\(\int \frac {1}{(c+d x)^3 (a+b x^2)^{5/2}} \, dx\) [1303]

Optimal result

Integrand size = 19, antiderivative size = 299 \[ \int \frac {1}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {b \left (a d \left (3 b c^2-a d^2\right )+b c \left (b c^2-3 a d^2\right ) x\right )}{3 a \left (b c^2+a d^2\right )^3 \left (a+b x^2\right )^{3/2}}+\frac {2 b \left (3 a d^3 \left (5 b c^2-a d^2\right )+b c \left (\frac {b^2 c^4}{a}+7 b c^2 d^2-12 a d^4\right ) x\right )}{3 a \left (b c^2+a d^2\right )^4 \sqrt {a+b x^2}}-\frac {d^5 \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^3 (c+d x)^2}-\frac {11 b c d^5 \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^4 (c+d x)}-\frac {5 b d^4 \left (6 b c^2-a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{9/2}} \] Output:

1/3*b*(a*d*(-a*d^2+3*b*c^2)+b*c*(-3*a*d^2+b*c^2)*x)/a/(a*d^2+b*c^2)^3/(b*x 
^2+a)^(3/2)+2/3*b*(3*a*d^3*(-a*d^2+5*b*c^2)+b*c*(b^2*c^4/a+7*b*c^2*d^2-12* 
a*d^4)*x)/a/(a*d^2+b*c^2)^4/(b*x^2+a)^(1/2)-1/2*d^5*(b*x^2+a)^(1/2)/(a*d^2 
+b*c^2)^3/(d*x+c)^2-11/2*b*c*d^5*(b*x^2+a)^(1/2)/(a*d^2+b*c^2)^4/(d*x+c)-5 
/2*b*d^4*(-a*d^2+6*b*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+ 
a)^(1/2))/(a*d^2+b*c^2)^(9/2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {\sqrt {a+b x^2} \left (-\frac {3 d^5 \left (b c^2+a d^2\right )}{(c+d x)^2}-\frac {33 b c d^5}{c+d x}+\frac {4 b \left (-3 a^3 d^5+b^3 c^5 x+7 a b^2 c^3 d^2 x+3 a^2 b c d^3 (5 c-4 d x)\right )}{a^2 \left (a+b x^2\right )}+\frac {2 b \left (b c^2+a d^2\right ) \left (-a^2 d^3+b^2 c^3 x+3 a b c d (c-d x)\right )}{a \left (a+b x^2\right )^2}\right )}{\left (b c^2+a d^2\right )^4}+\frac {15 b d^4 \left (6 b c^2-a d^2\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{9/2}}+\frac {15 b d^4 \left (-6 b c^2+a d^2\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{9/2}}\right ) \] Input:

Integrate[1/((c + d*x)^3*(a + b*x^2)^(5/2)),x]
 

Output:

((Sqrt[a + b*x^2]*((-3*d^5*(b*c^2 + a*d^2))/(c + d*x)^2 - (33*b*c*d^5)/(c 
+ d*x) + (4*b*(-3*a^3*d^5 + b^3*c^5*x + 7*a*b^2*c^3*d^2*x + 3*a^2*b*c*d^3* 
(5*c - 4*d*x)))/(a^2*(a + b*x^2)) + (2*b*(b*c^2 + a*d^2)*(-(a^2*d^3) + b^2 
*c^3*x + 3*a*b*c*d*(c - d*x)))/(a*(a + b*x^2)^2)))/(b*c^2 + a*d^2)^4 + (15 
*b*d^4*(6*b*c^2 - a*d^2)*Log[c + d*x])/(b*c^2 + a*d^2)^(9/2) + (15*b*d^4*( 
-6*b*c^2 + a*d^2)*Log[a*d - b*c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/ 
(b*c^2 + a*d^2)^(9/2))/6
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {496, 25, 686, 27, 688, 25, 27, 679, 488, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} (c+d x)^3} \, dx\)

\(\Big \downarrow \) 496

\(\displaystyle \frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}-\frac {\int -\frac {2 b c^2+4 b d x c+5 a d^2}{(c+d x)^3 \left (b x^2+a\right )^{3/2}}dx}{3 a \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {2 b c^2+4 b d x c+5 a d^2}{(c+d x)^3 \left (b x^2+a\right )^{3/2}}dx}{3 a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 686

