\(\int (f x)^m (d+e x^2)^3 (a+b \text {sech}^{-1}(c x)) \, dx\) [176]

Optimal result

Integrand size = 23, antiderivative size = 593 \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b e \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^6 f (2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}-\frac {b e^2 \left (e (5+m)^2+3 c^2 d \left (42+13 m+m^2\right )\right ) (f x)^{3+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^4 f^3 (4+m) (5+m) (6+m) (7+m)}-\frac {b e^3 (f x)^{5+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{c^2 f^5 (6+m) (7+m)}+\frac {d^3 (f x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f (1+m)}+\frac {3 d^2 e (f x)^{3+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {3 d e^2 (f x)^{5+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (5+m)}+\frac {e^3 (f x)^{7+m} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (7+m)}+\frac {b \left (c^6 d^3 (2+m) (4+m) (6+m)+\frac {e (1+m)^2 \left (e^2 \left (15+8 m+m^2\right )^2+3 c^2 d e (3+m)^2 \left (42+13 m+m^2\right )+3 c^4 d^2 \left (840+638 m+179 m^2+22 m^3+m^4\right )\right )}{(3+m) (5+m) (7+m)}\right ) (f x)^{1+m} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{c^6 f (1+m)^2 (2+m) (4+m) (6+m)} \] Output:

-b*e*(e^2*(m^2+8*m+15)^2+3*c^2*d*e*(3+m)^2*(m^2+13*m+42)+3*c^4*d^2*(m^4+22 
*m^3+179*m^2+638*m+840))*(f*x)^(1+m)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2 
*x^2+1)^(1/2)/c^6/f/(2+m)/(3+m)/(4+m)/(5+m)/(6+m)/(7+m)-b*e^2*(e*(5+m)^2+3 
*c^2*d*(m^2+13*m+42))*(f*x)^(3+m)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^ 
2+1)^(1/2)/c^4/f^3/(4+m)/(5+m)/(6+m)/(7+m)-b*e^3*(f*x)^(5+m)*(1/(c*x+1))^( 
1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/f^5/(6+m)/(7+m)+d^3*(f*x)^(1+m)* 
(a+b*arcsech(c*x))/f/(1+m)+3*d^2*e*(f*x)^(3+m)*(a+b*arcsech(c*x))/f^3/(3+m 
)+3*d*e^2*(f*x)^(5+m)*(a+b*arcsech(c*x))/f^5/(5+m)+e^3*(f*x)^(7+m)*(a+b*ar 
csech(c*x))/f^7/(7+m)+b*(c^6*d^3*(2+m)*(4+m)*(6+m)+e*(1+m)^2*(e^2*(m^2+8*m 
+15)^2+3*c^2*d*e*(3+m)^2*(m^2+13*m+42)+3*c^4*d^2*(m^4+22*m^3+179*m^2+638*m 
+840))/(3+m)/(5+m)/(7+m))*(f*x)^(1+m)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*hype 
rgeom([1/2, 1/2+1/2*m],[3/2+1/2*m],c^2*x^2)/c^6/f/(1+m)^2/(2+m)/(4+m)/(6+m 
)
 

Mathematica [A] (verified)

Time = 2.90 (sec) , antiderivative size = 441, normalized size of antiderivative = 0.74 \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=x (f x)^m \left (\frac {a d^3}{1+m}+\frac {3 a d^2 e x^2}{3+m}+\frac {3 a d e^2 x^4}{5+m}+\frac {a e^3 x^6}{7+m}+\frac {b d^3 \text {sech}^{-1}(c x)}{1+m}+\frac {3 b d^2 e x^2 \text {sech}^{-1}(c x)}{3+m}+\frac {3 b d e^2 x^4 \text {sech}^{-1}(c x)}{5+m}+\frac {b e^3 x^6 \text {sech}^{-1}(c x)}{7+m}-\frac {b d^3 \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{(1+m)^2 (-1+c x)}-\frac {3 b d^2 e x^2 \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {5+m}{2},c^2 x^2\right )}{(3+m)^2 (-1+c x)}-\frac {3 b d e^2 x^4 \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+m}{2},\frac {7+m}{2},c^2 x^2\right )}{(5+m)^2 (-1+c x)}-\frac {b e^3 x^6 \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7+m}{2},\frac {9+m}{2},c^2 x^2\right )}{(7+m)^2 (-1+c x)}\right ) \] Input:

Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSech[c*x]),x]
 

Output:

x*(f*x)^m*((a*d^3)/(1 + m) + (3*a*d^2*e*x^2)/(3 + m) + (3*a*d*e^2*x^4)/(5 
+ m) + (a*e^3*x^6)/(7 + m) + (b*d^3*ArcSech[c*x])/(1 + m) + (3*b*d^2*e*x^2 
*ArcSech[c*x])/(3 + m) + (3*b*d*e^2*x^4*ArcSech[c*x])/(5 + m) + (b*e^3*x^6 
*ArcSech[c*x])/(7 + m) - (b*d^3*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2 
]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((1 + m)^2*(-1 + 
c*x)) - (3*b*d^2*e*x^2*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*Hyperge 
ometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/((3 + m)^2*(-1 + c*x)) - ( 
3*b*d*e^2*x^4*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*Hypergeometric2F 
1[1/2, (5 + m)/2, (7 + m)/2, c^2*x^2])/((5 + m)^2*(-1 + c*x)) - (b*e^3*x^6 
*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (7 + m 
)/2, (9 + m)/2, c^2*x^2])/((7 + m)^2*(-1 + c*x)))
 

Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 527, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6855, 2340, 25, 1590, 25, 363, 278}

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.

\(\displaystyle \int \left (d+e x^2\right )^3 (f x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\)

\(\Big \downarrow \) 6855

\(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {(f x)^m \left (\frac {e^3 x^6}{m+7}+\frac {3 d e^2 x^4}{m+5}+\frac {3 d^2 e x^2}{m+3}+\frac {d^3}{m+1}\right )}{\sqrt {1-c^2 x^2}}dx+\frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}\)

\(\Big \downarrow \) 2340

\(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {\int -\frac {(f x)^m \left (\frac {e^2 \left (3 d \left (m^2+13 m+42\right ) c^2+e (m+5)^2\right ) x^4}{(m+5) (m+7)}+\frac {3 c^2 d^2 e (m+6) x^2}{m+3}+\frac {c^2 d^3 (m+6)}{m+1}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+6)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)}\right )+\frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}\)

\(\Big \downarrow \) 25

\(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {(f x)^m \left (\frac {e^2 \left (3 d \left (m^2+13 m+42\right ) c^2+e (m+5)^2\right ) x^4}{(m+5) (m+7)}+\frac {3 c^2 d^2 e (m+6) x^2}{m+3}+\frac {c^2 d^3 (m+6)}{m+1}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+6)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)}\right )+\frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}\)

\(\Big \downarrow \) 1590

\(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {-\frac {\int -\frac {(f x)^m \left (\frac {d^3 (m+4) (m+6) c^4}{m+1}+\frac {e \left (3 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right ) c^4+3 d e (m+3)^2 \left (m^2+13 m+42\right ) c^2+e^2 \left (m^2+8 m+15\right )^2\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+4)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^2 f^3 (m+4) (m+5) (m+7)}}{c^2 (m+6)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)}\right )+\frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}\)

\(\Big \downarrow \) 25

\(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\int \frac {(f x)^m \left (\frac {d^3 (m+4) (m+6) c^4}{m+1}+\frac {e \left (3 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right ) c^4+3 d e (m+3)^2 \left (m^2+13 m+42\right ) c^2+e^2 \left (m^2+8 m+15\right )^2\right ) x^2}{(m+3) (m+5) (m+7)}\right )}{\sqrt {1-c^2 x^2}}dx}{c^2 (m+4)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^2 f^3 (m+4) (m+5) (m+7)}}{c^2 (m+6)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)}\right )+\frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}\)

