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

Optimal. Leaf size=216 \[ \frac{5 a^3 d \left (8 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 d x \sqrt{a+c x^2} \left (8 c d^2-3 a e^2\right )}{128 c}+\frac{e \left (a+c x^2\right )^{7/2} \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right )}{504 c^2}+\frac{d x \left (a+c x^2\right )^{5/2} \left (8 c d^2-3 a e^2\right )}{48 c}+\frac{5 a d x \left (a+c x^2\right )^{3/2} \left (8 c d^2-3 a e^2\right )}{192 c}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]

[Out]

(5*a^2*d*(8*c*d^2 - 3*a*e^2)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*d*(8*c*d^2 - 3*a*e^2)*x*(a + c*x^2)^(3/2))/(192
*c) + (d*(8*c*d^2 - 3*a*e^2)*x*(a + c*x^2)^(5/2))/(48*c) + (e*(d + e*x)^2*(a + c*x^2)^(7/2))/(9*c) + (e*(16*(1
0*c*d^2 - a*e^2) + 77*c*d*e*x)*(a + c*x^2)^(7/2))/(504*c^2) + (5*a^3*d*(8*c*d^2 - 3*a*e^2)*ArcTanh[(Sqrt[c]*x)
/Sqrt[a + c*x^2]])/(128*c^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.163836, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {743, 780, 195, 217, 206} \[ \frac{5 a^3 d \left (8 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 d x \sqrt{a+c x^2} \left (8 c d^2-3 a e^2\right )}{128 c}+\frac{e \left (a+c x^2\right )^{7/2} \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right )}{504 c^2}+\frac{d x \left (a+c x^2\right )^{5/2} \left (8 c d^2-3 a e^2\right )}{48 c}+\frac{5 a d x \left (a+c x^2\right )^{3/2} \left (8 c d^2-3 a e^2\right )}{192 c}+\frac{e \left (a+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + c*x^2)^(5/2),x]

[Out]

(5*a^2*d*(8*c*d^2 - 3*a*e^2)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*d*(8*c*d^2 - 3*a*e^2)*x*(a + c*x^2)^(3/2))/(192
*c) + (d*(8*c*d^2 - 3*a*e^2)*x*(a + c*x^2)^(5/2))/(48*c) + (e*(d + e*x)^2*(a + c*x^2)^(7/2))/(9*c) + (e*(16*(1
0*c*d^2 - a*e^2) + 77*c*d*e*x)*(a + c*x^2)^(7/2))/(504*c^2) + (5*a^3*d*(8*c*d^2 - 3*a*e^2)*ArcTanh[(Sqrt[c]*x)
/Sqrt[a + c*x^2]])/(128*c^(3/2))

