Optimal. Leaf size=618 \[ -\frac {15 b c d^2 x^{m+2} \sqrt {c^2 d x^2+d} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;-c^2 x^2\right )}{(m+1) (m+2)^2 (m+4) (m+6) \sqrt {c^2 x^2+1}}+\frac {15 d^2 x^{m+1} \sqrt {c^2 d x^2+d} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{(m+6) \left (m^3+7 m^2+14 m+8\right ) \sqrt {c^2 x^2+1}}+\frac {15 d^2 x^{m+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{(m+6) \left (m^2+6 m+8\right )}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{m+6}+\frac {5 d x^{m+1} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{(m+4) (m+6)}-\frac {5 b c d^2 x^{m+2} \sqrt {c^2 d x^2+d}}{(m+6) \left (m^2+6 m+8\right ) \sqrt {c^2 x^2+1}}-\frac {b c d^2 x^{m+2} \sqrt {c^2 d x^2+d}}{\left (m^2+8 m+12\right ) \sqrt {c^2 x^2+1}}-\frac {15 b c d^2 x^{m+2} \sqrt {c^2 d x^2+d}}{(m+2)^2 (m+4) (m+6) \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^{m+6} \sqrt {c^2 d x^2+d}}{(m+6)^2 \sqrt {c^2 x^2+1}}-\frac {2 b c^3 d^2 x^{m+4} \sqrt {c^2 d x^2+d}}{(m+4) (m+6) \sqrt {c^2 x^2+1}}-\frac {5 b c^3 d^2 x^{m+4} \sqrt {c^2 d x^2+d}}{(m+4)^2 (m+6) \sqrt {c^2 x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.56, antiderivative size = 618, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5744, 5742, 5762, 30, 14, 270} \[ -\frac {15 b c d^2 x^{m+2} \sqrt {c^2 d x^2+d} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;-c^2 x^2\right )}{(m+1) (m+2)^2 (m+4) (m+6) \sqrt {c^2 x^2+1}}+\frac {15 d^2 x^{m+1} \sqrt {c^2 d x^2+d} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{(m+6) \left (m^3+7 m^2+14 m+8\right ) \sqrt {c^2 x^2+1}}+\frac {15 d^2 x^{m+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{(m+6) \left (m^2+6 m+8\right )}+\frac {x^{m+1} \left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{m+6}+\frac {5 d x^{m+1} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{(m+4) (m+6)}-\frac {5 b c d^2 x^{m+2} \sqrt {c^2 d x^2+d}}{(m+6) \left (m^2+6 m+8\right ) \sqrt {c^2 x^2+1}}-\frac {b c d^2 x^{m+2} \sqrt {c^2 d x^2+d}}{\left (m^2+8 m+12\right ) \sqrt {c^2 x^2+1}}-\frac {15 b c d^2 x^{m+2} \sqrt {c^2 d x^2+d}}{(m+2)^2 (m+4) (m+6) \sqrt {c^2 x^2+1}}-\frac {2 b c^3 d^2 x^{m+4} \sqrt {c^2 d x^2+d}}{(m+4) (m+6) \sqrt {c^2 x^2+1}}-\frac {5 b c^3 d^2 x^{m+4} \sqrt {c^2 d x^2+d}}{(m+4)^2 (m+6) \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^{m+6} \sqrt {c^2 d x^2+d}}{(m+6)^2 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 270
Rule 5742
Rule 5744
Rule 5762
Rubi steps
\begin {align*} \int x^m \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {x^{1+m} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{6+m}+\frac {(5 d) \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{6+m}-\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int x^{1+m} \left (1+c^2 x^2\right )^2 \, dx}{(6+m) \sqrt {1+c^2 x^2}}\\ &=\frac {5 d x^{1+m} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{(4+m) (6+m)}+\frac {x^{1+m} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{6+m}+\frac {\left (15 d^2\right ) \int x^m \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{(4+m) (6+m)}-\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^{1+m}+2 c^2 x^{3+m}+c^4 x^{5+m}\right ) \, dx}{(6+m) \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int x^{1+m} \left (1+c^2 x^2\right ) \, dx}{(4+m) (6+m) \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 x^{2+m} \sqrt {d+c^2 d x^2}}{\left (12+8 m+m^2\right ) \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d^2 x^{4+m} \sqrt {d+c^2 d x^2}}{(4+m) (6+m) \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^{6+m} \sqrt {d+c^2 d x^2}}{(6+m)^2 \sqrt {1+c^2 x^2}}+\frac {15 d^2 x^{1+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(2+m) (4+m) (6+m)}+\frac {5 d x^{1+m} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{(4+m) (6+m)}+\frac {x^{1+m} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{6+m}-\frac {\left (5 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^{1+m}+c^2 x^{3+m}\right ) \, dx}{(4+m) (6+m) \sqrt {1+c^2 x^2}}+\frac {\left (15 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^m \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{(2+m) (4+m) (6+m) \sqrt {1+c^2 x^2}}-\frac {\left (15 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int x^{1+m} \, dx}{(2+m) (4+m) (6+m) \sqrt {1+c^2 x^2}}\\ &=-\frac {15 b c d^2 x^{2+m} \sqrt {d+c^2 d x^2}}{(2+m)^2 (4+m) (6+m) \sqrt {1+c^2 x^2}}-\frac {5 b c d^2 x^{2+m} \sqrt {d+c^2 d x^2}}{(2+m) (4+m) (6+m) \sqrt {1+c^2 x^2}}-\frac {b c d^2 x^{2+m} \sqrt {d+c^2 d x^2}}{\left (12+8 m+m^2\right ) \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 x^{4+m} \sqrt {d+c^2 d x^2}}{(4+m)^2 (6+m) \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d^2 x^{4+m} \sqrt {d+c^2 d x^2}}{(4+m) (6+m) \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^{6+m} \sqrt {d+c^2 d x^2}}{(6+m)^2 \sqrt {1+c^2 x^2}}+\frac {15 d^2 x^{1+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(2+m) (4+m) (6+m)}+\frac {5 d x^{1+m} \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{(4+m) (6+m)}+\frac {x^{1+m} \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{6+m}+\frac {15 d^2 x^{1+m} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};-c^2 x^2\right )}{(1+m) (2+m) (4+m) (6+m) \sqrt {1+c^2 x^2}}-\frac {15 b c d^2 x^{2+m} \sqrt {d+c^2 d x^2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};-c^2 x^2\right )}{(1+m) (2+m)^2 (4+m) (6+m) \sqrt {1+c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.28, size = 332, normalized size = 0.54 \[ \frac {d^2 x^{m+1} \sqrt {c^2 d x^2+d} \left (-\frac {5 \left (3 (m+4) \left (b c x \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;-c^2 x^2\right )-(m+2) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};-c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )-(m+1) (m+2) \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+b c (m+1) x\right )-\left ((m+1) (m+4) (m+2)^2 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )\right )+b c (m+1) (m+2) x \left (c^2 (m+2) x^2+m+4\right )\right )}{(m+1) (m+2)^2 (m+4)^2 \sqrt {c^2 x^2+1}}+\left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c x \left (c^4 (m+2) (m+4) x^4+2 c^2 (m+2) (m+6) x^2+(m+4) (m+6)\right )}{(m+2) (m+4) (m+6) \sqrt {c^2 x^2+1}}\right )}{m+6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} x^{4} + 2 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{4} d^{2} x^{4} + 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.64, size = 0, normalized size = 0.00 \[ \int x^{m} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^m\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________