\(\int \frac {(A+B x) (d+e x)^{7/2}}{(a-c x^2)^3} \, dx\) [134]

Optimal result

Integrand size = 25, antiderivative size = 396 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\frac {(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 A c d^2-14 a B d e-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (c d^2+a e^2\right )\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (7 a B e \left (2 \sqrt {c} d+3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2+18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{11/4}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \left (7 a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2-18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{11/4}} \] Output:

1/4*(e*x+d)^(5/2)*(a*(A*e+B*d)+(A*c*d+B*a*e)*x)/a/c/(-c*x^2+a)^2+1/16*(e*x 
+d)^(1/2)*(a*e*(-5*A*a*e^2+7*A*c*d^2-14*B*a*d*e)+(2*A*c*d*(-2*a*e^2+3*c*d^ 
2)-7*a*B*e*(a*e^2+c*d^2))*x)/a^2/c^2/(-c*x^2+a)+1/32*(c^(1/2)*d-a^(1/2)*e) 
^(3/2)*(7*a*B*e*(2*c^(1/2)*d+3*a^(1/2)*e)-A*(12*c^(3/2)*d^2+18*a^(1/2)*c*d 
*e+5*a*c^(1/2)*e^2))*arctanh(c^(1/4)*(e*x+d)^(1/2)/(c^(1/2)*d-a^(1/2)*e)^( 
1/2))/a^(5/2)/c^(11/4)-1/32*(c^(1/2)*d+a^(1/2)*e)^(3/2)*(7*a*B*e*(2*c^(1/2 
)*d-3*a^(1/2)*e)-A*(12*c^(3/2)*d^2-18*a^(1/2)*c*d*e+5*a*c^(1/2)*e^2))*arct 
anh(c^(1/4)*(e*x+d)^(1/2)/(c^(1/2)*d+a^(1/2)*e)^(1/2))/a^(5/2)/c^(11/4)
 

Mathematica [A] (verified)

Time = 6.03 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} c \sqrt {d+e x} \left (6 A c^3 d^3 x^3+a^3 e^2 (5 A e+7 B (2 d+e x))-a c^2 d x \left (7 B d e x^2+A \left (10 d^2+d e x+8 e^2 x^2\right )\right )-a^2 c \left (A e \left (11 d^2+4 d e x+9 e^2 x^2\right )+B \left (4 d^3+5 d^2 e x+26 d e^2 x^2+11 e^3 x^3\right )\right )\right )}{\left (a-c x^2\right )^2}-\left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (7 a B e \left (-2 \sqrt {c} d+3 \sqrt {a} e\right )+A \left (12 c^{3/2} d^2-18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )-\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (-7 a B e \left (2 \sqrt {c} d+3 \sqrt {a} e\right )+A \left (12 c^{3/2} d^2+18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2} c^3} \] Input:

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

Output:

((-2*Sqrt[a]*c*Sqrt[d + e*x]*(6*A*c^3*d^3*x^3 + a^3*e^2*(5*A*e + 7*B*(2*d 
+ e*x)) - a*c^2*d*x*(7*B*d*e*x^2 + A*(10*d^2 + d*e*x + 8*e^2*x^2)) - a^2*c 
*(A*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2) + B*(4*d^3 + 5*d^2*e*x + 26*d*e^2*x^2 
 + 11*e^3*x^3))))/(a - c*x^2)^2 - (Sqrt[c]*d + Sqrt[a]*e)*Sqrt[-(c*d) - Sq 
rt[a]*Sqrt[c]*e]*(7*a*B*e*(-2*Sqrt[c]*d + 3*Sqrt[a]*e) + A*(12*c^(3/2)*d^2 
 - 18*Sqrt[a]*c*d*e + 5*a*Sqrt[c]*e^2))*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt 
[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)] - (Sqrt[c]*(Sqrt[c]*d - Sqr 
t[a]*e)^2*(-7*a*B*e*(2*Sqrt[c]*d + 3*Sqrt[a]*e) + A*(12*c^(3/2)*d^2 + 18*S 
qrt[a]*c*d*e + 5*a*Sqrt[c]*e^2))*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]* 
Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]) 
/(32*a^(5/2)*c^3)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {684, 27, 684, 27, 654, 27, 1480, 221}

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.

