\(\int \frac {(a+c x^2)^{3/2}}{(d+e x)^7} \, dx\) [544]

Optimal result

Integrand size = 19, antiderivative size = 269 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {a c^2 \left (6 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac {a^2 c^3 \left (6 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \]

[Out]

-1/24*c*(-a*e^2+6*c*d^2)*(-c*d*x+a*e)*(c*x^2+a)^(3/2)/(a*e^2+c*d^2)^3/(e*x+d)^4-1/6*e*(c*x^2+a)^(5/2)/(a*e^2+c
*d^2)/(e*x+d)^6-7/30*c*d*e*(c*x^2+a)^(5/2)/(a*e^2+c*d^2)^2/(e*x+d)^5-1/16*a^2*c^3*(-a*e^2+6*c*d^2)*arctanh((-c
*d*x+a*e)/(a*e^2+c*d^2)^(1/2)/(c*x^2+a)^(1/2))/(a*e^2+c*d^2)^(9/2)-1/16*a*c^2*(-a*e^2+6*c*d^2)*(-c*d*x+a*e)*(c
*x^2+a)^(1/2)/(a*e^2+c*d^2)^4/(e*x+d)^2

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {759, 821, 735, 739, 212} \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {a^2 c^3 \left (6 c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac {a c^2 \sqrt {a+c x^2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 (d+e x)^5 \left (a e^2+c d^2\right )^2}-\frac {c \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac {e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

[In]

Int[(a + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

-1/16*(a*c^2*(6*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)^4*(d + e*x)^2) - (c*(6*c*d^2 -
a*e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(24*(c*d^2 + a*e^2)^3*(d + e*x)^4) - (e*(a + c*x^2)^(5/2))/(6*(c*d^2 +
a*e^2)*(d + e*x)^6) - (7*c*d*e*(a + c*x^2)^(5/2))/(30*(c*d^2 + a*e^2)^2*(d + e*x)^5) - (a^2*c^3*(6*c*d^2 - a*
e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(9/2))

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 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(-2*a*e + (2*c
*d)*x)*((a + c*x^2)^p/(2*(m + 1)*(c*d^2 + a*e^2))), x] - Dist[4*a*c*(p/(2*(m + 1)*(c*d^2 + a*e^2))), Int[(d +
e*x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2
, 0] && GtQ[p, 0]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 759

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)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*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] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {c \int \frac {(-6 d+e x) \left (a+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{6 \left (c d^2+a e^2\right )} \\ & = -\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}+\frac {\left (c \left (6 c d^2-a e^2\right )\right ) \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{6 \left (c d^2+a e^2\right )^2} \\ & = -\frac {c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}+\frac {\left (a c^2 \left (6 c d^2-a e^2\right )\right ) \int \frac {\sqrt {a+c x^2}}{(d+e x)^3} \, dx}{8 \left (c d^2+a e^2\right )^3} \\ & = -\frac {a c^2 \left (6 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}+\frac {\left (a^2 c^3 \left (6 c d^2-a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{16 \left (c d^2+a e^2\right )^4} \\ & = -\frac {a c^2 \left (6 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac {\left (a^2 c^3 \left (6 c d^2-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^4} \\ & = -\frac {a c^2 \left (6 c d^2-a e^2\right ) (a e-c d x) \sqrt {a+c x^2}}{16 \left (c d^2+a e^2\right )^4 (d+e x)^2}-\frac {c \left (6 c d^2-a e^2\right ) (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^3 (d+e x)^4}-\frac {e \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac {7 c d e \left (a+c x^2\right )^{5/2}}{30 \left (c d^2+a e^2\right )^2 (d+e x)^5}-\frac {a^2 c^3 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.65 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {1}{240} \left (-\frac {\sqrt {a+c x^2} \left (40 \left (c d^2+a e^2\right )^5-104 c d \left (c d^2+a e^2\right )^4 (d+e x)+2 c \left (c d^2+a e^2\right )^3 \left (38 c d^2+35 a e^2\right ) (d+e x)^2-2 c^2 d \left (c d^2+a e^2\right )^2 \left (2 c d^2+9 a e^2\right ) (d+e x)^3-c^2 \left (c d^2+a e^2\right ) \left (4 c^2 d^4+24 a c d^2 e^2-15 a^2 e^4\right ) (d+e x)^4-c^3 d \left (4 c^2 d^4+28 a c d^2 e^2-81 a^2 e^4\right ) (d+e x)^5\right )}{e^3 \left (c d^2+a e^2\right )^4 (d+e x)^6}+\frac {15 a^2 c^3 \left (6 c d^2-a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{9/2}}+\frac {15 a^2 c^3 \left (-6 c d^2+a e^2\right ) \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{9/2}}\right ) \]

