\(\int x^3 (d+e x^2)^{3/2} (a+b \csc ^{-1}(c x)) \, dx\) [128]

Optimal result

Integrand size = 23, antiderivative size = 374 \[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=-\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {c^2 x^2}}+\frac {b \left (13 c^2 d+25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {2 b c d^{7/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{35 e^2 \sqrt {c^2 x^2}}-\frac {b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt {c^2 x^2}} \] Output:

-1/560*b*(3*c^4*d^2-38*c^2*d*e-25*e^2)*x*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2) 
/c^5/e/(c^2*x^2)^(1/2)+1/840*b*(13*c^2*d+25*e)*x*(c^2*x^2-1)^(1/2)*(e*x^2+ 
d)^(3/2)/c^3/e/(c^2*x^2)^(1/2)+1/42*b*x*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(5/2)/ 
c/e/(c^2*x^2)^(1/2)-1/5*d*(e*x^2+d)^(5/2)*(a+b*arccsc(c*x))/e^2+1/7*(e*x^2 
+d)^(7/2)*(a+b*arccsc(c*x))/e^2+2/35*b*c*d^(7/2)*x*arctan((e*x^2+d)^(1/2)/ 
d^(1/2)/(c^2*x^2-1)^(1/2))/e^2/(c^2*x^2)^(1/2)-1/560*b*(35*c^6*d^3-35*c^4* 
d^2*e-63*c^2*d*e^2-25*e^3)*x*arctanh(e^(1/2)*(c^2*x^2-1)^(1/2)/c/(e*x^2+d) 
^(1/2))/c^6/e^(3/2)/(c^2*x^2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 1.45 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.81 \[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {96 a \left (d+e x^2\right )^3 \left (-2 d+5 e x^2\right )+\frac {2 b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right ) \left (75 e^2+2 c^2 e \left (82 d+25 e x^2\right )+c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )\right )}{c^5}+\frac {3 b \left (32 c^4 d^4 \sqrt {1+\frac {d}{e x^2}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+\frac {e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \sqrt {1-\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )}{\sqrt {1-c^2 x^2}}\right )}{c^5 x}+96 b \left (d+e x^2\right )^3 \left (-2 d+5 e x^2\right ) \csc ^{-1}(c x)}{3360 e^2 \sqrt {d+e x^2}} \] Input:

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

Output:

(96*a*(d + e*x^2)^3*(-2*d + 5*e*x^2) + (2*b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 
 e*x^2)*(75*e^2 + 2*c^2*e*(82*d + 25*e*x^2) + c^4*(57*d^2 + 106*d*e*x^2 + 
40*e^2*x^4)))/c^5 + (3*b*(32*c^4*d^4*Sqrt[1 + d/(e*x^2)]*AppellF1[1, 1/2, 
1/2, 2, 1/(c^2*x^2), -(d/(e*x^2))] + (e*(35*c^6*d^3 - 35*c^4*d^2*e - 63*c^ 
2*d*e^2 - 25*e^3)*Sqrt[1 - 1/(c^2*x^2)]*x^4*Sqrt[1 + (e*x^2)/d]*AppellF1[1 
, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)])/Sqrt[1 - c^2*x^2]))/(c^5*x) + 96*b* 
(d + e*x^2)^3*(-2*d + 5*e*x^2)*ArcCsc[c*x])/(3360*e^2*Sqrt[d + e*x^2])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.91, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {5762, 27, 435, 171, 27, 171, 27, 171, 27, 175, 66, 104, 217, 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.