\(\Big \downarrow \) 363

\(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\left (\frac {c^4 d^3 (m+4) (m+6)}{m+1}+\frac {e (m+1) \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^2 (m+2) (m+3) (m+5) (m+7)}\right ) \int \frac {(f x)^m}{\sqrt {1-c^2 x^2}}dx-\frac {e \sqrt {1-c^2 x^2} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^2 f (m+2) (m+3) (m+5) (m+7)}}{c^2 (m+4)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^2 f^3 (m+4) (m+5) (m+7)}}{c^2 (m+6)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)}\right )+\frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {d^3 (f x)^{m+1} \left (a+b \text {sech}^{-1}(c x)\right )}{f (m+1)}+\frac {3 d^2 e (f x)^{m+3} \left (a+b \text {sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {3 d e^2 (f x)^{m+5} \left (a+b \text {sech}^{-1}(c x)\right )}{f^5 (m+5)}+\frac {e^3 (f x)^{m+7} \left (a+b \text {sech}^{-1}(c x)\right )}{f^7 (m+7)}+b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\frac {(f x)^{m+1} \left (\frac {c^4 d^3 (m+4) (m+6)}{m+1}+\frac {e (m+1) \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^2 (m+2) (m+3) (m+5) (m+7)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},c^2 x^2\right )}{f (m+1)}-\frac {e \sqrt {1-c^2 x^2} (f x)^{m+1} \left (3 c^4 d^2 \left (m^4+22 m^3+179 m^2+638 m+840\right )+3 c^2 d e (m+3)^2 \left (m^2+13 m+42\right )+e^2 \left (m^2+8 m+15\right )^2\right )}{c^2 f (m+2) (m+3) (m+5) (m+7)}}{c^2 (m+4)}-\frac {e^2 \sqrt {1-c^2 x^2} (f x)^{m+3} \left (3 c^2 d \left (m^2+13 m+42\right )+e (m+5)^2\right )}{c^2 f^3 (m+4) (m+5) (m+7)}}{c^2 (m+6)}-\frac {e^3 \sqrt {1-c^2 x^2} (f x)^{m+5}}{c^2 f^5 (m+6) (m+7)}\right )\)

Input:

Int[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSech[c*x]),x]
 

Output:

(d^3*(f*x)^(1 + m)*(a + b*ArcSech[c*x]))/(f*(1 + m)) + (3*d^2*e*(f*x)^(3 + 
 m)*(a + b*ArcSech[c*x]))/(f^3*(3 + m)) + (3*d*e^2*(f*x)^(5 + m)*(a + b*Ar 
cSech[c*x]))/(f^5*(5 + m)) + (e^3*(f*x)^(7 + m)*(a + b*ArcSech[c*x]))/(f^7 
*(7 + m)) + b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(-((e^3*(f*x)^(5 + m)*Sqr 
t[1 - c^2*x^2])/(c^2*f^5*(6 + m)*(7 + m))) + (-((e^2*(e*(5 + m)^2 + 3*c^2* 
d*(42 + 13*m + m^2))*(f*x)^(3 + m)*Sqrt[1 - c^2*x^2])/(c^2*f^3*(4 + m)*(5 
+ m)*(7 + m))) + (-((e*(e^2*(15 + 8*m + m^2)^2 + 3*c^2*d*e*(3 + m)^2*(42 + 
 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^3 + m^4))*(f*x)^(1 
+ m)*Sqrt[1 - c^2*x^2])/(c^2*f*(2 + m)*(3 + m)*(5 + m)*(7 + m))) + (((c^4* 
d^3*(4 + m)*(6 + m))/(1 + m) + (e*(1 + m)*(e^2*(15 + 8*m + m^2)^2 + 3*c^2* 
d*e*(3 + m)^2*(42 + 13*m + m^2) + 3*c^4*d^2*(840 + 638*m + 179*m^2 + 22*m^ 
3 + m^4)))/(c^2*(2 + m)*(3 + m)*(5 + m)*(7 + m)))*(f*x)^(1 + m)*Hypergeome 
tric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(f*(1 + m)))/(c^2*(4 + m)))/( 
c^2*(6 + m)))
 