\(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 684

\(\displaystyle \frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}-\frac {\int -\frac {(d+e x)^{3/2} \left (6 A c d^2-a e (7 B d+5 A e)+e (A c d-7 a B e) x\right )}{2 \left (a-c x^2\right )^2}dx}{4 a c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(d+e x)^{3/2} \left (6 A c d^2-a e (7 B d+5 A e)+e (A c d-7 a B e) x\right )}{\left (a-c x^2\right )^2}dx}{8 a c}+\frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\)

\(\Big \downarrow \) 684

\(\displaystyle \frac {\frac {\sqrt {d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {14 a B d e \left (c d^2-2 a e^2\right )-A \left (12 c^2 d^4-19 a c e^2 d^2+5 a^2 e^4\right )-e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right ) x}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {14 a B d e \left (c d^2-2 a e^2\right )-A \left (12 c^2 d^4-19 a c e^2 d^2+5 a^2 e^4\right )-e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a c}}{8 a c}+\frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\)

\(\Big \downarrow \) 654

\(\displaystyle \frac {\frac {\sqrt {d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {e \left (\left (c d^2-a e^2\right ) \left (6 A c d^2-7 a B e d-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {e \int \frac {\left (c d^2-a e^2\right ) \left (6 A c d^2-7 a B e d-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {\sqrt {d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {e \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (-21 a^{3/2} B e^2+18 \sqrt {a} A c d e+5 a A \sqrt {c} e^2-14 a B \sqrt {c} d e+12 A c^{3/2} d^2\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (7 a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (-18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{2 a c \left (a-c x^2\right )}-\frac {e \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (-21 a^{3/2} B e^2+18 \sqrt {a} A c d e+5 a A \sqrt {c} e^2-14 a B \sqrt {c} d e+12 A c^{3/2} d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} c^{3/4} e}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (7 a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (-18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} c^{3/4} e}\right )}{2 a c}}{8 a c}+\frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2}\)

Input:

Int[((A + B*x)*(d + e*x)^(7/2))/(a - c*x^2)^3,x]
 

Output:

((d + e*x)^(5/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2 
) + ((Sqrt[d + e*x]*(a*e*(7*A*c*d^2 - 14*a*B*d*e - 5*a*A*e^2) + (2*A*c*d*( 
3*c*d^2 - 2*a*e^2) - 7*a*B*e*(c*d^2 + a*e^2))*x))/(2*a*c*(a - c*x^2)) - (e 
*(((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(12*A*c^(3/2)*d^2 - 14*a*B*Sqrt[c]*d*e + 
18*Sqrt[a]*A*c*d*e - 21*a^(3/2)*B*e^2 + 5*a*A*Sqrt[c]*e^2)*ArcTanh[(c^(1/4 
)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)*e) + ((S 
qrt[c]*d + Sqrt[a]*e)^(3/2)*(7*a*B*e*(2*Sqrt[c]*d - 3*Sqrt[a]*e) - A*(12*c 
^(3/2)*d^2 - 18*Sqrt[a]*c*d*e + 5*a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d 
+ e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(3/4)*e)))/(2*a*c))/(8* 
a*c)
 

Defintions of rubi rules used

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

rule 684

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 + 1)*((a*(e*f + d*g 
) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1))   Int[ 
(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 
2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a 
, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] 
 && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 2.37 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.26

