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

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

[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

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Rubi [A]
time = 0.20, antiderivative size = 307, normalized size of antiderivative = 1.00, 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 = {757, 847, 794, 201, 223, 212} \begin {gather*} \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} \end {gather*}

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 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*} \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 ) \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*}

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Mathematica [A]
time = 0.71, size = 283, normalized size = 0.92 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[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))

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Maple [A]
time = 0.48, size = 344, normalized size = 1.12

method	result	size
default	\(e^{4} \left (\frac {x^{3} \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{10 c}-\frac {3 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{8 c}\right )}{10 c}\right )+4 d \,e^{3} \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{9 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{63 c^{2}}\right )+6 d^{2} e^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{8 c}\right )+\frac {4 d^{3} e \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 c}+d^{4} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )\)	\(344\)
risch	\(-\frac {\left (-8064 e^{4} c^{4} x^{9}-35840 d \,e^{3} c^{4} x^{8}-21168 e^{4} c^{3} a \,x^{7}-60480 d^{2} e^{2} c^{4} x^{7}-97280 d \,e^{3} c^{3} a \,x^{6}-46080 d^{3} e \,c^{4} x^{6}-15624 e^{4} a^{2} c^{2} x^{5}-171360 d^{2} e^{2} c^{3} a \,x^{5}-13440 c^{4} d^{4} x^{5}-76800 d \,e^{3} a^{2} c^{2} x^{4}-138240 d^{3} e \,c^{3} a \,x^{4}-630 e^{4} c \,a^{3} x^{3}-148680 d^{2} e^{2} a^{2} c^{2} x^{3}-43680 d^{4} c^{3} a \,x^{3}-5120 d \,e^{3} c \,a^{3} x^{2}-138240 d^{3} e \,a^{2} c^{2} x^{2}+945 e^{4} a^{4} x -18900 d^{2} e^{2} c \,a^{3} x -55440 d^{4} a^{2} c^{2} x +10240 a^{4} d \,e^{3}-46080 d^{3} e c \,a^{3}\right ) \sqrt {c \,x^{2}+a}}{80640 c^{2}}+\frac {3 a^{5} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) e^{4}}{256 c^{\frac {5}{2}}}-\frac {15 a^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) d^{2} e^{2}}{64 c^{\frac {3}{2}}}+\frac {5 a^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right ) d^{4}}{16 \sqrt {c}}\)	\(365\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^4*(1/10*x^3*(c*x^2+a)^(7/2)/c-3/10*a/c*(1/8*x*(c*x^2+a)^(7/2)/c-1/8*a/c*(1/6*x*(c*x^2+a)^(5/2)+5/6*a*(1/4*x*
(c*x^2+a)^(3/2)+3/4*a*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2)))))))+4*d*e^3*(1/9*x^2
*(c*x^2+a)^(7/2)/c-2/63*a/c^2*(c*x^2+a)^(7/2))+6*d^2*e^2*(1/8*x*(c*x^2+a)^(7/2)/c-1/8*a/c*(1/6*x*(c*x^2+a)^(5/
2)+5/6*a*(1/4*x*(c*x^2+a)^(3/2)+3/4*a*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))))))+4
/7*d^3*e*(c*x^2+a)^(7/2)/c+d^4*(1/6*x*(c*x^2+a)^(5/2)+5/6*a*(1/4*x*(c*x^2+a)^(3/2)+3/4*a*(1/2*x*(c*x^2+a)^(1/2
)+1/2*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2)))))

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Maxima [A]
time = 0.32, size = 352, normalized size = 1.15 \begin {gather*} \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 {5 \, a^{3} d^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} x^{3} e^{4}}{10 \, c} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d x^{2} e^{3}}{9 \, c} + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d^{2} x e^{2}}{4 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a d^{2} x e^{2}}{8 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{2} x e^{2}}{32 \, c} - \frac {15 \, \sqrt {c x^{2} + a} a^{3} d^{2} x e^{2}}{64 \, c} - \frac {15 \, a^{4} d^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{2}}{64 \, c^{\frac {3}{2}}} + \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d^{3} e}{7 \, c} - \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a x e^{4}}{80 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a^{2} x e^{4}}{160 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3} x e^{4}}{128 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} a^{4} x e^{4}}{256 \, c^{2}} + \frac {3 \, a^{5} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) e^{4}}{256 \, c^{\frac {5}{2}}} - \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a d e^{3}}{63 \, c^{2}} \end {gather*}

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]