Defintions of rubi rules used

rule 25

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

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 1590

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ 
(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q 
 + 1))   Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + 
b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 
0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

rule 6855

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( 
x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si 
mp[(a + b*ArcSech[c*x])   u, x] + Simp[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)] 
Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; Fre 
eQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && 
 GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2 
*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

Maple [F]

\[\int \left (f x \right )^{m} \left (x^{2} e +d \right )^{3} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )d x\]

Input:

int((f*x)^m*(e*x^2+d)^3*(a+b*arcsech(c*x)),x)
 

Output:

int((f*x)^m*(e*x^2+d)^3*(a+b*arcsech(c*x)),x)
 

Fricas [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arcsech(c*x)),x, algorithm="fricas")
 

Output:

integral((a*e^3*x^6 + 3*a*d*e^2*x^4 + 3*a*d^2*e*x^2 + a*d^3 + (b*e^3*x^6 + 
 3*b*d*e^2*x^4 + 3*b*d^2*e*x^2 + b*d^3)*arcsech(c*x))*(f*x)^m, x)
 

Sympy [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (f x\right )^{m} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{3}\, dx \] Input:

integrate((f*x)**m*(e*x**2+d)**3*(a+b*asech(c*x)),x)
 

Output:

Integral((f*x)**m*(a + b*asech(c*x))*(d + e*x**2)**3, x)
 

Maxima [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arcsech(c*x)),x, algorithm="maxima")
 

Output:

a*e^3*f^m*x^7*x^m/(m + 7) + 3*a*d*e^2*f^m*x^5*x^m/(m + 5) + 3*a*d^2*e*f^m* 
x^3*x^m/(m + 3) + (f*x)^(m + 1)*a*d^3/(f*(m + 1)) + (((m^3 + 9*m^2 + 23*m 
+ 15)*b*e^3*f^m*x^7*x^m + 3*(m^3 + 11*m^2 + 31*m + 21)*b*d*e^2*f^m*x^5*x^m 
 + 3*(m^3 + 13*m^2 + 47*m + 35)*b*d^2*e*f^m*x^3*x^m + (m^3 + 15*m^2 + 71*m 
 + 105)*b*d^3*f^m*x*x^m)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) - ((m^3 + 9 
*m^2 + 23*m + 15)*b*e^3*f^m*x^7*x^m + 3*(m^3 + 11*m^2 + 31*m + 21)*b*d*e^2 
*f^m*x^5*x^m + 3*(m^3 + 13*m^2 + 47*m + 35)*b*d^2*e*f^m*x^3*x^m + (m^3 + 1 
5*m^2 + 71*m + 105)*b*d^3*f^m*x*x^m)*log(x))/(m^4 + 16*m^3 + 86*m^2 + 176* 
m + 105) - integrate((b*c^2*e^3*f^m*(m + 7)*x^2*log(c) - (e^3*f^m*(m + 7)* 
log(c) - e^3*f^m)*b)*x^6*x^m/(c^2*(m + 7)*x^2 - m - 7), x) - integrate(3*( 
b*c^2*d*e^2*f^m*(m + 5)*x^2*log(c) - (d*e^2*f^m*(m + 5)*log(c) - d*e^2*f^m 
)*b)*x^4*x^m/(c^2*(m + 5)*x^2 - m - 5), x) - integrate(3*(b*c^2*d^2*e*f^m* 
(m + 3)*x^2*log(c) - (d^2*e*f^m*(m + 3)*log(c) - d^2*e*f^m)*b)*x^2*x^m/(c^ 
2*(m + 3)*x^2 - m - 3), x) - integrate((b*c^2*d^3*f^m*(m + 1)*x^2*log(c) - 
 (d^3*f^m*(m + 1)*log(c) - d^3*f^m)*b)*x^m/(c^2*(m + 1)*x^2 - m - 1), x) + 
 integrate(((m^3 + 9*m^2 + 23*m + 15)*b*c^2*e^3*f^m*x^8*x^m + 3*(m^3 + 11* 
m^2 + 31*m + 21)*b*c^2*d*e^2*f^m*x^6*x^m + 3*(m^3 + 13*m^2 + 47*m + 35)*b* 
c^2*d^2*e*f^m*x^4*x^m + (m^3 + 15*m^2 + 71*m + 105)*b*c^2*d^3*f^m*x^2*x^m) 
/((m^4 + 16*m^3 + 86*m^2 + 176*m + 105)*c^2*x^2 - m^4 - 16*m^3 + ((m^4 + 1 
6*m^3 + 86*m^2 + 176*m + 105)*c^2*x^2 - m^4 - 16*m^3 - 86*m^2 - 176*m -...
 

