3.554 \(\int \frac{(a+c x^2)^{5/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=314 \[ -\frac{c^2 \sqrt{a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{8 e^5 (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{c^3 d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 e^6 \left (a e^2+c d^2\right )^{5/2}}+\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^6}-\frac{c \left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^3 (d+e x)^4 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5} \]

[Out]

-(c^2*(d*(8*c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + e*(12*c^2*d^4 + 23*a*c*d^2*e^2 + 8*a^2*e^4)*x)*Sqrt[a + c*x^
2])/(8*e^5*(c*d^2 + a*e^2)^2*(d + e*x)^2) - (c*(d*(4*c*d^2 + a*e^2) + e*(7*c*d^2 + 4*a*e^2)*x)*(a + c*x^2)^(3/
2))/(12*e^3*(c*d^2 + a*e^2)*(d + e*x)^4) - (a + c*x^2)^(5/2)/(5*e*(d + e*x)^5) + (c^(5/2)*ArcTanh[(Sqrt[c]*x)/
Sqrt[a + c*x^2]])/e^6 + (c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a
*e^2]*Sqrt[a + c*x^2])])/(8*e^6*(c*d^2 + a*e^2)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.343272, antiderivative size = 314, normalized size of antiderivative = 1., 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 = {733, 811, 844, 217, 206, 725} \[ -\frac{c^2 \sqrt{a+c x^2} \left (e x \left (8 a^2 e^4+23 a c d^2 e^2+12 c^2 d^4\right )+d \left (a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right )\right )}{8 e^5 (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{c^3 d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 e^6 \left (a e^2+c d^2\right )^{5/2}}+\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^6}-\frac{c \left (a+c x^2\right )^{3/2} \left (e x \left (4 a e^2+7 c d^2\right )+d \left (a e^2+4 c d^2\right )\right )}{12 e^3 (d+e x)^4 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^6,x]

[Out]

-(c^2*(d*(8*c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + e*(12*c^2*d^4 + 23*a*c*d^2*e^2 + 8*a^2*e^4)*x)*Sqrt[a + c*x^
2])/(8*e^5*(c*d^2 + a*e^2)^2*(d + e*x)^2) - (c*(d*(4*c*d^2 + a*e^2) + e*(7*c*d^2 + 4*a*e^2)*x)*(a + c*x^2)^(3/
2))/(12*e^3*(c*d^2 + a*e^2)*(d + e*x)^4) - (a + c*x^2)^(5/2)/(5*e*(d + e*x)^5) + (c^(5/2)*ArcTanh[(Sqrt[c]*x)/
Sqrt[a + c*x^2]])/e^6 + (c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a
*e^2]*Sqrt[a + c*x^2])])/(8*e^6*(c*d^2 + a*e^2)^(5/2))

Rule 733

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

Rule 811

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

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^6} \, dx &=-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}+\frac{c \int \frac{x \left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{e}\\ &=-\frac{c \left (d \left (4 c d^2+a e^2\right )+e \left (7 c d^2+4 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}-\frac{c \int \frac{\left (6 a c d e-8 c \left (c d^2+a e^2\right ) x\right ) \sqrt{a+c x^2}}{(d+e x)^3} \, dx}{8 e^3 \left (c d^2+a e^2\right )}\\ &=-\frac{c^2 \left (d \left (8 c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+e \left (12 c^2 d^4+23 a c d^2 e^2+8 a^2 e^4\right ) x\right ) \sqrt{a+c x^2}}{8 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c \left (d \left (4 c d^2+a e^2\right )+e \left (7 c d^2+4 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}+\frac{c \int \frac{-4 a c^2 d e \left (4 c d^2+7 a e^2\right )+32 c^2 \left (c d^2+a e^2\right )^2 x}{(d+e x) \sqrt{a+c x^2}} \, dx}{32 e^5 \left (c d^2+a e^2\right )^2}\\ &=-\frac{c^2 \left (d \left (8 c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+e \left (12 c^2 d^4+23 a c d^2 e^2+8 a^2 e^4\right ) x\right ) \sqrt{a+c x^2}}{8 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c \left (d \left (4 c d^2+a e^2\right )+e \left (7 c d^2+4 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}+\frac{c^3 \int \frac{1}{\sqrt{a+c x^2}} \, dx}{e^6}-\frac{\left (c^3 d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{8 e^6 \left (c d^2+a e^2\right )^2}\\ &=-\frac{c^2 \left (d \left (8 c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+e \left (12 c^2 d^4+23 a c d^2 e^2+8 a^2 e^4\right ) x\right ) \sqrt{a+c x^2}}{8 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c \left (d \left (4 c d^2+a e^2\right )+e \left (7 c d^2+4 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{e^6}+\frac{\left (c^3 d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right )\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 )}{8 e^6 \left (c d^2+a e^2\right )^2}\\ &=-\frac{c^2 \left (d \left (8 c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+e \left (12 c^2 d^4+23 a c d^2 e^2+8 a^2 e^4\right ) x\right ) \sqrt{a+c x^2}}{8 e^5 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac{c \left (d \left (4 c d^2+a e^2\right )+e \left (7 c d^2+4 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 e^3 \left (c d^2+a e^2\right ) (d+e x)^4}-\frac{\left (a+c x^2\right )^{5/2}}{5 e (d+e x)^5}+\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^6}+\frac{c^3 d \left (8 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{8 e^6 \left (c d^2+a e^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.559167, size = 359, normalized size = 1.14 \[ \frac{-\frac{e \sqrt{a+c x^2} \left (c^2 (d+e x)^4 \left (184 a^2 e^4+503 a c d^2 e^2+274 c^2 d^4\right )-c^2 d (d+e x)^3 \left (311 a e^2+326 c d^2\right ) \left (a e^2+c d^2\right )-126 c d (d+e x) \left (a e^2+c d^2\right )^3+2 c (d+e x)^2 \left (44 a e^2+137 c d^2\right ) \left (a e^2+c d^2\right )^2+24 \left (a e^2+c d^2\right )^4\right )}{(d+e x)^5 \left (a e^2+c d^2\right )^2}+\frac{15 c^3 d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{5/2}}-\frac{15 c^3 d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{5/2}}+120 c^{5/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{120 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^6,x]