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

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Fricas [A]
time = 2.44, size = 642, normalized size = 2.09 \begin {gather*} \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 (13440 \, c^{5} d^{4} x^{5} + 43680 \, a c^{4} d^{4} x^{3} + 55440 \, a^{2} c^{3} d^{4} x + 63 \, {\left (128 \, c^{5} x^{9} + 336 \, a c^{4} x^{7} + 248 \, a^{2} c^{3} x^{5} + 10 \, a^{3} c^{2} x^{3} - 15 \, a^{4} c x\right )} e^{4} + 5120 \, {\left (7 \, c^{5} d x^{8} + 19 \, a c^{4} d x^{6} + 15 \, a^{2} c^{3} d x^{4} + a^{3} c^{2} d x^{2} - 2 \, a^{4} c d\right )} e^{3} + 1260 \, {\left (48 \, c^{5} d^{2} x^{7} + 136 \, a c^{4} d^{2} x^{5} + 118 \, a^{2} c^{3} d^{2} x^{3} + 15 \, a^{3} c^{2} d^{2} x\right )} e^{2} + 46080 \, {\left (c^{5} d^{3} x^{6} + 3 \, a c^{4} d^{3} x^{4} + 3 \, a^{2} c^{3} d^{3} x^{2} + a^{3} c^{2} d^{3}\right )} e\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 (13440 \, c^{5} d^{4} x^{5} + 43680 \, a c^{4} d^{4} x^{3} + 55440 \, a^{2} c^{3} d^{4} x + 63 \, {\left (128 \, c^{5} x^{9} + 336 \, a c^{4} x^{7} + 248 \, a^{2} c^{3} x^{5} + 10 \, a^{3} c^{2} x^{3} - 15 \, a^{4} c x\right )} e^{4} + 5120 \, {\left (7 \, c^{5} d x^{8} + 19 \, a c^{4} d x^{6} + 15 \, a^{2} c^{3} d x^{4} + a^{3} c^{2} d x^{2} - 2 \, a^{4} c d\right )} e^{3} + 1260 \, {\left (48 \, c^{5} d^{2} x^{7} + 136 \, a c^{4} d^{2} x^{5} + 118 \, a^{2} c^{3} d^{2} x^{3} + 15 \, a^{3} c^{2} d^{2} x\right )} e^{2} + 46080 \, {\left (c^{5} d^{3} x^{6} + 3 \, a c^{4} d^{3} x^{4} + 3 \, a^{2} c^{3} d^{3} x^{2} + a^{3} c^{2} d^{3}\right )} e\right )} \sqrt {c x^{2} + a}}{80640 \, c^{3}}\right ] \end {gather*}

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*(13440*c^5*d^4*x^5 + 43680*a*c^4*d^4*x^3 + 55440*a^2*c^3*d^4*x + 63*(128*c^5*x^9 + 336*a*c^4*x^7
+ 248*a^2*c^3*x^5 + 10*a^3*c^2*x^3 - 15*a^4*c*x)*e^4 + 5120*(7*c^5*d*x^8 + 19*a*c^4*d*x^6 + 15*a^2*c^3*d*x^4 +
a^3*c^2*d*x^2 - 2*a^4*c*d)*e^3 + 1260*(48*c^5*d^2*x^7 + 136*a*c^4*d^2*x^5 + 118*a^2*c^3*d^2*x^3 + 15*a^3*c^2*
d^2*x)*e^2 + 46080*(c^5*d^3*x^6 + 3*a*c^4*d^3*x^4 + 3*a^2*c^3*d^3*x^2 + a^3*c^2*d^3)*e)*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)) - (1
3440*c^5*d^4*x^5 + 43680*a*c^4*d^4*x^3 + 55440*a^2*c^3*d^4*x + 63*(128*c^5*x^9 + 336*a*c^4*x^7 + 248*a^2*c^3*x
^5 + 10*a^3*c^2*x^3 - 15*a^4*c*x)*e^4 + 5120*(7*c^5*d*x^8 + 19*a*c^4*d*x^6 + 15*a^2*c^3*d*x^4 + a^3*c^2*d*x^2
- 2*a^4*c*d)*e^3 + 1260*(48*c^5*d^2*x^7 + 136*a*c^4*d^2*x^5 + 118*a^2*c^3*d^2*x^3 + 15*a^3*c^2*d^2*x)*e^2 + 46
080*(c^5*d^3*x^6 + 3*a*c^4*d^3*x^4 + 3*a^2*c^3*d^3*x^2 + a^3*c^2*d^3)*e)*sqrt(c*x^2 + a))/c^3]

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Sympy [A]
time = 168.75, size = 1062, normalized size = 3.46 \begin {gather*} - \frac {3 a^{\frac {9}{2}} e^{4} x}{256 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {15 a^{\frac {7}{2}} d^{2} e^{2} x}{64 c \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {a^{\frac {7}{2}} e^{4} x^{3}}{256 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} d^{4} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} d^{4} x}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} d^{2} e^{2} x^{3}}{64 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {129 a^{\frac {5}{2}} e^{4} x^{5}}{640 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} c d^{4} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} c d^{2} e^{2} x^{5}}{32 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {73 a^{\frac {3}{2}} c e^{4} x^{7}}{160 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 \sqrt {a} c^{2} d^{4} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {23 \sqrt {a} c^{2} d^{2} e^{2} x^{7}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {29 \sqrt {a} c^{2} e^{4} x^{9}}{80 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 a^{5} e^{4} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{256 c^{\frac {5}{2}}} - \frac {15 a^{4} d^{2} e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{64 c^{\frac {3}{2}}} + \frac {5 a^{3} d^{4} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 \sqrt {c}} + 4 a^{2} d^{3} e \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + 4 a^{2} d e^{3} \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 8 a c d^{3} e \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 8 a c d e^{3} \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 4 c^{2} d^{3} e \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 4 c^{2} d e^{3} \left (\begin {cases} - \frac {16 a^{4} \sqrt {a + c x^{2}}}{315 c^{4}} + \frac {8 a^{3} x^{2} \sqrt {a + c x^{2}}}{315 c^{3}} - \frac {2 a^{2} x^{4} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{6} \sqrt {a + c x^{2}}}{63 c} + \frac {x^{8} \sqrt {a + c x^{2}}}{9} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + \frac {c^{3} d^{4} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 c^{3} d^{2} e^{2} x^{9}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{3} e^{4} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}

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 = 1.85, size = 360, normalized size = 1.17 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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