3.791 \(\int (d x)^m (a^2+2 a b x^2+b^2 x^4)^{5/2} \, dx\)

Optimal. Leaf size=313 \[ \frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+11}}{d^{11} (m+11) \left (a+b x^2\right )}+\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+9}}{d^9 (m+9) \left (a+b x^2\right )}+\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )}+\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )} \]

[Out]

a^5*(d*x)^(1+m)*((b*x^2+a)^2)^(1/2)/d/(1+m)/(b*x^2+a)+5*a^4*b*(d*x)^(3+m)*((b*x^2+a)^2)^(1/2)/d^3/(3+m)/(b*x^2
+a)+10*a^3*b^2*(d*x)^(5+m)*((b*x^2+a)^2)^(1/2)/d^5/(5+m)/(b*x^2+a)+10*a^2*b^3*(d*x)^(7+m)*((b*x^2+a)^2)^(1/2)/
d^7/(7+m)/(b*x^2+a)+5*a*b^4*(d*x)^(9+m)*((b*x^2+a)^2)^(1/2)/d^9/(9+m)/(b*x^2+a)+b^5*(d*x)^(11+m)*((b*x^2+a)^2)
^(1/2)/d^11/(11+m)/(b*x^2+a)

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Rubi [A]  time = 0.12, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1112, 270} \[ \frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac {10 a^3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )}+\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac {5 a b^4 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+9}}{d^9 (m+9) \left (a+b x^2\right )}+\frac {b^5 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+11}}{d^{11} (m+11) \left (a+b x^2\right )}+\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(a^5*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d*(1 + m)*(a + b*x^2)) + (5*a^4*b*(d*x)^(3 + m)*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(d^3*(3 + m)*(a + b*x^2)) + (10*a^3*b^2*(d*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
/(d^5*(5 + m)*(a + b*x^2)) + (10*a^2*b^3*(d*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^7*(7 + m)*(a + b*x^
2)) + (5*a*b^4*(d*x)^(9 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^9*(9 + m)*(a + b*x^2)) + (b^5*(d*x)^(11 + m)*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(d^11*(11 + m)*(a + b*x^2))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

\begin {align*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (d x)^m \left (a b+b^2 x^2\right )^5 \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^5 b^5 (d x)^m+\frac {5 a^4 b^6 (d x)^{2+m}}{d^2}+\frac {10 a^3 b^7 (d x)^{4+m}}{d^4}+\frac {10 a^2 b^8 (d x)^{6+m}}{d^6}+\frac {5 a b^9 (d x)^{8+m}}{d^8}+\frac {b^{10} (d x)^{10+m}}{d^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {a^5 (d x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d (1+m) \left (a+b x^2\right )}+\frac {5 a^4 b (d x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 (3+m) \left (a+b x^2\right )}+\frac {10 a^3 b^2 (d x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 (5+m) \left (a+b x^2\right )}+\frac {10 a^2 b^3 (d x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^7 (7+m) \left (a+b x^2\right )}+\frac {5 a b^4 (d x)^{9+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^9 (9+m) \left (a+b x^2\right )}+\frac {b^5 (d x)^{11+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^{11} (11+m) \left (a+b x^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 111, normalized size = 0.35 \[ \frac {x \left (\left (a+b x^2\right )^2\right )^{5/2} (d x)^m \left (\frac {a^5}{m+1}+\frac {5 a^4 b x^2}{m+3}+\frac {10 a^3 b^2 x^4}{m+5}+\frac {10 a^2 b^3 x^6}{m+7}+\frac {5 a b^4 x^8}{m+9}+\frac {b^5 x^{10}}{m+11}\right )}{\left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x*(d*x)^m*((a + b*x^2)^2)^(5/2)*(a^5/(1 + m) + (5*a^4*b*x^2)/(3 + m) + (10*a^3*b^2*x^4)/(5 + m) + (10*a^2*b^3
*x^6)/(7 + m) + (5*a*b^4*x^8)/(9 + m) + (b^5*x^10)/(11 + m)))/(a + b*x^2)^5