\(\displaystyle \frac {-\frac {\int \frac {b d \left (3 a d \left (2 b c^2-5 a d^2\right )-2 b c \left (2 b c^2+9 a d^2\right ) x\right )}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a b \left (a d^2+b c^2\right )}-\frac {a d \left (2 b c^2-5 a d^2\right )-b c x \left (9 a d^2+2 b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {d \int \frac {3 a d \left (2 b c^2-5 a d^2\right )-2 b c \left (2 b c^2+9 a d^2\right ) x}{(c+d x)^3 \sqrt {b x^2+a}}dx}{a \left (a d^2+b c^2\right )}-\frac {a d \left (2 b c^2-5 a d^2\right )-b c x \left (9 a d^2+2 b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {-\frac {d \left (-\frac {\int -\frac {b \left (2 a c d \left (2 b c^2-33 a d^2\right )-\left (4 b^2 c^4+24 a b d^2 c^2-15 a^2 d^4\right ) x\right )}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {a d \left (2 b c^2-5 a d^2\right )-b c x \left (9 a d^2+2 b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {d \left (\frac {\int \frac {b \left (2 a c d \left (2 b c^2-33 a d^2\right )-\left (4 b^2 c^4+24 a b d^2 c^2-15 a^2 d^4\right ) x\right )}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {a d \left (2 b c^2-5 a d^2\right )-b c x \left (9 a d^2+2 b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {d \left (\frac {b \int \frac {2 a c d \left (2 b c^2-33 a d^2\right )-\left (4 b^2 c^4+24 a b d^2 c^2-15 a^2 d^4\right ) x}{(c+d x)^2 \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {a d \left (2 b c^2-5 a d^2\right )-b c x \left (9 a d^2+2 b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {-\frac {d \left (\frac {b \left (-\frac {15 a^2 d^3 \left (6 b c^2-a d^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}-\frac {c \sqrt {a+b x^2} \left (-81 a^2 d^4+28 a b c^2 d^2+4 b^2 c^4\right )}{(c+d x) \left (a d^2+b c^2\right )}\right )}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {a d \left (2 b c^2-5 a d^2\right )-b c x \left (9 a d^2+2 b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {d \left (\frac {b \left (\frac {15 a^2 d^3 \left (6 b c^2-a d^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{a d^2+b c^2}-\frac {c \sqrt {a+b x^2} \left (-81 a^2 d^4+28 a b c^2 d^2+4 b^2 c^4\right )}{(c+d x) \left (a d^2+b c^2\right )}\right )}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {a d \left (2 b c^2-5 a d^2\right )-b c x \left (9 a d^2+2 b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {d \left (\frac {b \left (\frac {15 a^2 d^3 \left (6 b c^2-a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\left (a d^2+b c^2\right )^{3/2}}-\frac {c \sqrt {a+b x^2} \left (-81 a^2 d^4+28 a b c^2 d^2+4 b^2 c^4\right )}{(c+d x) \left (a d^2+b c^2\right )}\right )}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (-15 a^2 d^4+24 a b c^2 d^2+4 b^2 c^4\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {a d \left (2 b c^2-5 a d^2\right )-b c x \left (9 a d^2+2 b c^2\right )}{a \sqrt {a+b x^2} (c+d x)^2 \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}+\frac {a d+b c x}{3 a \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

Int[1/((c + d*x)^3*(a + b*x^2)^(5/2)),x]
 

Output:

(a*d + b*c*x)/(3*a*(b*c^2 + a*d^2)*(c + d*x)^2*(a + b*x^2)^(3/2)) + (-((a* 
d*(2*b*c^2 - 5*a*d^2) - b*c*(2*b*c^2 + 9*a*d^2)*x)/(a*(b*c^2 + a*d^2)*(c + 
 d*x)^2*Sqrt[a + b*x^2])) - (d*(-1/2*((4*b^2*c^4 + 24*a*b*c^2*d^2 - 15*a^2 
*d^4)*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)*(c + d*x)^2) + (b*(-((c*(4*b^2*c^4 
 + 28*a*b*c^2*d^2 - 81*a^2*d^4)*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)*(c + d*x 
))) + (15*a^2*d^3*(6*b*c^2 - a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a* 
d^2]*Sqrt[a + b*x^2])])/(b*c^2 + a*d^2)^(3/2)))/(2*(b*c^2 + a*d^2))))/(a*( 
b*c^2 + a*d^2)))/(3*a*(b*c^2 + a*d^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 488

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

rule 496

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 
 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a 
 + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 
*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad 
raticQ[a, 0, b, c, d, n, p, x]
 

rule 679

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] + Simp[(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] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 688

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)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1602\) vs. \(2(275)=550\).