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

\(\Big \downarrow \) 5762

\(\displaystyle \frac {b c x \int -\frac {\left (2 d-5 e x^2\right ) \left (e x^2+d\right )^{5/2}}{35 e^2 x \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \int \frac {\left (2 d-5 e x^2\right ) \left (e x^2+d\right )^{5/2}}{x \sqrt {c^2 x^2-1}}dx}{35 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 435

\(\displaystyle -\frac {b c x \int \frac {\left (2 d-5 e x^2\right ) \left (e x^2+d\right )^{5/2}}{x^2 \sqrt {c^2 x^2-1}}dx^2}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 171

\(\displaystyle -\frac {b c x \left (\frac {\int \frac {\left (e x^2+d\right )^{3/2} \left (12 c^2 d^2-e \left (13 d c^2+25 e\right ) x^2\right )}{2 x^2 \sqrt {c^2 x^2-1}}dx^2}{3 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \left (\frac {\int \frac {\left (e x^2+d\right )^{3/2} \left (12 c^2 d^2-e \left (13 d c^2+25 e\right ) x^2\right )}{x^2 \sqrt {c^2 x^2-1}}dx^2}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 171

\(\displaystyle -\frac {b c x \left (\frac {\frac {\int \frac {3 \sqrt {e x^2+d} \left (16 d^3 c^4+e \left (3 d^2 c^4-38 d e c^2-25 e^2\right ) x^2\right )}{2 x^2 \sqrt {c^2 x^2-1}}dx^2}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \left (\frac {\frac {3 \int \frac {\sqrt {e x^2+d} \left (16 d^3 c^4+e \left (3 d^2 c^4-38 d e c^2-25 e^2\right ) x^2\right )}{x^2 \sqrt {c^2 x^2-1}}dx^2}{4 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 171

\(\displaystyle -\frac {b c x \left (\frac {\frac {3 \left (\frac {\int \frac {32 d^4 c^6+e \left (35 d^3 c^6-35 d^2 e c^4-63 d e^2 c^2-25 e^3\right ) x^2}{2 x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{c^2}+\frac {e \sqrt {c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \left (\frac {\frac {3 \left (\frac {\int \frac {32 d^4 c^6+e \left (35 d^3 c^6-35 d^2 e c^4-63 d e^2 c^2-25 e^3\right ) x^2}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 175

\(\displaystyle -\frac {b c x \left (\frac {\frac {3 \left (\frac {32 c^6 d^4 \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {b c x \left (\frac {\frac {3 \left (\frac {32 c^6 d^4 \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+2 e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {b c x \left (\frac {\frac {3 \left (\frac {64 c^6 d^4 \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}+2 e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {b c x \left (\frac {\frac {3 \left (\frac {2 e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}-64 c^6 d^{7/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 c^2}+\frac {e \sqrt {c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {b c x \left (\frac {\frac {3 \left (\frac {\frac {2 \sqrt {e} \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c}-64 c^6 d^{7/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 c^2}+\frac {e \sqrt {c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{c^2}\right )}{4 c^2}-\frac {e \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{2 c^2}}{6 c^2}-\frac {5 e \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{3 c^2}\right )}{70 e^2 \sqrt {c^2 x^2}}\)

Input:

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

Output:

-1/5*(d*(d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/e^2 + ((d + e*x^2)^(7/2)*(a 
 + b*ArcCsc[c*x]))/(7*e^2) - (b*c*x*((-5*e*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^ 
(5/2))/(3*c^2) + (-1/2*(e*(13*c^2*d + 25*e)*Sqrt[-1 + c^2*x^2]*(d + e*x^2) 
^(3/2))/c^2 + (3*((e*(3*c^4*d^2 - 38*c^2*d*e - 25*e^2)*Sqrt[-1 + c^2*x^2]* 
Sqrt[d + e*x^2])/c^2 + (-64*c^6*d^(7/2)*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sq 
rt[-1 + c^2*x^2])] + (2*Sqrt[e]*(35*c^6*d^3 - 35*c^4*d^2*e - 63*c^2*d*e^2 
- 25*e^3)*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/c)/(2 
*c^2)))/(4*c^2))/(6*c^2)))/(70*e^2*Sqrt[c^2*x^2])
 

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

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 435

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

rule 5762

Int[((a_.) + ArcCsc[(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]}, Sim 
p[(a + b*ArcCsc[c*x])   u, x] + Simp[b*c*(x/Sqrt[c^2*x^2])   Int[SimplifyIn 
tegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{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]

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

Input:

int(x^3*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x)
 

Output:

int(x^3*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x)),x)
 

Fricas [A] (verification not implemented)

Time = 2.24 (sec) , antiderivative size = 1697, normalized size of antiderivative = 4.54 \[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