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 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)^3 \left (a+c x^2\right )^{5/2} \, dx &=\frac{e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac{\int (d+e x) \left (9 c d^2-2 a e^2+11 c d e x\right ) \left (a+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac{e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac{e \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right ) \left (a+c x^2\right )^{7/2}}{504 c^2}+\frac{\left (d \left (8 c d^2-3 a e^2\right )\right ) \int \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac{d \left (8 c d^2-3 a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac{e \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right ) \left (a+c x^2\right )^{7/2}}{504 c^2}+\frac{\left (5 a d \left (8 c d^2-3 a e^2\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c}\\ &=\frac{5 a d \left (8 c d^2-3 a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{d \left (8 c d^2-3 a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac{e \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right ) \left (a+c x^2\right )^{7/2}}{504 c^2}+\frac{\left (5 a^2 d \left (8 c d^2-3 a e^2\right )\right ) \int \sqrt{a+c x^2} \, dx}{64 c}\\ &=\frac{5 a^2 d \left (8 c d^2-3 a e^2\right ) x \sqrt{a+c x^2}}{128 c}+\frac{5 a d \left (8 c d^2-3 a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{d \left (8 c d^2-3 a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac{e \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right ) \left (a+c x^2\right )^{7/2}}{504 c^2}+\frac{\left (5 a^3 d \left (8 c d^2-3 a e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{128 c}\\ &=\frac{5 a^2 d \left (8 c d^2-3 a e^2\right ) x \sqrt{a+c x^2}}{128 c}+\frac{5 a d \left (8 c d^2-3 a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{d \left (8 c d^2-3 a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac{e \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right ) \left (a+c x^2\right )^{7/2}}{504 c^2}+\frac{\left (5 a^3 d \left (8 c d^2-3 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{128 c}\\ &=\frac{5 a^2 d \left (8 c d^2-3 a e^2\right ) x \sqrt{a+c x^2}}{128 c}+\frac{5 a d \left (8 c d^2-3 a e^2\right ) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{d \left (8 c d^2-3 a e^2\right ) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{7/2}}{9 c}+\frac{e \left (16 \left (10 c d^2-a e^2\right )+77 c d e x\right ) \left (a+c x^2\right )^{7/2}}{504 c^2}+\frac{5 a^3 d \left (8 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.16549, size = 216, normalized size = 1. \[ \frac{\sqrt{a+c x^2} \left (6 a^2 c^2 x \left (1728 d^2 e x+924 d^3+1239 d e^2 x^2+320 e^3 x^3\right )+a^3 c e \left (3456 d^2+945 d e x+128 e^2 x^2\right )-256 a^4 e^3+8 a c^3 x^3 \left (1296 d^2 e x+546 d^3+1071 d e^2 x^2+304 e^3 x^3\right )+16 c^4 x^5 \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )\right )-315 a^3 \sqrt{c} d \left (3 a e^2-8 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{8064 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^2]*(-256*a^4*e^3 + a^3*c*e*(3456*d^2 + 945*d*e*x + 128*e^2*x^2) + 16*c^4*x^5*(84*d^3 + 216*d^2*e
*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 8*a*c^3*x^3*(546*d^3 + 1296*d^2*e*x + 1071*d*e^2*x^2 + 304*e^3*x^3) + 6*a^2
*c^2*x*(924*d^3 + 1728*d^2*e*x + 1239*d*e^2*x^2 + 320*e^3*x^3)) - 315*a^3*Sqrt[c]*d*(-8*c*d^2 + 3*a*e^2)*Log[c
*x + Sqrt[c]*Sqrt[a + c*x^2]])/(8064*c^2)

________________________________________________________________________________________

Maple [A]  time = 0.053, size = 245, normalized size = 1.1 \begin{align*}{\frac{{e}^{3}{x}^{2}}{9\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,a{e}^{3}}{63\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,d{e}^{2}x}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{ad{e}^{2}x}{16\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,d{e}^{2}{a}^{2}x}{64\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,d{e}^{2}{a}^{3}x}{128\,c}\sqrt{c{x}^{2}+a}}-{\frac{15\,d{e}^{2}{a}^{4}}{128}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,{d}^{2}e}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{d}^{3}x}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{d}^{3}ax}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{3}{a}^{2}x}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,{a}^{3}{d}^{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)^3*(c*x^2+a)^(5/2),x)

[Out]

1/9*e^3*x^2*(c*x^2+a)^(7/2)/c-2/63*e^3*a/c^2*(c*x^2+a)^(7/2)+3/8*d*e^2*x*(c*x^2+a)^(7/2)/c-1/16*d*e^2*a/c*x*(c
*x^2+a)^(5/2)-5/64*d*e^2*a^2/c*x*(c*x^2+a)^(3/2)-15/128*d*e^2*a^3/c*x*(c*x^2+a)^(1/2)-15/128*d*e^2*a^4/c^(3/2)
*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+3/7*d^2*e*(c*x^2+a)^(7/2)/c+1/6*d^3*x*(c*x^2+a)^(5/2)+5/24*d^3*a*x*(c*x^2+a)^(3
/2)+5/16*d^3*a^2*x*(c*x^2+a)^(1/2)+5/16*d^3*a^3/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))

