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

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

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 216, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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])

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 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]

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 = 0.16, size = 216, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-256 a^4 e^3+a^3 c e \left (3456 d^2+945 d e x+128 e^2 x^2\right )+6 a^2 c^2 x \left (924 d^3+1728 d^2 e x+1239 d e^2 x^2+320 e^3 x^3\right )+8 a c^3 x^3 \left (546 d^3+1296 d^2 e x+1071 d e^2 x^2+304 e^3 x^3\right )+16 c^4 x^5 \left (84 d^3+216 d^2 e x+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} \end {gather*}

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)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.70, size = 262, normalized size = 1.21 \begin {gather*} \frac {5 \left (3 a^4 d e^2-8 a^3 c d^3\right ) \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{128 c^{3/2}}+\frac {\sqrt {a+c x^2} \left (-256 a^4 e^3+3456 a^3 c d^2 e+945 a^3 c d e^2 x+128 a^3 c e^3 x^2+5544 a^2 c^2 d^3 x+10368 a^2 c^2 d^2 e x^2+7434 a^2 c^2 d e^2 x^3+1920 a^2 c^2 e^3 x^4+4368 a c^3 d^3 x^3+10368 a c^3 d^2 e x^4+8568 a c^3 d e^2 x^5+2432 a c^3 e^3 x^6+1344 c^4 d^3 x^5+3456 c^4 d^2 e x^6+3024 c^4 d e^2 x^7+896 c^4 e^3 x^8\right )}{8064 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + c*x^2]*(3456*a^3*c*d^2*e - 256*a^4*e^3 + 5544*a^2*c^2*d^3*x + 945*a^3*c*d*e^2*x + 10368*a^2*c^2*d^2*
e*x^2 + 128*a^3*c*e^3*x^2 + 4368*a*c^3*d^3*x^3 + 7434*a^2*c^2*d*e^2*x^3 + 10368*a*c^3*d^2*e*x^4 + 1920*a^2*c^2
*e^3*x^4 + 1344*c^4*d^3*x^5 + 8568*a*c^3*d*e^2*x^5 + 3456*c^4*d^2*e*x^6 + 2432*a*c^3*e^3*x^6 + 3024*c^4*d*e^2*
x^7 + 896*c^4*e^3*x^8))/(8064*c^2) + (5*(-8*a^3*c*d^3 + 3*a^4*d*e^2)*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(128
*c^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 518, normalized size = 2.40 \begin {gather*} \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 {gather*}

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]

________________________________________________________________________________________

giac [A]  time = 0.25, size = 280, normalized size = 1.30 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 245, normalized size = 1.13 \begin {gather*} -\frac {15 a^{4} d \,e^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{128 c^{\frac {3}{2}}}+\frac {5 a^{3} d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 \sqrt {c}}-\frac {15 \sqrt {c \,x^{2}+a}\, a^{3} d \,e^{2} x}{128 c}+\frac {5 \sqrt {c \,x^{2}+a}\, a^{2} d^{3} x}{16}-\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} d \,e^{2} x}{64 c}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a \,d^{3} x}{24}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} a d \,e^{2} x}{16 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} e^{3} x^{2}}{9 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} d^{3} x}{6}+\frac {3 \left (c \,x^{2}+a \right )^{\frac {7}{2}} d \,e^{2} x}{8 c}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {7}{2}} a \,e^{3}}{63 c^{2}}+\frac {3 \left (c \,x^{2}+a \right )^{\frac {7}{2}} d^{2} e}{7 c} \end {gather*}

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(c^(1/2)*x+(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(c^(1/2)*x+(c*x^2+a)^(1/2))

________________________________________________________________________________________

maxima [A]  time = 1.45, size = 230, normalized size = 1.06 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} e^{3} x^{2}}{9 \, c} + \frac {1}{6} \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d^{3} x + \frac {5}{24} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a d^{3} x + \frac {5}{16} \, \sqrt {c x^{2} + a} a^{2} d^{3} x + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d e^{2} x}{8 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a d e^{2} x}{16 \, c} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} d e^{2} x}{64 \, c} - \frac {15 \, \sqrt {c x^{2} + a} a^{3} d e^{2} x}{128 \, c} + \frac {5 \, a^{3} d^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {c}} - \frac {15 \, a^{4} d e^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {3}{2}}} + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} d^{2} e}{7 \, c} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a e^{3}}{63 \, c^{2}} \end {gather*}

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]

1/9*(c*x^2 + a)^(7/2)*e^3*x^2/c + 1/6*(c*x^2 + a)^(5/2)*d^3*x + 5/24*(c*x^2 + a)^(3/2)*a*d^3*x + 5/16*sqrt(c*x
^2 + a)*a^2*d^3*x + 3/8*(c*x^2 + a)^(7/2)*d*e^2*x/c - 1/16*(c*x^2 + a)^(5/2)*a*d*e^2*x/c - 5/64*(c*x^2 + a)^(3
/2)*a^2*d*e^2*x/c - 15/128*sqrt(c*x^2 + a)*a^3*d*e^2*x/c + 5/16*a^3*d^3*arcsinh(c*x/sqrt(a*c))/sqrt(c) - 15/12
8*a^4*d*e^2*arcsinh(c*x/sqrt(a*c))/c^(3/2) + 3/7*(c*x^2 + a)^(7/2)*d^2*e/c - 2/63*(c*x^2 + a)^(7/2)*a*e^3/c^2

________________________________________________________________________________________

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 )}^3 \,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)^3,x)

[Out]

int((a + c*x^2)^(5/2)*(d + e*x)^3, x)

________________________________________________________________________________________

sympy [A]  time = 35.04, size = 843, normalized size = 3.90 \begin {gather*} \frac {15 a^{\frac {7}{2}} d e^{2} x}{128 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} d^{3} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} d^{3} x}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} d e^{2} x^{3}}{128 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} c d^{3} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} c d e^{2} x^{5}}{64 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 \sqrt {a} c^{2} d^{3} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {23 \sqrt {a} c^{2} d e^{2} x^{7}}{16 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {15 a^{4} d e^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{128 c^{\frac {3}{2}}} + \frac {5 a^{3} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 \sqrt {c}} + 3 a^{2} d^{2} 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 ) + a^{2} 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 ) + 6 a c d^{2} 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 ) + 2 a c 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 ) + 3 c^{2} d^{2} 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 ) + c^{2} 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^{3} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 c^{3} d e^{2} x^{9}}{8 \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)**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))

________________________________________________________________________________________