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fricas [A]  time = 0.86, size = 369, normalized size = 1.18 \[ \frac {{\left ({\left (b^{5} m^{5} + 25 \, b^{5} m^{4} + 230 \, b^{5} m^{3} + 950 \, b^{5} m^{2} + 1689 \, b^{5} m + 945 \, b^{5}\right )} x^{11} + 5 \, {\left (a b^{4} m^{5} + 27 \, a b^{4} m^{4} + 262 \, a b^{4} m^{3} + 1122 \, a b^{4} m^{2} + 2041 \, a b^{4} m + 1155 \, a b^{4}\right )} x^{9} + 10 \, {\left (a^{2} b^{3} m^{5} + 29 \, a^{2} b^{3} m^{4} + 302 \, a^{2} b^{3} m^{3} + 1366 \, a^{2} b^{3} m^{2} + 2577 \, a^{2} b^{3} m + 1485 \, a^{2} b^{3}\right )} x^{7} + 10 \, {\left (a^{3} b^{2} m^{5} + 31 \, a^{3} b^{2} m^{4} + 350 \, a^{3} b^{2} m^{3} + 1730 \, a^{3} b^{2} m^{2} + 3489 \, a^{3} b^{2} m + 2079 \, a^{3} b^{2}\right )} x^{5} + 5 \, {\left (a^{4} b m^{5} + 33 \, a^{4} b m^{4} + 406 \, a^{4} b m^{3} + 2262 \, a^{4} b m^{2} + 5353 \, a^{4} b m + 3465 \, a^{4} b\right )} x^{3} + {\left (a^{5} m^{5} + 35 \, a^{5} m^{4} + 470 \, a^{5} m^{3} + 3010 \, a^{5} m^{2} + 9129 \, a^{5} m + 10395 \, a^{5}\right )} x\right )} \left (d x\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

((b^5*m^5 + 25*b^5*m^4 + 230*b^5*m^3 + 950*b^5*m^2 + 1689*b^5*m + 945*b^5)*x^11 + 5*(a*b^4*m^5 + 27*a*b^4*m^4
+ 262*a*b^4*m^3 + 1122*a*b^4*m^2 + 2041*a*b^4*m + 1155*a*b^4)*x^9 + 10*(a^2*b^3*m^5 + 29*a^2*b^3*m^4 + 302*a^2
*b^3*m^3 + 1366*a^2*b^3*m^2 + 2577*a^2*b^3*m + 1485*a^2*b^3)*x^7 + 10*(a^3*b^2*m^5 + 31*a^3*b^2*m^4 + 350*a^3*
b^2*m^3 + 1730*a^3*b^2*m^2 + 3489*a^3*b^2*m + 2079*a^3*b^2)*x^5 + 5*(a^4*b*m^5 + 33*a^4*b*m^4 + 406*a^4*b*m^3
+ 2262*a^4*b*m^2 + 5353*a^4*b*m + 3465*a^4*b)*x^3 + (a^5*m^5 + 35*a^5*m^4 + 470*a^5*m^3 + 3010*a^5*m^2 + 9129*
a^5*m + 10395*a^5)*x)*(d*x)^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

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giac [B]  time = 0.28, size = 900, normalized size = 2.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