Time = 0.39 (sec) , antiderivative size = 1603, normalized size of antiderivative = 5.36

method	result	size
default	\(\text {Expression too large to display}\)	\(1603\)

Input:

int(1/(d*x+c)^3/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/d^3*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^ 
2+b*c^2)/d^2)^(3/2)+7/2*b*c*d/(a*d^2+b*c^2)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)/ 
(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+5*b*c*d/(a*d^2+b*c^2 
)*(1/3/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^( 
3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2 
-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+16/3 
*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d)/(b*(x+c/d 
)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))+1/(a*d^2+b*c^2)*d^2*(1/(a*d^ 
2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b*c*d 
/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2) 
/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(a*d^2+b*c^2)*d^2 
/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d 
^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2) 
)/(x+c/d))))-4*b/(a*d^2+b*c^2)*d^2*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+ 
b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^ 
(3/2)+16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d) 
/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)))-5/2*b/(a*d^2+b*c^ 
2)*d^2*(1/3/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d 
^2)^(3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2 
)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1301 vs. \(2 (276) = 552\).

Time = 0.96 (sec) , antiderivative size = 2628, normalized size of antiderivative = 8.79 \[ \int \frac {1}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[-1/12*(15*(6*a^4*b^2*c^4*d^4 - a^5*b*c^2*d^6 + (6*a^2*b^4*c^2*d^6 - a^3*b 
^3*d^8)*x^6 + 2*(6*a^2*b^4*c^3*d^5 - a^3*b^3*c*d^7)*x^5 + (6*a^2*b^4*c^4*d 
^4 + 11*a^3*b^3*c^2*d^6 - 2*a^4*b^2*d^8)*x^4 + 4*(6*a^3*b^3*c^3*d^5 - a^4* 
b^2*c*d^7)*x^3 + (12*a^3*b^3*c^4*d^4 + 4*a^4*b^2*c^2*d^6 - a^5*b*d^8)*x^2 
+ 2*(6*a^4*b^2*c^3*d^5 - a^5*b*c*d^7)*x)*sqrt(b*c^2 + a*d^2)*log((2*a*b*c* 
d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 + 2*sqrt(b*c^2 + a*d 
^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - 2*(6*a^2*b 
^4*c^8*d + 70*a^3*b^3*c^6*d^3 + 14*a^4*b^2*c^4*d^5 - 53*a^5*b*c^2*d^7 - 3* 
a^6*d^9 + (4*b^6*c^7*d^2 + 32*a*b^5*c^5*d^4 - 53*a^2*b^4*c^3*d^6 - 81*a^3* 
b^3*c*d^8)*x^5 + (8*b^6*c^8*d + 64*a*b^5*c^6*d^3 - 16*a^2*b^4*c^4*d^5 - 87 
*a^3*b^3*c^2*d^7 - 15*a^4*b^2*d^9)*x^4 + 2*(2*b^6*c^9 + 19*a*b^5*c^7*d^2 + 
 65*a^2*b^4*c^5*d^4 - 24*a^3*b^3*c^3*d^6 - 72*a^4*b^2*c*d^8)*x^3 + 2*(6*a* 
b^5*c^8*d + 63*a^2*b^4*c^6*d^3 - 7*a^3*b^3*c^4*d^5 - 74*a^4*b^2*c^2*d^7 - 
10*a^5*b*d^9)*x^2 + (6*a*b^5*c^9 + 42*a^2*b^4*c^7*d^2 + 110*a^3*b^3*c^5*d^ 
4 + 13*a^4*b^2*c^3*d^6 - 61*a^5*b*c*d^8)*x)*sqrt(b*x^2 + a))/(a^4*b^5*c^12 
 + 5*a^5*b^4*c^10*d^2 + 10*a^6*b^3*c^8*d^4 + 10*a^7*b^2*c^6*d^6 + 5*a^8*b* 
c^4*d^8 + a^9*c^2*d^10 + (a^2*b^7*c^10*d^2 + 5*a^3*b^6*c^8*d^4 + 10*a^4*b^ 
5*c^6*d^6 + 10*a^5*b^4*c^4*d^8 + 5*a^6*b^3*c^2*d^10 + a^7*b^2*d^12)*x^6 + 
2*(a^2*b^7*c^11*d + 5*a^3*b^6*c^9*d^3 + 10*a^4*b^5*c^7*d^5 + 10*a^5*b^4*c^ 
5*d^7 + 5*a^6*b^3*c^3*d^9 + a^7*b^2*c*d^11)*x^5 + (a^2*b^7*c^12 + 7*a^3...
 

