### 3.556 $$\int \frac{(a+c x^2)^{5/2}}{(d+e x)^8} \, dx$$

Optimal. Leaf size=246 $-\frac{5 a^2 c^3 d \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac{5 a^3 c^4 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac{5 a c^2 d \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac{c d \left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )}$

[Out]

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

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Rubi [A]  time = 0.124018, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.21, Rules used = {731, 721, 725, 206} $-\frac{5 a^2 c^3 d \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac{5 a^3 c^4 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac{5 a c^2 d \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac{c d \left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

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

Rule 731

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*d)/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x]
/; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0]

Rule 721

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 725

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 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 \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx &=-\frac{e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac{(c d) \int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx}{c d^2+a e^2}\\ &=-\frac{c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac{\left (5 a c^2 d\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{5 a c^2 d (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{c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac{\left (5 a^2 c^3 d\right ) \int \frac{\sqrt{a+c x^2}}{(d+e x)^3} \, dx}{8 \left (c d^2+a e^2\right )^3}\\ &=-\frac{5 a^2 c^3 d (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{5 a c^2 d (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{c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}+\frac{\left (5 a^3 c^4 d\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{16 \left (c d^2+a e^2\right )^4}\\ &=-\frac{5 a^2 c^3 d (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{5 a c^2 d (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{c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}-\frac{\left (5 a^3 c^4 d\right ) \operatorname{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{5 a^2 c^3 d (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{5 a c^2 d (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{c d (a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right )^2 (d+e x)^6}-\frac{e \left (a+c x^2\right )^{7/2}}{7 \left (c d^2+a e^2\right ) (d+e x)^7}-\frac{5 a^3 c^4 d \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]  time = 0.562648, size = 403, normalized size = 1.64 $-\frac{\sqrt{a+c x^2} \left (2 c^2 (d+e x)^4 \left (72 a^2 e^4+159 a c d^2 e^2+80 c^2 d^4\right ) \left (a e^2+c d^2\right )^2-c^3 d (d+e x)^5 \left (57 a^2 e^4+30 a c d^2 e^2+8 c^2 d^4\right ) \left (a e^2+c d^2\right )-c^3 (d+e x)^6 \left (87 a^2 c d^2 e^4-48 a^3 e^6+38 a c^2 d^4 e^2+8 c^3 d^6\right )-2 c^2 d (d+e x)^3 \left (197 a e^2+200 c d^2\right ) \left (a e^2+c d^2\right )^3-232 c d (d+e x) \left (a e^2+c d^2\right )^5+8 c (d+e x)^2 \left (18 a e^2+55 c d^2\right ) \left (a e^2+c d^2\right )^4+48 \left (a e^2+c d^2\right )^6\right )}{336 e^5 (d+e x)^7 \left (a e^2+c d^2\right )^4}-\frac{5 a^3 c^4 d \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{16 \left (a e^2+c d^2\right )^{9/2}}+\frac{5 a^3 c^4 d \log (d+e x)}{16 \left (a e^2+c d^2\right )^{9/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

-(Sqrt[a + c*x^2]*(48*(c*d^2 + a*e^2)^6 - 232*c*d*(c*d^2 + a*e^2)^5*(d + e*x) + 8*c*(c*d^2 + a*e^2)^4*(55*c*d^
2 + 18*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2)^3*(200*c*d^2 + 197*a*e^2)*(d + e*x)^3 + 2*c^2*(c*d^2 + a*e
^2)^2*(80*c^2*d^4 + 159*a*c*d^2*e^2 + 72*a^2*e^4)*(d + e*x)^4 - c^3*d*(c*d^2 + a*e^2)*(8*c^2*d^4 + 30*a*c*d^2*
e^2 + 57*a^2*e^4)*(d + e*x)^5 - c^3*(8*c^3*d^6 + 38*a*c^2*d^4*e^2 + 87*a^2*c*d^2*e^4 - 48*a^3*e^6)*(d + e*x)^6
))/(336*e^5*(c*d^2 + a*e^2)^4*(d + e*x)^7) + (5*a^3*c^4*d*Log[d + e*x])/(16*(c*d^2 + a*e^2)^(9/2)) - (5*a^3*c^
4*d*Log[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))

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Maple [B]  time = 0.24, size = 7718, normalized size = 31.4 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^8,x)

[Out]

result too large to display

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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((c*x^2+a)^(5/2)/(e*x+d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**8,x)

[Out]

Timed out

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Giac [B]  time = 2.2945, size = 3178, normalized size = 12.92 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