[In]

Integrate[(a + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

(-((Sqrt[a + c*x^2]*(40*(c*d^2 + a*e^2)^5 - 104*c*d*(c*d^2 + a*e^2)^4*(d + e*x) + 2*c*(c*d^2 + a*e^2)^3*(38*c*
d^2 + 35*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)^2*(2*c*d^2 + 9*a*e^2)*(d + e*x)^3 - c^2*(c*d^2 + a*e^2)*
(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^4)*(d + e*x)^4 - c^3*d*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*a^2*e^4)*(d + e
*x)^5))/(e^3*(c*d^2 + a*e^2)^4*(d + e*x)^6)) + (15*a^2*c^3*(6*c*d^2 - a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(9/
2) + (15*a^2*c^3*(-6*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(9
/2))/240

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(8230\) vs. \(2(245)=490\).

Time = 2.20 (sec) , antiderivative size = 8231, normalized size of antiderivative = 30.60

method	result	size
default	\(\text {Expression too large to display}\)	\(8231\)

[In]

int((c*x^2+a)^(3/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1229 vs. \(2 (246) = 492\).

Time = 7.90 (sec) , antiderivative size = 2485, normalized size of antiderivative = 9.24 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

[-1/480*(15*(6*a^2*c^4*d^8 - a^3*c^3*d^6*e^2 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*e^8)*x^6 + 6*(6*a^2*c^4*d^3*e^5 -
a^3*c^3*d*e^7)*x^5 + 15*(6*a^2*c^4*d^4*e^4 - a^3*c^3*d^2*e^6)*x^4 + 20*(6*a^2*c^4*d^5*e^3 - a^3*c^3*d^3*e^5)*x
^3 + 15*(6*a^2*c^4*d^6*e^2 - a^3*c^3*d^4*e^4)*x^2 + 6*(6*a^2*c^4*d^7*e - a^3*c^3*d^5*e^3)*x)*sqrt(c*d^2 + a*e^
2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sq
rt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(246*a^2*c^4*d^8*e + 513*a^3*c^3*d^6*e^3 + 433*a^4*c^2*d^4*e^5 +
206*a^5*c*d^2*e^7 + 40*a^6*e^9 - (4*c^6*d^7*e^2 + 32*a*c^5*d^5*e^4 - 53*a^2*c^4*d^3*e^6 - 81*a^3*c^3*d*e^8)*x
^5 - 3*(8*c^6*d^8*e + 64*a*c^5*d^6*e^3 - 76*a^2*c^4*d^4*e^5 - 137*a^3*c^3*d^2*e^7 - 5*a^4*c^2*e^9)*x^4 - 2*(30
*c^6*d^9 + 239*a*c^5*d^7*e^2 - 158*a^2*c^4*d^5*e^4 - 388*a^3*c^3*d^3*e^6 - 21*a^4*c^2*d*e^8)*x^3 - 2*(114*a*c^
5*d^8*e - 423*a^2*c^4*d^6*e^3 - 698*a^3*c^3*d^4*e^5 - 196*a^4*c^2*d^2*e^7 - 35*a^5*c*e^9)*x^2 - 3*(50*a*c^5*d^
9 - 117*a^2*c^4*d^7*e^2 - 221*a^3*c^3*d^5*e^4 - 66*a^4*c^2*d^3*e^6 - 12*a^5*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^5*
d^16 + 5*a*c^4*d^14*e^2 + 10*a^2*c^3*d^12*e^4 + 10*a^3*c^2*d^10*e^6 + 5*a^4*c*d^8*e^8 + a^5*d^6*e^10 + (c^5*d^
10*e^6 + 5*a*c^4*d^8*e^8 + 10*a^2*c^3*d^6*e^10 + 10*a^3*c^2*d^4*e^12 + 5*a^4*c*d^2*e^14 + a^5*e^16)*x^6 + 6*(c
^5*d^11*e^5 + 5*a*c^4*d^9*e^7 + 10*a^2*c^3*d^7*e^9 + 10*a^3*c^2*d^5*e^11 + 5*a^4*c*d^3*e^13 + a^5*d*e^15)*x^5