Giac [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x^{2} + d\right )}^{3} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m} \,d x } \] Input:

integrate((f*x)^m*(e*x^2+d)^3*(a+b*arcsech(c*x)),x, algorithm="giac")
 

Output:

integrate((e*x^2 + d)^3*(b*arcsech(c*x) + a)*(f*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int {\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \] Input:

int((f*x)^m*(d + e*x^2)^3*(a + b*acosh(1/(c*x))),x)
 

Output:

int((f*x)^m*(d + e*x^2)^3*(a + b*acosh(1/(c*x))), x)
 

Reduce [F]

\[ \int (f x)^m \left (d+e x^2\right )^3 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx =\text {Too large to display} \] Input:

int((f*x)^m*(e*x^2+d)^3*(a+b*asech(c*x)),x)
                                                                                    
                                                                                    
 

Output:

(f**m*(x**m*a*d**3*m**3*x + 15*x**m*a*d**3*m**2*x + 71*x**m*a*d**3*m*x + 1 
05*x**m*a*d**3*x + 3*x**m*a*d**2*e*m**3*x**3 + 39*x**m*a*d**2*e*m**2*x**3 
+ 141*x**m*a*d**2*e*m*x**3 + 105*x**m*a*d**2*e*x**3 + 3*x**m*a*d*e**2*m**3 
*x**5 + 33*x**m*a*d*e**2*m**2*x**5 + 93*x**m*a*d*e**2*m*x**5 + 63*x**m*a*d 
*e**2*x**5 + x**m*a*e**3*m**3*x**7 + 9*x**m*a*e**3*m**2*x**7 + 23*x**m*a*e 
**3*m*x**7 + 15*x**m*a*e**3*x**7 + int(x**m*asech(c*x)*x**6,x)*b*e**3*m**4 
 + 16*int(x**m*asech(c*x)*x**6,x)*b*e**3*m**3 + 86*int(x**m*asech(c*x)*x** 
6,x)*b*e**3*m**2 + 176*int(x**m*asech(c*x)*x**6,x)*b*e**3*m + 105*int(x**m 
*asech(c*x)*x**6,x)*b*e**3 + 3*int(x**m*asech(c*x)*x**4,x)*b*d*e**2*m**4 + 
 48*int(x**m*asech(c*x)*x**4,x)*b*d*e**2*m**3 + 258*int(x**m*asech(c*x)*x* 
*4,x)*b*d*e**2*m**2 + 528*int(x**m*asech(c*x)*x**4,x)*b*d*e**2*m + 315*int 
(x**m*asech(c*x)*x**4,x)*b*d*e**2 + 3*int(x**m*asech(c*x)*x**2,x)*b*d**2*e 
*m**4 + 48*int(x**m*asech(c*x)*x**2,x)*b*d**2*e*m**3 + 258*int(x**m*asech( 
c*x)*x**2,x)*b*d**2*e*m**2 + 528*int(x**m*asech(c*x)*x**2,x)*b*d**2*e*m + 
315*int(x**m*asech(c*x)*x**2,x)*b*d**2*e + int(x**m*asech(c*x),x)*b*d**3*m 
**4 + 16*int(x**m*asech(c*x),x)*b*d**3*m**3 + 86*int(x**m*asech(c*x),x)*b* 
d**3*m**2 + 176*int(x**m*asech(c*x),x)*b*d**3*m + 105*int(x**m*asech(c*x), 
x)*b*d**3))/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105)