Optimal. Leaf size=183 \[ \frac {2 b^2 (e (c+d x))^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};(c+d x)^2\right )}{d e^3 (m+1) (m+2) (m+3)}-\frac {2 b (e (c+d x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e^2 (m+1) (m+2)}+\frac {(e (c+d x))^{m+1} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4805, 4627, 4711} \[ \frac {2 b^2 (e (c+d x))^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};(c+d x)^2\right )}{d e^3 (m+1) (m+2) (m+3)}-\frac {2 b (e (c+d x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{d e^2 (m+1) (m+2)}+\frac {(e (c+d x))^{m+1} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4627
Rule 4711
Rule 4805
Rubi steps
\begin {align*} \int (c e+d e x)^m \left (a+b \sin ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^m \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e (c+d x))^{1+m} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e (1+m)}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(e x)^{1+m} \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d e (1+m)}\\ &=\frac {(e (c+d x))^{1+m} \left (a+b \sin ^{-1}(c+d x)\right )^2}{d e (1+m)}-\frac {2 b (e (c+d x))^{2+m} \left (a+b \sin ^{-1}(c+d x)\right ) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};(c+d x)^2\right )}{d e^2 (1+m) (2+m)}+\frac {2 b^2 (e (c+d x))^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};(c+d x)^2\right )}{d e^3 (1+m) (2+m) (3+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 151, normalized size = 0.83 \[ \frac {(c+d x) (e (c+d x))^m \left (\frac {2 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};(c+d x)^2\right )}{(m+2) (m+3)}-\frac {2 b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{m+2}+\left (a+b \sin ^{-1}(c+d x)\right )^2\right )}{d (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \arcsin \left (d x + c\right )^{2} + 2 \, a b \arcsin \left (d x + c\right ) + a^{2}\right )} {\left (d e x + c e\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )}^{2} {\left (d e x + c e\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.58, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{m} \left (a +b \arcsin \left (d x +c \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b^{2} d e^{m} x + b^{2} c e^{m}\right )} {\left (d x + c\right )}^{m} \arctan \left (d x + c, \sqrt {d x + c + 1} \sqrt {-d x - c + 1}\right )^{2} - 2 \, {\left (a b d e^{m} x + a b c e^{m}\right )} {\left (d x + c\right )}^{m} \arctan \left (\sqrt {d x + c + 1} {\left (d x + c\right )} \sqrt {-d x - c + 1}, d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right ) - 2 \, {\left (d m + d\right )} \int \frac {{\left (a b d e^{m} x + a b c e^{m}\right )} e^{\left (m \log \left (d x + c\right ) + \frac {1}{2} \, \log \left (d x + c + 1\right ) + \frac {1}{2} \, \log \left (-d x - c + 1\right )\right )}}{{\left (d^{4} m + d^{4}\right )} x^{4} + c^{4} + 4 \, {\left (c d^{3} m + c d^{3}\right )} x^{3} + 2 \, {\left ({\left (3 \, c^{2} - 1\right )} d^{2} m + {\left (3 \, c^{2} - 1\right )} d^{2}\right )} x^{2} - 2 \, c^{2} + {\left (c^{4} - 2 \, c^{2} + 1\right )} m + 4 \, {\left ({\left (c^{3} - c\right )} d m + {\left (c^{3} - c\right )} d\right )} x + {\left (c^{2} m + {\left (d^{2} m + d^{2}\right )} x^{2} + c^{2} + 2 \, {\left (c d m + c d\right )} x\right )} e^{\left (\log \left (d x + c + 1\right ) + \log \left (-d x - c + 1\right )\right )} + 1}\,{d x}}{d m + d} + \frac {{\left (d e x + c e\right )}^{m + 1} a^{2}}{d e {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,e+d\,e\,x\right )}^m\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________