+ 15*(c^5*d^12*e^4 + 5*a*c^4*d^10*e^6 + 10*a^2*c^3*d^8*e^8 + 10*a^3*c^2*d^6*e^10 + 5*a^4*c*d^4*e^12 + a^5*d^2*
e^14)*x^4 + 20*(c^5*d^13*e^3 + 5*a*c^4*d^11*e^5 + 10*a^2*c^3*d^9*e^7 + 10*a^3*c^2*d^7*e^9 + 5*a^4*c*d^5*e^11 +
a^5*d^3*e^13)*x^3 + 15*(c^5*d^14*e^2 + 5*a*c^4*d^12*e^4 + 10*a^2*c^3*d^10*e^6 + 10*a^3*c^2*d^8*e^8 + 5*a^4*c*
d^6*e^10 + a^5*d^4*e^12)*x^2 + 6*(c^5*d^15*e + 5*a*c^4*d^13*e^3 + 10*a^2*c^3*d^11*e^5 + 10*a^3*c^2*d^9*e^7 + 5
*a^4*c*d^7*e^9 + a^5*d^5*e^11)*x), -1/240*(15*(6*a^2*c^4*d^8 - a^3*c^3*d^6*e^2 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*
e^8)*x^6 + 6*(6*a^2*c^4*d^3*e^5 - a^3*c^3*d*e^7)*x^5 + 15*(6*a^2*c^4*d^4*e^4 - a^3*c^3*d^2*e^6)*x^4 + 20*(6*a^
2*c^4*d^5*e^3 - a^3*c^3*d^3*e^5)*x^3 + 15*(6*a^2*c^4*d^6*e^2 - a^3*c^3*d^4*e^4)*x^2 + 6*(6*a^2*c^4*d^7*e - a^3
*c^3*d^5*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2
*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) + (246*a^2*c^4*d^8*e + 513*a^3*c^3*d^6*e^3 + 433*a^4*c^2*d^4*e^5 + 206*a^5*c*
d^2*e^7 + 40*a^6*e^9 - (4*c^6*d^7*e^2 + 32*a*c^5*d^5*e^4 - 53*a^2*c^4*d^3*e^6 - 81*a^3*c^3*d*e^8)*x^5 - 3*(8*c
^6*d^8*e + 64*a*c^5*d^6*e^3 - 76*a^2*c^4*d^4*e^5 - 137*a^3*c^3*d^2*e^7 - 5*a^4*c^2*e^9)*x^4 - 2*(30*c^6*d^9 +
239*a*c^5*d^7*e^2 - 158*a^2*c^4*d^5*e^4 - 388*a^3*c^3*d^3*e^6 - 21*a^4*c^2*d*e^8)*x^3 - 2*(114*a*c^5*d^8*e - 4
23*a^2*c^4*d^6*e^3 - 698*a^3*c^3*d^4*e^5 - 196*a^4*c^2*d^2*e^7 - 35*a^5*c*e^9)*x^2 - 3*(50*a*c^5*d^9 - 117*a^2
*c^4*d^7*e^2 - 221*a^3*c^3*d^5*e^4 - 66*a^4*c^2*d^3*e^6 - 12*a^5*c*d*e^8)*x)*sqrt(c*x^2 + a))/(c^5*d^16 + 5*a*
c^4*d^14*e^2 + 10*a^2*c^3*d^12*e^4 + 10*a^3*c^2*d^10*e^6 + 5*a^4*c*d^8*e^8 + a^5*d^6*e^10 + (c^5*d^10*e^6 + 5*
a*c^4*d^8*e^8 + 10*a^2*c^3*d^6*e^10 + 10*a^3*c^2*d^4*e^12 + 5*a^4*c*d^2*e^14 + a^5*e^16)*x^6 + 6*(c^5*d^11*e^5
+ 5*a*c^4*d^9*e^7 + 10*a^2*c^3*d^7*e^9 + 10*a^3*c^2*d^5*e^11 + 5*a^4*c*d^3*e^13 + a^5*d*e^15)*x^5 + 15*(c^5*d
^12*e^4 + 5*a*c^4*d^10*e^6 + 10*a^2*c^3*d^8*e^8 + 10*a^3*c^2*d^6*e^10 + 5*a^4*c*d^4*e^12 + a^5*d^2*e^14)*x^4 +
20*(c^5*d^13*e^3 + 5*a*c^4*d^11*e^5 + 10*a^2*c^3*d^9*e^7 + 10*a^3*c^2*d^7*e^9 + 5*a^4*c*d^5*e^11 + a^5*d^3*e^
13)*x^3 + 15*(c^5*d^14*e^2 + 5*a*c^4*d^12*e^4 + 10*a^2*c^3*d^10*e^6 + 10*a^3*c^2*d^8*e^8 + 5*a^4*c*d^6*e^10 +
a^5*d^4*e^12)*x^2 + 6*(c^5*d^15*e + 5*a*c^4*d^13*e^3 + 10*a^2*c^3*d^11*e^5 + 10*a^3*c^2*d^9*e^7 + 5*a^4*c*d^7*
e^9 + a^5*d^5*e^11)*x)]

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{7}}\, dx \]

[In]

integrate((c*x**2+a)**(3/2)/(e*x+d)**7,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**7, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (246) = 492\).

