847

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])

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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] (verified)

Time = 0.95 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.92 \[ \int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx=\frac {\sqrt {c} \sqrt {a+c x^2} \left (-5 a^4 e^3 (2048 d+189 e x)+10 a^3 c e \left (4608 d^3+1890 d^2 e x+512 d e^2 x^2+63 e^3 x^3\right )+64 c^4 x^5 \left (210 d^4+720 d^3 e x+945 d^2 e^2 x^2+560 d e^3 x^3+126 e^4 x^4\right )+24 a^2 c^2 x \left (2310 d^4+5760 d^3 e x+6195 d^2 e^2 x^2+3200 d e^3 x^3+651 e^4 x^4\right )+16 a c^3 x^3 \left (2730 d^4+8640 d^3 e x+10710 d^2 e^2 x^2+6080 d e^3 x^3+1323 e^4 x^4\right )\right )-315 a^3 \left (80 c^2 d^4-60 a c d^2 e^2+3 a^2 e^4\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{80640 c^{5/2}} \]

[In]

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

[Out]

(Sqrt[c]*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*(2310*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 + 864
0*d^3*e*x + 10710*d^2*e^2*x^2 + 6080*d*e^3*x^3 + 1323*e^4*x^4)) - 315*a^3*(80*c^2*d^4 - 60*a*c*d^2*e^2 + 3*a^2
*e^4)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(80640*c^(5/2))

Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.08

method	result	size
risch	\(-\frac {\left (-8064 c^{4} e^{4} x^{9}-35840 c^{4} d \,e^{3} x^{8}-21168 a \,c^{3} e^{4} x^{7}-60480 d^{2} e^{2} c^{4} x^{7}-97280 a \,c^{3} d \,e^{3} x^{6}-46080 c^{4} d^{3} e \,x^{6}-15624 a^{2} c^{2} e^{4} x^{5}-171360 d^{2} e^{2} c^{3} a \,x^{5}-13440 c^{4} d^{4} x^{5}-76800 d \,e^{3} a^{2} c^{2} x^{4}-138240 a \,c^{3} d^{3} e \,x^{4}-630 a^{3} c \,e^{4} x^{3}-148680 a^{2} c^{2} d^{2} e^{2} x^{3}-43680 a \,c^{3} d^{4} x^{3}-5120 a^{3} c d \,e^{3} x^{2}-138240 a^{2} c^{2} d^{3} e \,x^{2}+945 e^{4} a^{4} x -18900 a^{3} c \,d^{2} e^{2} x -55440 d^{4} a^{2} c^{2} x +10240 a^{4} d \,e^{3}-46080 a^{3} c \,d^{3} e \right ) \sqrt {c \,x^{2}+a}}{80640 c^{2}}+\frac {a^{3} \left (3 a^{2} e^{4}-60 a c \,d^{2} e^{2}+80 c^{2} d^{4}\right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{256 c^{\frac {5}{2}}}\)	\(332\)
default	\(d^{4} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )+e^{4} \left (\frac {x^{3} \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{10 c}-\frac {3 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{8 c}\right )}{10 c}\right )+4 d \,e^{3} \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{9 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{63 c^{2}}\right )+6 d^{2} e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{8 c}\right )+\frac {4 d^{3} e \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 c}\)	\(344\)

[In]

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

[Out]

-1/80640/c^2*(-8064*c^4*e^4*x^9-35840*c^4*d*e^3*x^8-21168*a*c^3*e^4*x^7-60480*c^4*d^2*e^2*x^7-97280*a*c^3*d*e^
3*x^6-46080*c^4*d^3*e*x^6-15624*a^2*c^2*e^4*x^5-171360*a*c^3*d^2*e^2*x^5-13440*c^4*d^4*x^5-76800*a^2*c^2*d*e^3
*x^4-138240*a*c^3*d^3*e*x^4-630*a^3*c*e^4*x^3-148680*a^2*c^2*d^2*e^2*x^3-43680*a*c^3*d^4*x^3-5120*a^3*c*d*e^3*
x^2-138240*a^2*c^2*d^3*e*x^2+945*a^4*e^4*x-18900*a^3*c*d^2*e^2*x-55440*a^2*c^2*d^4*x+10240*a^4*d*e^3-46080*a^3
*c*d^3*e)*(c*x^2+a)^(1/2)+1/256*a^3*(3*a^2*e^4-60*a*c*d^2*e^2+80*c^2*d^4)/c^(5/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2)
)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.25 \[ \int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx=\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 ] \]

[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]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 858 vs. \(2 (298) = 596\).