[1/6720*(96*b*c^7*sqrt(-d)*d^3*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c 
^2*d^2 - d*e)*x^2 - 4*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 
 + d)*sqrt(-d) + 8*d^2)/x^4) - 3*(35*b*c^6*d^3 - 35*b*c^4*d^2*e - 63*b*c^2 
*d*e^2 - 25*b*e^3)*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^ 
4*d*e - c^2*e^2)*x^2 + 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqr 
t(e*x^2 + d)*sqrt(e) + e^2) + 4*(240*a*c^7*e^3*x^6 + 384*a*c^7*d*e^2*x^4 + 
 48*a*c^7*d^2*e*x^2 - 96*a*c^7*d^3 + 48*(5*b*c^7*e^3*x^6 + 8*b*c^7*d*e^2*x 
^4 + b*c^7*d^2*e*x^2 - 2*b*c^7*d^3)*arccsc(c*x) + (40*b*c^5*e^3*x^4 + 57*b 
*c^5*d^2*e + 164*b*c^3*d*e^2 + 75*b*c*e^3 + 2*(53*b*c^5*d*e^2 + 25*b*c^3*e 
^3)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^7*e^2), 1/6720*(192*b*c^7* 
d^(7/2)*arctan(-1/2*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + 
 d)*sqrt(d)/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) - 3*(35*b*c^6*d^3 - 
 35*b*c^4*d^2*e - 63*b*c^2*d*e^2 - 25*b*e^3)*sqrt(e)*log(8*c^4*e^2*x^4 + c 
^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c^2*e^2)*x^2 + 4*(2*c^3*e*x^2 + c^3*d - 
c*e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(e) + e^2) + 4*(240*a*c^7*e^3*x 
^6 + 384*a*c^7*d*e^2*x^4 + 48*a*c^7*d^2*e*x^2 - 96*a*c^7*d^3 + 48*(5*b*c^7 
*e^3*x^6 + 8*b*c^7*d*e^2*x^4 + b*c^7*d^2*e*x^2 - 2*b*c^7*d^3)*arccsc(c*x) 
+ (40*b*c^5*e^3*x^4 + 57*b*c^5*d^2*e + 164*b*c^3*d*e^2 + 75*b*c*e^3 + 2*(5 
3*b*c^5*d*e^2 + 25*b*c^3*e^3)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^ 
7*e^2), 1/3360*(48*b*c^7*sqrt(-d)*d^3*log(((c^4*d^2 - 6*c^2*d*e + e^2)*...
 

Sympy [F(-1)]

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

integrate(x**3*(e*x**2+d)**(3/2)*(a+b*acsc(c*x)),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

integrate((e*x^2 + d)^(3/2)*(b*arccsc(c*x) + a)*x^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {-2 \sqrt {e \,x^{2}+d}\, a \,d^{3}+\sqrt {e \,x^{2}+d}\, a \,d^{2} e \,x^{2}+8 \sqrt {e \,x^{2}+d}\, a d \,e^{2} x^{4}+5 \sqrt {e \,x^{2}+d}\, a \,e^{3} x^{6}+35 \left (\int \sqrt {e \,x^{2}+d}\, \mathit {acsc} \left (c x \right ) x^{5}d x \right ) b \,e^{3}+35 \left (\int \sqrt {e \,x^{2}+d}\, \mathit {acsc} \left (c x \right ) x^{3}d x \right ) b d \,e^{2}}{35 e^{2}} \] Input:

int(x^3*(e*x^2+d)^(3/2)*(a+b*acsc(c*x)),x)
 

Output:

( - 2*sqrt(d + e*x**2)*a*d**3 + sqrt(d + e*x**2)*a*d**2*e*x**2 + 8*sqrt(d 
+ e*x**2)*a*d*e**2*x**4 + 5*sqrt(d + e*x**2)*a*e**3*x**6 + 35*int(sqrt(d + 
 e*x**2)*acsc(c*x)*x**5,x)*b*e**3 + 35*int(sqrt(d + e*x**2)*acsc(c*x)*x**3 
,x)*b*d*e**2)/(35*e**2)