-5/8*a^3*c^4*d*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)) + 1/168*(1792*(sqrt(c)*x - sqrt
(c*x^2 + a))^8*c^(19/2)*d^12*e + 512*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^10*d^13 + 2688*(sqrt(c)*x - sqrt(c*x^2
+ a))^9*c^9*d^11*e^2 + 2240*(sqrt(c)*x - sqrt(c*x^2 + a))^10*c^(17/2)*d^10*e^3 - 1792*(sqrt(c)*x - sqrt(c*x^2
+ a))^6*a*c^(19/2)*d^12*e + 1120*(sqrt(c)*x - sqrt(c*x^2 + a))^11*c^8*d^9*e^4 - 2944*(sqrt(c)*x - sqrt(c*x^2 +
a))^7*a*c^9*d^11*e^2 + 336*(sqrt(c)*x - sqrt(c*x^2 + a))^12*c^(15/2)*d^8*e^5 + 1792*(sqrt(c)*x - sqrt(c*x^2 +
a))^8*a*c^(17/2)*d^10*e^3 + 8288*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^8*d^9*e^4 + 2688*(sqrt(c)*x - sqrt(c*x^2
+ a))^5*a^2*c^9*d^11*e^2 + 8960*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a*c^(15/2)*d^8*e^5 - 1792*(sqrt(c)*x - sqrt(
c*x^2 + a))^6*a^2*c^(17/2)*d^10*e^3 + 4480*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^7*d^7*e^6 - 13248*(sqrt(c)*x -
sqrt(c*x^2 + a))^7*a^2*c^8*d^9*e^4 + 1344*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a*c^(13/2)*d^6*e^7 - 9072*(sqrt(c)
*x - sqrt(c*x^2 + a))^8*a^2*c^(15/2)*d^8*e^5 - 2240*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(17/2)*d^10*e^3 + 62
72*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^7*d^7*e^6 + 8288*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^8*d^9*e^4 + 13
440*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(13/2)*d^6*e^7 + 9072*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(15/2)*
d^8*e^5 + 6720*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^6*d^5*e^8 - 30736*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^
7*d^7*e^6 + 1120*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^8*d^9*e^4 + 2016*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^2*c
^(11/2)*d^4*e^9 - 55832*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(13/2)*d^6*e^7 - 8960*(sqrt(c)*x - sqrt(c*x^2 +
a))^4*a^4*c^(15/2)*d^8*e^5 - 42588*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^6*d^5*e^8 + 11312*(sqrt(c)*x - sqrt(c
*x^2 + a))^5*a^4*c^7*d^7*e^6 - 13370*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(11/2)*d^4*e^9 + 80192*(sqrt(c)*x
- sqrt(c*x^2 + a))^6*a^4*c^(13/2)*d^6*e^7 - 336*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(15/2)*d^8*e^5 - 3010*(s
qrt(c)*x - sqrt(c*x^2 + a))^11*a^3*c^5*d^3*e^10 + 100016*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^6*d^5*e^8 + 548
8*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^7*d^7*e^6 - 21*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^3*c^(9/2)*d^2*e^11 +
70210*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(11/2)*d^4*e^9 - 19488*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^5*c^(13/
2)*d^6*e^7 - 105*(sqrt(c)*x - sqrt(c*x^2 + a))^13*a^3*c^4*d*e^12 + 27370*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c
^5*d^3*e^10 - 79128*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^6*d^5*e^8 + 112*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^
7*d^7*e^6 + 9940*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^4*c^(9/2)*d^2*e^11 - 82180*(sqrt(c)*x - sqrt(c*x^2 + a))^6
*a^5*c^(11/2)*d^4*e^9 - 1792*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(13/2)*d^6*e^7 + 1820*(sqrt(c)*x - sqrt(c*x
^2 + a))^11*a^4*c^4*d*e^12 - 52500*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^5*d^3*e^10 + 14448*(sqrt(c)*x - sqrt(
c*x^2 + a))^3*a^6*c^6*d^5*e^8 + 336*(sqrt(c)*x - sqrt(c*x^2 + a))^12*a^4*c^(7/2)*e^13 - 16485*(sqrt(c)*x - sqr
t(c*x^2 + a))^8*a^5*c^(9/2)*d^2*e^11 + 49252*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(11/2)*d^4*e^9 - 8*a^7*c^(1
3/2)*d^6*e^7 - 4445*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^5*c^4*d*e^12 + 44660*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^6
*c^5*d^3*e^10 + 532*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^6*d^5*e^8 + 26880*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^6*
c^(9/2)*d^2*e^11 - 5026*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*c^(11/2)*d^4*e^9 + 6720*(sqrt(c)*x - sqrt(c*x^2 +
a))^7*a^6*c^4*d*e^12 - 17738*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^5*d^3*e^10 + 1680*(sqrt(c)*x - sqrt(c*x^2 +
a))^8*a^6*c^(7/2)*e^13 - 12047*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^7*c^(9/2)*d^2*e^11 - 38*a^8*c^(11/2)*d^4*e^9
- 5635*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^7*c^4*d*e^12 + 1218*(sqrt(c)*x - sqrt(c*x^2 + a))*a^8*c^5*d^3*e^10 +
4620*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^8*c^(9/2)*d^2*e^11 + 2212*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^8*c^4*d*e^
12 + 1008*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^8*c^(7/2)*e^13 - 87*a^9*c^(9/2)*d^2*e^11 - 567*(sqrt(c)*x - sqrt(c
*x^2 + a))*a^9*c^4*d*e^12 + 48*a^10*c^(7/2)*e^13)/((c^4*d^8*e^6 + 4*a*c^3*d^6*e^8 + 6*a^2*c^2*d^4*e^10 + 4*a^3
*c*d^2*e^12 + a^4*e^14)*((sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^
7)