[Out]

(-((e*Sqrt[a + c*x^2]*(24*(c*d^2 + a*e^2)^4 - 126*c*d*(c*d^2 + a*e^2)^3*(d + e*x) + 2*c*(c*d^2 + a*e^2)^2*(137
*c*d^2 + 44*a*e^2)*(d + e*x)^2 - c^2*d*(c*d^2 + a*e^2)*(326*c*d^2 + 311*a*e^2)*(d + e*x)^3 + c^2*(274*c^2*d^4
+ 503*a*c*d^2*e^2 + 184*a^2*e^4)*(d + e*x)^4))/((c*d^2 + a*e^2)^2*(d + e*x)^5)) - (15*c^3*d*(8*c^2*d^4 + 20*a*
c*d^2*e^2 + 15*a^2*e^4)*Log[d + e*x])/(c*d^2 + a*e^2)^(5/2) + 120*c^(5/2)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] +
(15*c^3*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(
c*d^2 + a*e^2)^(5/2))/(120*e^6)

________________________________________________________________________________________

Maple [B]  time = 0.207, size = 5921, normalized size = 18.9 \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)^6,x)

[Out]

result too large to display

________________________________________________________________________________________

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)^6,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)^6,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**6,x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**6, x)

________________________________________________________________________________________

Giac [B]  time = 2.18228, size = 1875, normalized size = 5.97 \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)^6,x, algorithm="giac")

[Out]

-c^(5/2)*e^(-6)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) - 1/4*(8*c^5*d^5 + 20*a*c^4*d^3*e^2 + 15*a^2*c^3*d*e^4)
*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^2*d^4*e^6 + 2*a*c*d^2*e^8 + a
^2*e^10)*sqrt(-c*d^2 - a*e^2)) - 1/60*(10000*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(13/2)*d^8*e + 4384*(sqrt(c)*x
- sqrt(c*x^2 + a))^5*c^7*d^9 + 8800*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*d^7*e^2 + 3600*(sqrt(c)*x - sqrt(c*x^2
+ a))^8*c^(11/2)*d^6*e^3 - 11920*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a*c^(13/2)*d^8*e + 600*(sqrt(c)*x - sqrt(c*x
^2 + a))^9*c^5*d^5*e^4 - 13872*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^6*d^7*e^2 + 3800*(sqrt(c)*x - sqrt(c*x^2 +
a))^6*a*c^(11/2)*d^6*e^3 + 12200*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^5*d^5*e^4 + 13120*(sqrt(c)*x - sqrt(c*x^2
+ a))^3*a^2*c^6*d^7*e^2 + 6300*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*d^4*e^5 - 3560*(sqrt(c)*x - sqrt(c*x
^2 + a))^4*a^2*c^(11/2)*d^6*e^3 + 1140*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^4*d^3*e^6 - 29076*(sqrt(c)*x - sqrt
(c*x^2 + a))^5*a^2*c^5*d^5*e^4 - 18950*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(9/2)*d^4*e^5 - 7360*(sqrt(c)*x -
sqrt(c*x^2 + a))^2*a^3*c^(11/2)*d^6*e^3 - 1250*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*d^3*e^6 + 17080*(sqrt(
c)*x - sqrt(c*x^2 + a))^3*a^3*c^5*d^5*e^4 + 1935*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*d^2*e^7 + 23950*(
sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(9/2)*d^4*e^5 + 495*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*d*e^8 + 370*(
sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^4*d^3*e^6 + 2140*(sqrt(c)*x - sqrt(c*x^2 + a))*a^4*c^5*d^5*e^4 - 8250*(sq
rt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(7/2)*d^2*e^7 - 12450*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*c^(9/2)*d^4*e^5 -
3030*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^3*c^3*d*e^8 - 4150*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^4*d^3*e^6 - 3
60*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*e^9 + 8800*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(7/2)*d^2*e^7
- 274*a^5*c^(9/2)*d^4*e^5 + 5520*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^3*d*e^8 + 3890*(sqrt(c)*x - sqrt(c*x^2
+ a))*a^5*c^4*d^3*e^6 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^4*c^(5/2)*e^9 - 2910*(sqrt(c)*x - sqrt(c*x^2 + a
))^2*a^5*c^(7/2)*d^2*e^7 - 4330*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^5*c^3*d*e^8 - 1120*(sqrt(c)*x - sqrt(c*x^2 +
a))^4*a^5*c^(5/2)*e^9 - 503*a^6*c^(7/2)*d^2*e^7 + 1345*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^3*d*e^8 + 560*(sqr
t(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(5/2)*e^9 - 184*a^7*c^(5/2)*e^9)/((c^2*d^4*e^6 + 2*a*c*d^2*e^8 + a^2*e^10)*(
(sqrt(c)*x - sqrt(c*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^5)