method	result	size
pseudoelliptic	\(-\frac {5 \left (-\frac {e \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\frac {\left (-6 A \,c^{2} d^{3}+8 e \left (A e +\frac {7 B d}{8}\right ) d a c -21 B \,e^{3} a^{2}\right ) \sqrt {a c \,e^{2}}}{5}+c \left (\frac {12 A \,c^{2} d^{4}}{5}-\frac {19 e \,d^{2} \left (A e +\frac {14 B d}{19}\right ) a c}{5}+a^{2} e^{3} \left (A e +\frac {28 B d}{5}\right )\right )\right ) \left (-c \,x^{2}+a \right )^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\frac {e \left (\frac {\left (6 A \,c^{2} d^{3}-8 e \left (A e +\frac {7 B d}{8}\right ) d a c +21 B \,e^{3} a^{2}\right ) \sqrt {a c \,e^{2}}}{5}+c \left (\frac {12 A \,c^{2} d^{4}}{5}-\frac {19 e \,d^{2} \left (A e +\frac {14 B d}{19}\right ) a c}{5}+a^{2} e^{3} \left (A e +\frac {28 B d}{5}\right )\right )\right ) \left (-c \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {e x +d}\, \sqrt {a c \,e^{2}}\, \left (\frac {6 A \,c^{3} d^{3} x^{3}}{5}-2 d \left (A \,d^{2}+\frac {e x \left (7 B x +A \right ) d}{10}+\frac {4 A \,e^{2} x^{2}}{5}\right ) x a \,c^{2}-\frac {11 \left (\frac {4 B \,d^{3}}{11}+e \left (\frac {5 B x}{11}+A \right ) d^{2}+\frac {4 e^{2} \left (\frac {13 B x}{2}+A \right ) x d}{11}+\frac {9 e^{3} x^{2} \left (\frac {11 B x}{9}+A \right )}{11}\right ) a^{2} c}{5}+e^{2} a^{3} \left (\frac {14 B d}{5}+e \left (\frac {7 B x}{5}+A \right )\right )\right )\right )\right )}{16 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a^{2} c^{2} \left (-c \,x^{2}+a \right )^{2}}\)	\(499\)
default	\(2 e^{4} \left (\frac {\frac {\left (8 A a c d \,e^{2}-6 A \,c^{2} d^{3}+11 B \,e^{3} a^{2}+7 B a c \,d^{2} e \right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{3} c}+\frac {\left (9 A \,a^{2} e^{4}-23 A a c \,d^{2} e^{2}+18 A \,c^{2} d^{4}-7 B \,a^{2} d \,e^{3}-21 B a c \,d^{3} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{32 c \,a^{2} e^{3}}-\frac {\left (14 A \,a^{2} c d \,e^{4}-32 A a \,c^{2} d^{3} e^{2}+18 A \,d^{5} c^{3}+7 B \,e^{5} a^{3}+14 B \,a^{2} c \,d^{2} e^{3}-21 B a \,c^{2} d^{4} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{32 c^{2} a^{2} e^{3}}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (5 A \,a^{2} e^{4}-11 A a c \,d^{2} e^{2}+6 A \,c^{2} d^{4}+7 B \,a^{2} d \,e^{3}-7 B a c \,d^{3} e \right ) \sqrt {e x +d}}{32 a^{2} c^{2} e^{3}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {-\frac {\left (-5 A \,a^{2} c \,e^{4}+19 A a \,c^{2} d^{2} e^{2}-12 A \,d^{4} c^{3}-28 B \,a^{2} c d \,e^{3}+14 B a \,c^{2} d^{3} e +8 A \sqrt {a c \,e^{2}}\, a c d \,e^{2}-6 A \sqrt {a c \,e^{2}}\, c^{2} d^{3}-21 B \sqrt {a c \,e^{2}}\, a^{2} e^{3}+7 B \sqrt {a c \,e^{2}}\, a c \,d^{2} e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 A \,a^{2} c \,e^{4}-19 A a \,c^{2} d^{2} e^{2}+12 A \,d^{4} c^{3}+28 B \,a^{2} c d \,e^{3}-14 B a \,c^{2} d^{3} e +8 A \sqrt {a c \,e^{2}}\, a c d \,e^{2}-6 A \sqrt {a c \,e^{2}}\, c^{2} d^{3}-21 B \sqrt {a c \,e^{2}}\, a^{2} e^{3}+7 B \sqrt {a c \,e^{2}}\, a c \,d^{2} e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} c \,e^{3}}\right )\)	\(698\)
derivativedivides	\(-2 e^{4} \left (-\frac {\frac {\left (8 A a c d \,e^{2}-6 A \,c^{2} d^{3}+11 B \,e^{3} a^{2}+7 B a c \,d^{2} e \right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{3} c}+\frac {\left (9 A \,a^{2} e^{4}-23 A a c \,d^{2} e^{2}+18 A \,c^{2} d^{4}-7 B \,a^{2} d \,e^{3}-21 B a c \,d^{3} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{32 c \,a^{2} e^{3}}-\frac {\left (14 A \,a^{2} c d \,e^{4}-32 A a \,c^{2} d^{3} e^{2}+18 A \,d^{5} c^{3}+7 B \,e^{5} a^{3}+14 B \,a^{2} c \,d^{2} e^{3}-21 B a \,c^{2} d^{4} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{32 c^{2} a^{2} e^{3}}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (5 A \,a^{2} e^{4}-11 A a c \,d^{2} e^{2}+6 A \,c^{2} d^{4}+7 B \,a^{2} d \,e^{3}-7 B a c \,d^{3} e \right ) \sqrt {e x +d}}{32 a^{2} c^{2} e^{3}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )^{2}}-\frac {-\frac {\left (-5 A \,a^{2} c \,e^{4}+19 A a \,c^{2} d^{2} e^{2}-12 A \,d^{4} c^{3}-28 B \,a^{2} c d \,e^{3}+14 B a \,c^{2} d^{3} e +8 A \sqrt {a c \,e^{2}}\, a c d \,e^{2}-6 A \sqrt {a c \,e^{2}}\, c^{2} d^{3}-21 B \sqrt {a c \,e^{2}}\, a^{2} e^{3}+7 B \sqrt {a c \,e^{2}}\, a c \,d^{2} e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 A \,a^{2} c \,e^{4}-19 A a \,c^{2} d^{2} e^{2}+12 A \,d^{4} c^{3}+28 B \,a^{2} c d \,e^{3}-14 B a \,c^{2} d^{3} e +8 A \sqrt {a c \,e^{2}}\, a c d \,e^{2}-6 A \sqrt {a c \,e^{2}}\, c^{2} d^{3}-21 B \sqrt {a c \,e^{2}}\, a^{2} e^{3}+7 B \sqrt {a c \,e^{2}}\, a c \,d^{2} e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} c \,e^{3}}\right )\)	\(699\)