________________________________________________________________________________________

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+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.99163, size = 1185, normalized size = 5.49 \begin{align*} \left [\frac{315 \,{\left (8 \, a^{3} c d^{3} - 3 \, a^{4} d e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (896 \, c^{4} e^{3} x^{8} + 3024 \, c^{4} d e^{2} x^{7} + 3456 \, a^{3} c d^{2} e - 256 \, a^{4} e^{3} + 128 \,{\left (27 \, c^{4} d^{2} e + 19 \, a c^{3} e^{3}\right )} x^{6} + 168 \,{\left (8 \, c^{4} d^{3} + 51 \, a c^{3} d e^{2}\right )} x^{5} + 384 \,{\left (27 \, a c^{3} d^{2} e + 5 \, a^{2} c^{2} e^{3}\right )} x^{4} + 42 \,{\left (104 \, a c^{3} d^{3} + 177 \, a^{2} c^{2} d e^{2}\right )} x^{3} + 128 \,{\left (81 \, a^{2} c^{2} d^{2} e + a^{3} c e^{3}\right )} x^{2} + 63 \,{\left (88 \, a^{2} c^{2} d^{3} + 15 \, a^{3} c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{16128 \, c^{2}}, -\frac{315 \,{\left (8 \, a^{3} c d^{3} - 3 \, a^{4} d e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (896 \, c^{4} e^{3} x^{8} + 3024 \, c^{4} d e^{2} x^{7} + 3456 \, a^{3} c d^{2} e - 256 \, a^{4} e^{3} + 128 \,{\left (27 \, c^{4} d^{2} e + 19 \, a c^{3} e^{3}\right )} x^{6} + 168 \,{\left (8 \, c^{4} d^{3} + 51 \, a c^{3} d e^{2}\right )} x^{5} + 384 \,{\left (27 \, a c^{3} d^{2} e + 5 \, a^{2} c^{2} e^{3}\right )} x^{4} + 42 \,{\left (104 \, a c^{3} d^{3} + 177 \, a^{2} c^{2} d e^{2}\right )} x^{3} + 128 \,{\left (81 \, a^{2} c^{2} d^{2} e + a^{3} c e^{3}\right )} x^{2} + 63 \,{\left (88 \, a^{2} c^{2} d^{3} + 15 \, a^{3} c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{8064 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16128*(315*(8*a^3*c*d^3 - 3*a^4*d*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(896*c^4
*e^3*x^8 + 3024*c^4*d*e^2*x^7 + 3456*a^3*c*d^2*e - 256*a^4*e^3 + 128*(27*c^4*d^2*e + 19*a*c^3*e^3)*x^6 + 168*(
8*c^4*d^3 + 51*a*c^3*d*e^2)*x^5 + 384*(27*a*c^3*d^2*e + 5*a^2*c^2*e^3)*x^4 + 42*(104*a*c^3*d^3 + 177*a^2*c^2*d
*e^2)*x^3 + 128*(81*a^2*c^2*d^2*e + a^3*c*e^3)*x^2 + 63*(88*a^2*c^2*d^3 + 15*a^3*c*d*e^2)*x)*sqrt(c*x^2 + a))/
c^2, -1/8064*(315*(8*a^3*c*d^3 - 3*a^4*d*e^2)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (896*c^4*e^3*x^8 +
3024*c^4*d*e^2*x^7 + 3456*a^3*c*d^2*e - 256*a^4*e^3 + 128*(27*c^4*d^2*e + 19*a*c^3*e^3)*x^6 + 168*(8*c^4*d^3
+ 51*a*c^3*d*e^2)*x^5 + 384*(27*a*c^3*d^2*e + 5*a^2*c^2*e^3)*x^4 + 42*(104*a*c^3*d^3 + 177*a^2*c^2*d*e^2)*x^3
+ 128*(81*a^2*c^2*d^2*e + a^3*c*e^3)*x^2 + 63*(88*a^2*c^2*d^3 + 15*a^3*c*d*e^2)*x)*sqrt(c*x^2 + a))/c^2]

________________________________________________________________________________________

Sympy [A]  time = 42.5977, size = 843, normalized size = 3.9 \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+a)**(5/2),x)

[Out]