Sympy [F]

\[ \int \frac {1}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x\right )^{3}}\, dx \] Input:

integrate(1/(d*x+c)**3/(b*x**2+a)**(5/2),x)
 

Output:

Integral(1/((a + b*x**2)**(5/2)*(c + d*x)**3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1176 vs. \(2 (276) = 552\).

Time = 0.13 (sec) , antiderivative size = 1176, normalized size of antiderivative = 3.93 \[ \int \frac {1}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

35/6*b^3*c^3*x/((b*x^2 + a)^(3/2)*a*b^3*c^6 + 3*(b*x^2 + a)^(3/2)*a^2*b^2* 
c^4*d^2 + 3*(b*x^2 + a)^(3/2)*a^3*b*c^2*d^4 + (b*x^2 + a)^(3/2)*a^4*d^6) + 
 35/2*b^3*c^3*x/(4*sqrt(b*x^2 + a)*a^2*b^3*c^6 + sqrt(b*x^2 + a)*a*b^4*c^8 
/d^2 + 6*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^2 + 4*sqrt(b*x^2 + a)*a^4*b*c^2*d^4 
 + sqrt(b*x^2 + a)*a^5*d^6) + 35/3*b^3*c^3*x/(sqrt(b*x^2 + a)*a^2*b^3*c^6 
+ 3*sqrt(b*x^2 + a)*a^3*b^2*c^4*d^2 + 3*sqrt(b*x^2 + a)*a^4*b*c^2*d^4 + sq 
rt(b*x^2 + a)*a^5*d^6) + 35/2*b^2*c^2/(sqrt(b*x^2 + a)*b^4*c^8/d^3 + 4*sqr 
t(b*x^2 + a)*a*b^3*c^6/d + 6*sqrt(b*x^2 + a)*a^2*b^2*c^4*d + 4*sqrt(b*x^2 
+ a)*a^3*b*c^2*d^3 + sqrt(b*x^2 + a)*a^4*d^5) + 35/6*b^2*c^2/((b*x^2 + a)^ 
(3/2)*b^3*c^6/d + 3*(b*x^2 + a)^(3/2)*a*b^2*c^4*d + 3*(b*x^2 + a)^(3/2)*a^ 
2*b*c^2*d^3 + (b*x^2 + a)^(3/2)*a^3*d^5) - 11/2*b^2*c*x/((b*x^2 + a)^(3/2) 
*a*b^2*c^4 + 2*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^2 + (b*x^2 + a)^(3/2)*a^3*d^4 
) - 5/2*b^2*c*x/(3*sqrt(b*x^2 + a)*a^2*b^2*c^4 + sqrt(b*x^2 + a)*a*b^3*c^6 
/d^2 + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^2 + sqrt(b*x^2 + a)*a^4*d^4) - 11*b^2 
*c*x/(sqrt(b*x^2 + a)*a^2*b^2*c^4 + 2*sqrt(b*x^2 + a)*a^3*b*c^2*d^2 + sqrt 
(b*x^2 + a)*a^4*d^4) - 7/2*b*c/((b*x^2 + a)^(3/2)*b^2*c^4*x + 2*(b*x^2 + a 
)^(3/2)*a*b*c^2*d^2*x + (b*x^2 + a)^(3/2)*a^2*d^4*x + (b*x^2 + a)^(3/2)*b^ 
2*c^5/d + 2*(b*x^2 + a)^(3/2)*a*b*c^3*d + (b*x^2 + a)^(3/2)*a^2*c*d^3) - 5 
/2*b/(sqrt(b*x^2 + a)*b^3*c^6/d^3 + 3*sqrt(b*x^2 + a)*a*b^2*c^4/d + 3*sqrt 
(b*x^2 + a)*a^2*b*c^2*d + sqrt(b*x^2 + a)*a^3*d^3) - 5/6*b/((b*x^2 + a)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2179 vs. \(2 (276) = 552\).