Time = 0.36 (sec) , antiderivative size = 1875, normalized size of antiderivative = 6.97 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+a)^(3/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/8*(6*a^2*c^4*d^2 - a^3*c^3*e^2)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/
((c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)*sqrt(-c*d^2 - a*e^2)) - 1/120*(90
*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^4*d^2*e^9 - 15*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^3*e^11 + 990*(sq
rt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2)*d^3*e^8 - 165*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(7/2)*d*e^10 -
320*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*d^8*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*d^6*e^5 + 2520*(s
qrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^5*d^4*e^7 - 2530*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^4*d^2*e^9 - 235*(sq
rt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^3*e^11 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(15/2)*d^9*e^2 - 1920*(sqrt(
c)*x - sqrt(c*x^2 + a))^8*a*c^(13/2)*d^7*e^4 + 7380*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*d^5*e^6 - 822
0*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(9/2)*d^3*e^8 + 285*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(7/2)*d*e^10
- 384*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*d^10*e - 1728*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^7*d^8*e^3 + 9456*
(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*d^6*e^5 - 20760*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^5*d^4*e^7 + 2700
*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^4*d^2*e^9 - 390*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^3*e^11 - 128*(sqr
t(c)*x - sqrt(c*x^2 + a))^6*c^(17/2)*d^11 + 64*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(15/2)*d^9*e^2 + 8592*(sqrt
(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*d^7*e^4 - 24440*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*d^5*e^6 +
14860*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(9/2)*d^3*e^8 - 810*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(7/2)*d
*e^10 + 384*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^8*d^10*e + 1728*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^7*d^8*e^
3 - 19056*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^6*d^6*e^5 + 21480*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^5*d^4*
e^7 - 7020*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*d^2*e^9 - 390*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6*c^3*e^11
- 480*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(15/2)*d^9*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(13/2)
*d^7*e^4 + 18720*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(11/2)*d^5*e^6 - 11640*(sqrt(c)*x - sqrt(c*x^2 + a))^4*
a^5*c^(9/2)*d^3*e^8 + 930*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(7/2)*d*e^10 + 320*(sqrt(c)*x - sqrt(c*x^2 + a
))^3*a^3*c^7*d^8*e^3 + 2960*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^6*d^6*e^5 - 12120*(sqrt(c)*x - sqrt(c*x^2 +
a))^3*a^5*c^5*d^4*e^7 + 2330*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^4*d^2*e^9 - 235*(sqrt(c)*x - sqrt(c*x^2 + a
))^3*a^7*c^3*e^11 - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(13/2)*d^7*e^4 - 1656*(sqrt(c)*x - sqrt(c*x^2 +
a))^2*a^5*c^(11/2)*d^5*e^6 + 4038*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(9/2)*d^3*e^8 - 321*(sqrt(c)*x - sqrt(
c*x^2 + a))^2*a^7*c^(7/2)*d*e^10 + 48*(sqrt(c)*x - sqrt(c*x^2 + a))*a^5*c^6*d^6*e^5 + 336*(sqrt(c)*x - sqrt(c*
x^2 + a))*a^6*c^5*d^4*e^7 - 882*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^4*d^2*e^9 - 15*(sqrt(c)*x - sqrt(c*x^2 + a
))*a^8*c^3*e^11 - 4*a^6*c^(11/2)*d^5*e^6 - 28*a^7*c^(9/2)*d^3*e^8 + 81*a^8*c^(7/2)*d*e^10)/((c^4*d^8*e^4 + 4*a
*c^3*d^6*e^6 + 6*a^2*c^2*d^4*e^8 + 4*a^3*c*d^2*e^10 + a^4*e^12)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c
)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^6)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^7} \,d x \]

[In]

int((a + c*x^2)^(3/2)/(d + e*x)^7,x)

[Out]

int((a + c*x^2)^(3/2)/(d + e*x)^7, x)