((d*x)^m*b^5*m^5*x^11*sgn(b*x^2 + a) + 25*(d*x)^m*b^5*m^4*x^11*sgn(b*x^2 + a) + 5*(d*x)^m*a*b^4*m^5*x^9*sgn(b*
x^2 + a) + 230*(d*x)^m*b^5*m^3*x^11*sgn(b*x^2 + a) + 135*(d*x)^m*a*b^4*m^4*x^9*sgn(b*x^2 + a) + 950*(d*x)^m*b^
5*m^2*x^11*sgn(b*x^2 + a) + 10*(d*x)^m*a^2*b^3*m^5*x^7*sgn(b*x^2 + a) + 1310*(d*x)^m*a*b^4*m^3*x^9*sgn(b*x^2 +
a) + 1689*(d*x)^m*b^5*m*x^11*sgn(b*x^2 + a) + 290*(d*x)^m*a^2*b^3*m^4*x^7*sgn(b*x^2 + a) + 5610*(d*x)^m*a*b^4
*m^2*x^9*sgn(b*x^2 + a) + 945*(d*x)^m*b^5*x^11*sgn(b*x^2 + a) + 10*(d*x)^m*a^3*b^2*m^5*x^5*sgn(b*x^2 + a) + 30
20*(d*x)^m*a^2*b^3*m^3*x^7*sgn(b*x^2 + a) + 10205*(d*x)^m*a*b^4*m*x^9*sgn(b*x^2 + a) + 310*(d*x)^m*a^3*b^2*m^4
*x^5*sgn(b*x^2 + a) + 13660*(d*x)^m*a^2*b^3*m^2*x^7*sgn(b*x^2 + a) + 5775*(d*x)^m*a*b^4*x^9*sgn(b*x^2 + a) + 5
*(d*x)^m*a^4*b*m^5*x^3*sgn(b*x^2 + a) + 3500*(d*x)^m*a^3*b^2*m^3*x^5*sgn(b*x^2 + a) + 25770*(d*x)^m*a^2*b^3*m*
x^7*sgn(b*x^2 + a) + 165*(d*x)^m*a^4*b*m^4*x^3*sgn(b*x^2 + a) + 17300*(d*x)^m*a^3*b^2*m^2*x^5*sgn(b*x^2 + a) +
14850*(d*x)^m*a^2*b^3*x^7*sgn(b*x^2 + a) + (d*x)^m*a^5*m^5*x*sgn(b*x^2 + a) + 2030*(d*x)^m*a^4*b*m^3*x^3*sgn(
b*x^2 + a) + 34890*(d*x)^m*a^3*b^2*m*x^5*sgn(b*x^2 + a) + 35*(d*x)^m*a^5*m^4*x*sgn(b*x^2 + a) + 11310*(d*x)^m*
a^4*b*m^2*x^3*sgn(b*x^2 + a) + 20790*(d*x)^m*a^3*b^2*x^5*sgn(b*x^2 + a) + 470*(d*x)^m*a^5*m^3*x*sgn(b*x^2 + a)
+ 26765*(d*x)^m*a^4*b*m*x^3*sgn(b*x^2 + a) + 3010*(d*x)^m*a^5*m^2*x*sgn(b*x^2 + a) + 17325*(d*x)^m*a^4*b*x^3*
sgn(b*x^2 + a) + 9129*(d*x)^m*a^5*m*x*sgn(b*x^2 + a) + 10395*(d*x)^m*a^5*x*sgn(b*x^2 + a))/(m^6 + 36*m^5 + 505
*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

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maple [A]  time = 0.01, size = 453, normalized size = 1.45 \[ \frac {\left (b^{5} m^{5} x^{10}+25 b^{5} m^{4} x^{10}+5 a \,b^{4} m^{5} x^{8}+230 b^{5} m^{3} x^{10}+135 a \,b^{4} m^{4} x^{8}+950 b^{5} m^{2} x^{10}+10 a^{2} b^{3} m^{5} x^{6}+1310 a \,b^{4} m^{3} x^{8}+1689 b^{5} m \,x^{10}+290 a^{2} b^{3} m^{4} x^{6}+5610 a \,b^{4} m^{2} x^{8}+945 b^{5} x^{10}+10 a^{3} b^{2} m^{5} x^{4}+3020 a^{2} b^{3} m^{3} x^{6}+10205 a \,b^{4} m \,x^{8}+310 a^{3} b^{2} m^{4} x^{4}+13660 a^{2} b^{3} m^{2} x^{6}+5775 a \,b^{4} x^{8}+5 a^{4} b \,m^{5} x^{2}+3500 a^{3} b^{2} m^{3} x^{4}+25770 a^{2} b^{3} m \,x^{6}+165 a^{4} b \,m^{4} x^{2}+17300 a^{3} b^{2} m^{2} x^{4}+14850 a^{2} b^{3} x^{6}+a^{5} m^{5}+2030 a^{4} b \,m^{3} x^{2}+34890 a^{3} b^{2} m \,x^{4}+35 a^{5} m^{4}+11310 a^{4} b \,m^{2} x^{2}+20790 a^{3} b^{2} x^{4}+470 a^{5} m^{3}+26765 a^{4} b m \,x^{2}+3010 a^{5} m^{2}+17325 a^{4} b \,x^{2}+9129 a^{5} m +10395 a^{5}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} x \left (d x \right )^{m}}{\left (m +11\right ) \left (m +9\right ) \left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right ) \left (b \,x^{2}+a \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