Time = 0.72 (sec) , antiderivative size = 858, normalized size of antiderivative = 2.79 \[ \int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx=\begin {cases} \sqrt {a + c x^{2}} \cdot \left (\frac {4 c^{2} d e^{3} x^{8}}{9} + \frac {c^{2} e^{4} x^{9}}{10} + \frac {x^{7} \cdot \left (\frac {21 a c^{2} e^{4}}{10} + 6 c^{3} d^{2} e^{2}\right )}{8 c} + \frac {x^{6} \cdot \left (\frac {76 a c^{2} d e^{3}}{9} + 4 c^{3} d^{3} e\right )}{7 c} + \frac {x^{5} \cdot \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} - \frac {7 a \left (\frac {21 a c^{2} e^{4}}{10} + 6 c^{3} d^{2} e^{2}\right )}{8 c} + c^{3} d^{4}\right )}{6 c} + \frac {x^{4} \cdot \left (12 a^{2} c d e^{3} + 12 a c^{2} d^{3} e - \frac {6 a \left (\frac {76 a c^{2} d e^{3}}{9} + 4 c^{3} d^{3} e\right )}{7 c}\right )}{5 c} + \frac {x^{3} \left (a^{3} e^{4} + 18 a^{2} c d^{2} e^{2} + 3 a c^{2} d^{4} - \frac {5 a \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} - \frac {7 a \left (\frac {21 a c^{2} e^{4}}{10} + 6 c^{3} d^{2} e^{2}\right )}{8 c} + c^{3} d^{4}\right )}{6 c}\right )}{4 c} + \frac {x^{2} \cdot \left (4 a^{3} d e^{3} + 12 a^{2} c d^{3} e - \frac {4 a \left (12 a^{2} c d e^{3} + 12 a c^{2} d^{3} e - \frac {6 a \left (\frac {76 a c^{2} d e^{3}}{9} + 4 c^{3} d^{3} e\right )}{7 c}\right )}{5 c}\right )}{3 c} + \frac {x \left (6 a^{3} d^{2} e^{2} + 3 a^{2} c d^{4} - \frac {3 a \left (a^{3} e^{4} + 18 a^{2} c d^{2} e^{2} + 3 a c^{2} d^{4} - \frac {5 a \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} - \frac {7 a \left (\frac {21 a c^{2} e^{4}}{10} + 6 c^{3} d^{2} e^{2}\right )}{8 c} + c^{3} d^{4}\right )}{6 c}\right )}{4 c}\right )}{2 c} + \frac {4 a^{3} d^{3} e - \frac {2 a \left (4 a^{3} d e^{3} + 12 a^{2} c d^{3} e - \frac {4 a \left (12 a^{2} c d e^{3} + 12 a c^{2} d^{3} e - \frac {6 a \left (\frac {76 a c^{2} d e^{3}}{9} + 4 c^{3} d^{3} e\right )}{7 c}\right )}{5 c}\right )}{3 c}}{c}\right ) + \left (a^{3} d^{4} - \frac {a \left (6 a^{3} d^{2} e^{2} + 3 a^{2} c d^{4} - \frac {3 a \left (a^{3} e^{4} + 18 a^{2} c d^{2} e^{2} + 3 a c^{2} d^{4} - \frac {5 a \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} - \frac {7 a \left (\frac {21 a c^{2} e^{4}}{10} + 6 c^{3} d^{2} e^{2}\right )}{8 c} + c^{3} d^{4}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\a^{\frac {5}{2}} \left (\begin {cases} d^{4} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{5}}{5 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((sqrt(a + c*x**2)*(4*c**2*d*e**3*x**8/9 + c**2*e**4*x**9/10 + x**7*(21*a*c**2*e**4/10 + 6*c**3*d**2*
e**2)/(8*c) + x**6*(76*a*c**2*d*e**3/9 + 4*c**3*d**3*e)/(7*c) + x**5*(3*a**2*c*e**4 + 18*a*c**2*d**2*e**2 - 7*
a*(21*a*c**2*e**4/10 + 6*c**3*d**2*e**2)/(8*c) + c**3*d**4)/(6*c) + x**4*(12*a**2*c*d*e**3 + 12*a*c**2*d**3*e
- 6*a*(76*a*c**2*d*e**3/9 + 4*c**3*d**3*e)/(7*c))/(5*c) + x**3*(a**3*e**4 + 18*a**2*c*d**2*e**2 + 3*a*c**2*d**
4 - 5*a*(3*a**2*c*e**4 + 18*a*c**2*d**2*e**2 - 7*a*(21*a*c**2*e**4/10 + 6*c**3*d**2*e**2)/(8*c) + c**3*d**4)/(
6*c))/(4*c) + x**2*(4*a**3*d*e**3 + 12*a**2*c*d**3*e - 4*a*(12*a**2*c*d*e**3 + 12*a*c**2*d**3*e - 6*a*(76*a*c*
*2*d*e**3/9 + 4*c**3*d**3*e)/(7*c))/(5*c))/(3*c) + x*(6*a**3*d**2*e**2 + 3*a**2*c*d**4 - 3*a*(a**3*e**4 + 18*a
**2*c*d**2*e**2 + 3*a*c**2*d**4 - 5*a*(3*a**2*c*e**4 + 18*a*c**2*d**2*e**2 - 7*a*(21*a*c**2*e**4/10 + 6*c**3*d
**2*e**2)/(8*c) + c**3*d**4)/(6*c))/(4*c))/(2*c) + (4*a**3*d**3*e - 2*a*(4*a**3*d*e**3 + 12*a**2*c*d**3*e - 4*
a*(12*a**2*c*d*e**3 + 12*a*c**2*d**3*e - 6*a*(76*a*c**2*d*e**3/9 + 4*c**3*d**3*e)/(7*c))/(5*c))/(3*c))/c) + (a