Input:

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

Output:

-5/16/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)/(a*c*e^2)^(1/2)*(-1/2*e*((c*d+(a*c*e 
^2)^(1/2))*c)^(1/2)*(1/5*(-6*A*c^2*d^3+8*e*(A*e+7/8*B*d)*d*a*c-21*B*e^3*a^ 
2)*(a*c*e^2)^(1/2)+c*(12/5*A*c^2*d^4-19/5*e*d^2*(A*e+14/19*B*d)*a*c+a^2*e^ 
3*(A*e+28/5*B*d)))*(-c*x^2+a)^2*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1 
/2))*c)^(1/2))+((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(-1/2*e*(1/5*(6*A*c^2*d^3- 
8*e*(A*e+7/8*B*d)*d*a*c+21*B*e^3*a^2)*(a*c*e^2)^(1/2)+c*(12/5*A*c^2*d^4-19 
/5*e*d^2*(A*e+14/19*B*d)*a*c+a^2*e^3*(A*e+28/5*B*d)))*(-c*x^2+a)^2*arctanh 
(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+((c*d+(a*c*e^2)^(1/2))*c 
)^(1/2)*(e*x+d)^(1/2)*(a*c*e^2)^(1/2)*(6/5*A*c^3*d^3*x^3-2*d*(A*d^2+1/10*e 
*x*(7*B*x+A)*d+4/5*A*e^2*x^2)*x*a*c^2-11/5*(4/11*B*d^3+e*(5/11*B*x+A)*d^2+ 
4/11*e^2*(13/2*B*x+A)*x*d+9/11*e^3*x^2*(11/9*B*x+A))*a^2*c+e^2*a^3*(14/5*B 
*d+e*(7/5*B*x+A)))))/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)/a^2/c^2/(-c*x^2+a)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6669 vs. \(2 (326) = 652\).

Time = 10.86 (sec) , antiderivative size = 6669, normalized size of antiderivative = 16.84 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 - a)^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1106 vs. \(2 (326) = 652\).