x*(b^5*m^5*x^10+25*b^5*m^4*x^10+5*a*b^4*m^5*x^8+230*b^5*m^3*x^10+135*a*b^4*m^4*x^8+950*b^5*m^2*x^10+10*a^2*b^3
*m^5*x^6+1310*a*b^4*m^3*x^8+1689*b^5*m*x^10+290*a^2*b^3*m^4*x^6+5610*a*b^4*m^2*x^8+945*b^5*x^10+10*a^3*b^2*m^5
*x^4+3020*a^2*b^3*m^3*x^6+10205*a*b^4*m*x^8+310*a^3*b^2*m^4*x^4+13660*a^2*b^3*m^2*x^6+5775*a*b^4*x^8+5*a^4*b*m
^5*x^2+3500*a^3*b^2*m^3*x^4+25770*a^2*b^3*m*x^6+165*a^4*b*m^4*x^2+17300*a^3*b^2*m^2*x^4+14850*a^2*b^3*x^6+a^5*
m^5+2030*a^4*b*m^3*x^2+34890*a^3*b^2*m*x^4+35*a^5*m^4+11310*a^4*b*m^2*x^2+20790*a^3*b^2*x^4+470*a^5*m^3+26765*
a^4*b*m*x^2+3010*a^5*m^2+17325*a^4*b*x^2+9129*a^5*m+10395*a^5)*(d*x)^m*((b*x^2+a)^2)^(5/2)/(m+11)/(m+9)/(m+7)/
(m+5)/(m+3)/(m+1)/(b*x^2+a)^5

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maxima [A]  time = 1.41, size = 243, normalized size = 0.78 \[ \frac {{\left ({\left (m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945\right )} b^{5} d^{m} x^{11} + 5 \, {\left (m^{5} + 27 \, m^{4} + 262 \, m^{3} + 1122 \, m^{2} + 2041 \, m + 1155\right )} a b^{4} d^{m} x^{9} + 10 \, {\left (m^{5} + 29 \, m^{4} + 302 \, m^{3} + 1366 \, m^{2} + 2577 \, m + 1485\right )} a^{2} b^{3} d^{m} x^{7} + 10 \, {\left (m^{5} + 31 \, m^{4} + 350 \, m^{3} + 1730 \, m^{2} + 3489 \, m + 2079\right )} a^{3} b^{2} d^{m} x^{5} + 5 \, {\left (m^{5} + 33 \, m^{4} + 406 \, m^{3} + 2262 \, m^{2} + 5353 \, m + 3465\right )} a^{4} b d^{m} x^{3} + {\left (m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395\right )} a^{5} d^{m} x\right )} x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

((m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)*b^5*d^m*x^11 + 5*(m^5 + 27*m^4 + 262*m^3 + 1122*m^2 + 2041*
m + 1155)*a*b^4*d^m*x^9 + 10*(m^5 + 29*m^4 + 302*m^3 + 1366*m^2 + 2577*m + 1485)*a^2*b^3*d^m*x^7 + 10*(m^5 + 3
1*m^4 + 350*m^3 + 1730*m^2 + 3489*m + 2079)*a^3*b^2*d^m*x^5 + 5*(m^5 + 33*m^4 + 406*m^3 + 2262*m^2 + 5353*m +
3465)*a^4*b*d^m*x^3 + (m^5 + 35*m^4 + 470*m^3 + 3010*m^2 + 9129*m + 10395)*a^5*d^m*x)*x^m/(m^6 + 36*m^5 + 505*
m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d\,x\right )}^m\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)

[Out]

int((d*x)^m*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral((d*x)**m*((a + b*x**2)**2)**(5/2), x)

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