**3*d**4 - a*(6*a**3*d**2*e**2 + 3*a**2*c*d**4 - 3*a*(a**3*e**4 + 18*a**2*c*d**2*e**2 + 3*a*c**2*d**4 - 5*a*(3
*a**2*c*e**4 + 18*a*c**2*d**2*e**2 - 7*a*(21*a*c**2*e**4/10 + 6*c**3*d**2*e**2)/(8*c) + c**3*d**4)/(6*c))/(4*c
))/(2*c))*Piecewise((log(2*sqrt(c)*sqrt(a + c*x**2) + 2*c*x)/sqrt(c), Ne(a, 0)), (x*log(x)/sqrt(c*x**2), True)
), Ne(c, 0)), (a**(5/2)*Piecewise((d**4*x, Eq(e, 0)), ((d + e*x)**5/(5*e), True)), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.19 \[ \int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx=\frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} e^{4} x^{3}}{10 \, c} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d e^{3} x^{2}}{9 \, c} + \frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{4} x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d^{4} x + \frac {5}{16} \, \sqrt {c x^{2} + a} a^{2} d^{4} x + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d^{2} e^{2} x}{4 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a d^{2} e^{2} x}{8 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{2} e^{2} x}{32 \, c} - \frac {15 \, \sqrt {c x^{2} + a} a^{3} d^{2} e^{2} x}{64 \, c} - \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a e^{4} x}{80 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a^{2} e^{4} x}{160 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3} e^{4} x}{128 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} a^{4} e^{4} x}{256 \, c^{2}} + \frac {5 \, a^{3} d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} - \frac {15 \, a^{4} d^{2} e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{64 \, c^{\frac {3}{2}}} + \frac {3 \, a^{5} e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{256 \, c^{\frac {5}{2}}} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d^{3} e}{7 \, c} - \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a d e^{3}}{63 \, c^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.21 \[ \int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx=\frac {1}{80640} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (9 \, c^{2} e^{4} x + 40 \, c^{2} d e^{3}\right )} x + \frac {27 \, {\left (20 \, c^{10} d^{2} e^{2} + 7 \, a c^{9} e^{4}\right )}}{c^{8}}\right )} x + \frac {320 \, {\left (9 \, c^{10} d^{3} e + 19 \, a c^{9} d e^{3}\right )}}{c^{8}}\right )} x + \frac {21 \, {\left (80 \, c^{10} d^{4} + 1020 \, a c^{9} d^{2} e^{2} + 93 \, a^{2} c^{8} e^{4}\right )}}{c^{8}}\right )} x + \frac {1920 \, {\left (9 \, a c^{9} d^{3} e + 5 \, a^{2} c^{8} d e^{3}\right )}}{c^{8}}\right )} x + \frac {105 \, {\left (208 \, a c^{9} d^{4} + 708 \, a^{2} c^{8} d^{2} e^{2} + 3 \, a^{3} c^{7} e^{4}\right )}}{c^{8}}\right )} x + \frac {2560 \, {\left (27 \, a^{2} c^{8} d^{3} e + a^{3} c^{7} d e^{3}\right )}}{c^{8}}\right )} x + \frac {315 \, {\left (176 \, a^{2} c^{8} d^{4} + 60 \, a^{3} c^{7} d^{2} e^{2} - 3 \, a^{4} c^{6} e^{4}\right )}}{c^{8}}\right )} x + \frac {5120 \, {\left (9 \, a^{3} c^{7} d^{3} e - 2 \, a^{4} c^{6} d e^{3}\right )}}{c^{8}}\right )} - \frac {{\left (80 \, a^{3} c^{2} d^{4} - 60 \, a^{4} c d^{2} e^{2} + 3 \, a^{5} e^{4}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{256 \, c^{\frac {5}{2}}} \]

[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*e^4*x + 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)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^4 \left (a+c x^2\right )^{5/2} \, dx=\int {\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^4 \,d x \]

[In]

int((a + c*x^2)^(5/2)*(d + e*x)^4,x)

[Out]

int((a + c*x^2)^(5/2)*(d + e*x)^4, x)