15*a**(7/2)*d*e**2*x/(128*c*sqrt(1 + c*x**2/a)) + a**(5/2)*d**3*x*sqrt(1 + c*x**2/a)/2 + 3*a**(5/2)*d**3*x/(16
*sqrt(1 + c*x**2/a)) + 133*a**(5/2)*d*e**2*x**3/(128*sqrt(1 + c*x**2/a)) + 35*a**(3/2)*c*d**3*x**3/(48*sqrt(1
+ c*x**2/a)) + 127*a**(3/2)*c*d*e**2*x**5/(64*sqrt(1 + c*x**2/a)) + 17*sqrt(a)*c**2*d**3*x**5/(24*sqrt(1 + c*x
**2/a)) + 23*sqrt(a)*c**2*d*e**2*x**7/(16*sqrt(1 + c*x**2/a)) - 15*a**4*d*e**2*asinh(sqrt(c)*x/sqrt(a))/(128*c
**(3/2)) + 5*a**3*d**3*asinh(sqrt(c)*x/sqrt(a))/(16*sqrt(c)) + 3*a**2*d**2*e*Piecewise((sqrt(a)*x**2/2, Eq(c,
0)), ((a + c*x**2)**(3/2)/(3*c), True)) + a**2*e**3*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqr
t(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + 6*a*c*d**2*e*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)) + 2*a*c*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)) + 3*c
**2*d**2*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*sq
rt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0)), (sqrt(a)*x**6/6, True)) + c**2*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)/(10
5*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**3*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + 3*c**3*d*e**2*x**9/(8*sqrt(a)*sqrt(1 + c*x**2/a))

________________________________________________________________________________________

Giac [A]  time = 1.3225, size = 378, normalized size = 1.75 \begin{align*} \frac{1}{8064} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (2 \,{\left (7 \,{\left (8 \, c^{2} x e^{3} + 27 \, c^{2} d e^{2}\right )} x + \frac{8 \,{\left (27 \, c^{9} d^{2} e + 19 \, a c^{8} e^{3}\right )}}{c^{7}}\right )} x + \frac{21 \,{\left (8 \, c^{9} d^{3} + 51 \, a c^{8} d e^{2}\right )}}{c^{7}}\right )} x + \frac{48 \,{\left (27 \, a c^{8} d^{2} e + 5 \, a^{2} c^{7} e^{3}\right )}}{c^{7}}\right )} x + \frac{21 \,{\left (104 \, a c^{8} d^{3} + 177 \, a^{2} c^{7} d e^{2}\right )}}{c^{7}}\right )} x + \frac{64 \,{\left (81 \, a^{2} c^{7} d^{2} e + a^{3} c^{6} e^{3}\right )}}{c^{7}}\right )} x + \frac{63 \,{\left (88 \, a^{2} c^{7} d^{3} + 15 \, a^{3} c^{6} d e^{2}\right )}}{c^{7}}\right )} x + \frac{128 \,{\left (27 \, a^{3} c^{6} d^{2} e - 2 \, a^{4} c^{5} e^{3}\right )}}{c^{7}}\right )} - \frac{5 \,{\left (8 \, a^{3} c d^{3} - 3 \, a^{4} d e^{2}\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8064*sqrt(c*x^2 + a)*((2*((4*((2*(7*(8*c^2*x*e^3 + 27*c^2*d*e^2)*x + 8*(27*c^9*d^2*e + 19*a*c^8*e^3)/c^7)*x
+ 21*(8*c^9*d^3 + 51*a*c^8*d*e^2)/c^7)*x + 48*(27*a*c^8*d^2*e + 5*a^2*c^7*e^3)/c^7)*x + 21*(104*a*c^8*d^3 + 17
7*a^2*c^7*d*e^2)/c^7)*x + 64*(81*a^2*c^7*d^2*e + a^3*c^6*e^3)/c^7)*x + 63*(88*a^2*c^7*d^3 + 15*a^3*c^6*d*e^2)/
c^7)*x + 128*(27*a^3*c^6*d^2*e - 2*a^4*c^5*e^3)/c^7) - 5/128*(8*a^3*c*d^3 - 3*a^4*d*e^2)*log(abs(-sqrt(c)*x +
sqrt(c*x^2 + a)))/c^(3/2)