Time = 0.32 (sec) , antiderivative size = 1106, normalized size of antiderivative = 2.79 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/32*(2*(3*a*c^2*d^3*e - 4*a^2*c*d*e^3)*A*e^2*abs(c) - 7*(a^2*c*d^2*e^2 - 
 3*a^3*e^4)*B*e^2*abs(c) + (6*sqrt(a*c)*c^2*d^4*e - 11*sqrt(a*c)*a*c*d^2*e 
^3 + 5*sqrt(a*c)*a^2*e^5)*A*abs(c)*abs(e) - 7*(sqrt(a*c)*a*c*d^3*e^2 - sqr 
t(a*c)*a^2*d*e^4)*B*abs(c)*abs(e) - (12*c^3*d^5*e - 19*a*c^2*d^3*e^3 + 5*a 
^2*c*d*e^5)*A*abs(c) + 14*(a*c^2*d^4*e^2 - 2*a^2*c*d^2*e^4)*B*abs(c))*arct 
an(sqrt(e*x + d)/sqrt(-(a^2*c^3*d + sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3* 
c^2*e^2)*a^2*c^3))/(a^2*c^3)))/((a^3*c^3*e - sqrt(a*c)*a^2*c^3*d)*sqrt(-c^ 
2*d - sqrt(a*c)*c*e)*abs(e)) - 1/32*(2*(3*a*c^2*d^3*e - 4*a^2*c*d*e^3)*A*e 
^2*abs(c) - 7*(a^2*c*d^2*e^2 - 3*a^3*e^4)*B*e^2*abs(c) - (6*sqrt(a*c)*c^2* 
d^4*e - 11*sqrt(a*c)*a*c*d^2*e^3 + 5*sqrt(a*c)*a^2*e^5)*A*abs(c)*abs(e) + 
7*(sqrt(a*c)*a*c*d^3*e^2 - sqrt(a*c)*a^2*d*e^4)*B*abs(c)*abs(e) - (12*c^3* 
d^5*e - 19*a*c^2*d^3*e^3 + 5*a^2*c*d*e^5)*A*abs(c) + 14*(a*c^2*d^4*e^2 - 2 
*a^2*c*d^2*e^4)*B*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^3*d - sqrt(a^4 
*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2*e^2)*a^2*c^3))/(a^2*c^3)))/((a^3*c^3*e + 
 sqrt(a*c)*a^2*c^3*d)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e)) - 1/16*(6*(e*x 
+ d)^(7/2)*A*c^3*d^3*e - 18*(e*x + d)^(5/2)*A*c^3*d^4*e + 18*(e*x + d)^(3/ 
2)*A*c^3*d^5*e - 6*sqrt(e*x + d)*A*c^3*d^6*e - 7*(e*x + d)^(7/2)*B*a*c^2*d 
^2*e^2 + 21*(e*x + d)^(5/2)*B*a*c^2*d^3*e^2 - 21*(e*x + d)^(3/2)*B*a*c^2*d 
^4*e^2 + 7*sqrt(e*x + d)*B*a*c^2*d^5*e^2 - 8*(e*x + d)^(7/2)*A*a*c^2*d*e^3 
 + 23*(e*x + d)^(5/2)*A*a*c^2*d^2*e^3 - 32*(e*x + d)^(3/2)*A*a*c^2*d^3*...
 

Mupad [B] (verification not implemented)

Time = 6.71 (sec) , antiderivative size = 11687, normalized size of antiderivative = 29.51 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(((A + B*x)*(d + e*x)^(7/2))/(a - c*x^2)^3,x)
 

Output:

- atan(((((20480*A*a^7*c^6*e^7 + 28672*B*a^7*c^6*d*e^6 + 24576*A*a^5*c^8*d 
^4*e^3 - 45056*A*a^6*c^7*d^2*e^5 - 28672*B*a^6*c^7*d^3*e^4)/(4096*a^6*c^5) 
 - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((144*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e^7 
*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 1 
96*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 2 
1*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9* 
c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c 
^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) 
 + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c*d*e^6 
*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2))*((144*A^2*a^5*c^10*d^7 - 441* 
B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8* 
d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9* 
c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 7 
35*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3* 
e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15* 
c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 266*A* 
B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) + ((d + e*x)^(1/2)* 
(441*B^2*a^5*e^10 + 144*A^2*c^5*d^8*e^2 + 25*A^2*a^4*c*e^10 + 385*A^2*a^2* 
c^3*d^4*e^6 - 126*A^2*a^3*c^2*d^2*e^8 + 196*B^2*a^2*c^3*d^6*e^4 - 735*B^2* 
a^3*c^2*d^4*e^6 - 420*A^2*a*c^4*d^6*e^4 + 490*B^2*a^4*c*d^2*e^8 - 56*A*...
 

Reduce [B] (verification not implemented)

Time = 6.86 (sec) , antiderivative size = 3452, normalized size of antiderivative = 8.72 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int((B*x+A)*(e*x+d)^(7/2)/(-c*x^2+a)^3,x)
 

Output:

( - 42*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt( 
c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a**3*b*e**3 + 26*sqrt(a)*sqrt(sqrt(c)*s 
qrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c 
*d)))*a**3*c*d*e**2 + 28*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt( 
d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a**2*b*c*d**2*e + 84* 
sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt 
(sqrt(c)*sqrt(a)*e - c*d)))*a**2*b*c*e**3*x**2 - 24*sqrt(a)*sqrt(sqrt(c)*s 
qrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c 
*d)))*a**2*c**2*d**3 - 52*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt 
(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a**2*c**2*d*e**2*x** 
2 - 56*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt( 
c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a*b*c**2*d**2*e*x**2 - 42*sqrt(a)*sqrt( 
sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt 
(a)*e - c*d)))*a*b*c**2*e**3*x**4 + 48*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c* 
d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a*c**3* 
d**3*x**2 + 26*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c 
)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a*c**3*d*e**2*x**4 + 28*sqrt(a) 
*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c 
)*sqrt(a)*e - c*d)))*b*c**3*d**2*e*x**4 - 24*sqrt(a)*sqrt(sqrt(c)*sqrt(a)* 
e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d))...