Time = 0.32 (sec) , antiderivative size = 2179, normalized size of antiderivative = 7.29 \[ \int \frac {1}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(d*x+c)^3/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

5*(6*b^2*c^2*d^4 - a*b*d^6)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqr 
t(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4* 
d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8)*sqrt(-b*c^2 - a*d^2)) + 1/3*((2*((b^18*c^ 
29 + 19*a*b^17*c^27*d^2 + 138*a^2*b^16*c^25*d^4 + 538*a^3*b^15*c^23*d^6 + 
1243*a^4*b^14*c^21*d^8 + 1617*a^5*b^13*c^19*d^10 + 528*a^6*b^12*c^17*d^12 
- 2244*a^7*b^11*c^15*d^14 - 5049*a^8*b^10*c^13*d^16 - 5819*a^9*b^9*c^11*d^ 
18 - 4334*a^10*b^8*c^9*d^20 - 2166*a^11*b^7*c^7*d^22 - 707*a^12*b^6*c^5*d^ 
24 - 137*a^13*b^5*c^3*d^26 - 12*a^14*b^4*c*d^28)*x/(a^2*b^17*c^32 + 16*a^3 
*b^16*c^30*d^2 + 120*a^4*b^15*c^28*d^4 + 560*a^5*b^14*c^26*d^6 + 1820*a^6* 
b^13*c^24*d^8 + 4368*a^7*b^12*c^22*d^10 + 8008*a^8*b^11*c^20*d^12 + 11440* 
a^9*b^10*c^18*d^14 + 12870*a^10*b^9*c^16*d^16 + 11440*a^11*b^8*c^14*d^18 + 
 8008*a^12*b^7*c^12*d^20 + 4368*a^13*b^6*c^10*d^22 + 1820*a^14*b^5*c^8*d^2 
4 + 560*a^15*b^4*c^6*d^26 + 120*a^16*b^3*c^4*d^28 + 16*a^17*b^2*c^2*d^30 + 
 a^18*b*d^32) + 3*(5*a^2*b^16*c^26*d^3 + 59*a^3*b^15*c^24*d^5 + 318*a^4*b^ 
14*c^22*d^7 + 1034*a^5*b^13*c^20*d^9 + 2255*a^6*b^12*c^18*d^11 + 3465*a^7* 
b^11*c^16*d^13 + 3828*a^8*b^10*c^14*d^15 + 3036*a^9*b^9*c^12*d^17 + 1683*a 
^10*b^8*c^10*d^19 + 605*a^11*b^7*c^8*d^21 + 110*a^12*b^6*c^6*d^23 - 6*a^13 
*b^5*c^4*d^25 - 7*a^14*b^4*c^2*d^27 - a^15*b^3*d^29)/(a^2*b^17*c^32 + 16*a 
^3*b^16*c^30*d^2 + 120*a^4*b^15*c^28*d^4 + 560*a^5*b^14*c^26*d^6 + 1820*a^ 
6*b^13*c^24*d^8 + 4368*a^7*b^12*c^22*d^10 + 8008*a^8*b^11*c^20*d^12 + 1...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (c+d\,x\right )}^3} \,d x \] Input:

int(1/((a + b*x^2)^(5/2)*(c + d*x)^3),x)
 

Output:

int(1/((a + b*x^2)^(5/2)*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 2772, normalized size of antiderivative = 9.27 \[ \int \frac {1}{(c+d x)^3 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/(d*x+c)^3/(b*x^2+a)^(5/2),x)
 

Output:

(15*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
a*d + b*c*x)*a**5*b*c**2*d**6 + 30*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b 
*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**5*b*c*d**7*x + 15*sqrt(a*d* 
*2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)* 
a**5*b*d**8*x**2 - 90*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a 
*d**2 + b*c**2) - a*d + b*c*x)*a**4*b**2*c**4*d**4 - 180*sqrt(a*d**2 + b*c 
**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*b** 
2*c**3*d**5*x - 60*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d* 
*2 + b*c**2) - a*d + b*c*x)*a**4*b**2*c**2*d**6*x**2 + 60*sqrt(a*d**2 + b* 
c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*b* 
*2*c*d**7*x**3 + 30*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d 
**2 + b*c**2) - a*d + b*c*x)*a**4*b**2*d**8*x**4 - 180*sqrt(a*d**2 + b*c** 
2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b**3* 
c**4*d**4*x**2 - 360*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a* 
d**2 + b*c**2) - a*d + b*c*x)*a**3*b**3*c**3*d**5*x**3 - 165*sqrt(a*d**2 + 
 b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3 
*b**3*c**2*d**6*x**4 + 30*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sq 
rt(a*d**2 + b*c**2) - a*d + b*c*x)*a**3*b**3*c*d**7*x**5 + 15*sqrt(a*d**2 
+ b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a** 
3*b**3*d**8*x**6 - 90*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqr...