3.1.87 \(\int (f x)^m (d+e x^2) (a^2+2 a b x^2+b^2 x^4)^{5/2} \, dx\) [87]

Optimal. Leaf size=400 \[ \frac {a^5 d (f x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f (1+m) \left (a+b x^2\right )}+\frac {a^4 (5 b d+a e) (f x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^3 (3+m) \left (a+b x^2\right )}+\frac {5 a^3 b (2 b d+a e) (f x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^5 (5+m) \left (a+b x^2\right )}+\frac {10 a^2 b^2 (b d+a e) (f x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^7 (7+m) \left (a+b x^2\right )}+\frac {5 a b^3 (b d+2 a e) (f x)^{9+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^9 (9+m) \left (a+b x^2\right )}+\frac {b^4 (b d+5 a e) (f x)^{11+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^{11} (11+m) \left (a+b x^2\right )}+\frac {b^5 e (f x)^{13+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^{13} (13+m) \left (a+b x^2\right )} \]

[Out]

a^5*d*(f*x)^(1+m)*((b*x^2+a)^2)^(1/2)/f/(1+m)/(b*x^2+a)+a^4*(a*e+5*b*d)*(f*x)^(3+m)*((b*x^2+a)^2)^(1/2)/f^3/(3
+m)/(b*x^2+a)+5*a^3*b*(a*e+2*b*d)*(f*x)^(5+m)*((b*x^2+a)^2)^(1/2)/f^5/(5+m)/(b*x^2+a)+10*a^2*b^2*(a*e+b*d)*(f*
x)^(7+m)*((b*x^2+a)^2)^(1/2)/f^7/(7+m)/(b*x^2+a)+5*a*b^3*(2*a*e+b*d)*(f*x)^(9+m)*((b*x^2+a)^2)^(1/2)/f^9/(9+m)
/(b*x^2+a)+b^4*(5*a*e+b*d)*(f*x)^(11+m)*((b*x^2+a)^2)^(1/2)/f^11/(11+m)/(b*x^2+a)+b^5*e*(f*x)^(13+m)*((b*x^2+a
)^2)^(1/2)/f^13/(13+m)/(b*x^2+a)

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Rubi [A]
time = 0.16, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1264, 459} \begin {gather*} \frac {10 a^2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+7} (a e+b d)}{f^7 (m+7) \left (a+b x^2\right )}+\frac {b^5 e \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+13}}{f^{13} (m+13) \left (a+b x^2\right )}+\frac {b^4 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+11} (5 a e+b d)}{f^{11} (m+11) \left (a+b x^2\right )}+\frac {5 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+9} (2 a e+b d)}{f^9 (m+9) \left (a+b x^2\right )}+\frac {a^5 d \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+1}}{f (m+1) \left (a+b x^2\right )}+\frac {a^4 \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+3} (a e+5 b d)}{f^3 (m+3) \left (a+b x^2\right )}+\frac {5 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4} (f x)^{m+5} (a e+2 b d)}{f^5 (m+5) \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*d*(f*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f*(1 + m)*(a + b*x^2)) + (a^4*(5*b*d + a*e)*(f*x)^(3 +
m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f^3*(3 + m)*(a + b*x^2)) + (5*a^3*b*(2*b*d + a*e)*(f*x)^(5 + m)*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(f^5*(5 + m)*(a + b*x^2)) + (10*a^2*b^2*(b*d + a*e)*(f*x)^(7 + m)*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4])/(f^7*(7 + m)*(a + b*x^2)) + (5*a*b^3*(b*d + 2*a*e)*(f*x)^(9 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
/(f^9*(9 + m)*(a + b*x^2)) + (b^4*(b*d + 5*a*e)*(f*x)^(11 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f^11*(11 + m)
*(a + b*x^2)) + (b^5*e*(f*x)^(13 + m)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(f^13*(13 + m)*(a + b*x^2))

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 1264

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

Rubi steps

\begin {align*} \int (f x)^m \left (d+e x^2\right ) \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 (f x)^m \left (a b+b^2 x^2\right )^5 \left (d+e x^2\right ) \, 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 (f x)^m+\frac {a^4 b^5 (5 b d+a e) (f x)^{2+m}}{f^2}+\frac {5 a^3 b^6 (2 b d+a e) (f x)^{4+m}}{f^4}+\frac {10 a^2 b^7 (b d+a e) (f x)^{6+m}}{f^6}+\frac {5 a b^8 (b d+2 a e) (f x)^{8+m}}{f^8}+\frac {b^9 (b d+5 a e) (f x)^{10+m}}{f^{10}}+\frac {b^{10} e (f x)^{12+m}}{f^{12}}\right ) \, dx}{b^4 \left (a b+b^2 x^2\right )}\\ &=\frac {a^5 d (f x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f (1+m) \left (a+b x^2\right )}+\frac {a^4 (5 b d+a e) (f x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^3 (3+m) \left (a+b x^2\right )}+\frac {5 a^3 b (2 b d+a e) (f x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^5 (5+m) \left (a+b x^2\right )}+\frac {10 a^2 b^2 (b d+a e) (f x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^7 (7+m) \left (a+b x^2\right )}+\frac {5 a b^3 (b d+2 a e) (f x)^{9+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^9 (9+m) \left (a+b x^2\right )}+\frac {b^4 (b d+5 a e) (f x)^{11+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^{11} (11+m) \left (a+b x^2\right )}+\frac {b^5 e (f x)^{13+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{f^{13} (13+m) \left (a+b x^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.45, size = 160, normalized size = 0.40 \begin {gather*} \frac {x (f x)^m \sqrt {\left (a+b x^2\right )^2} \left (\frac {a^5 d}{1+m}+\frac {a^4 (5 b d+a e) x^2}{3+m}+\frac {5 a^3 b (2 b d+a e) x^4}{5+m}+\frac {10 a^2 b^2 (b d+a e) x^6}{7+m}+\frac {5 a b^3 (b d+2 a e) x^8}{9+m}+\frac {b^4 (b d+5 a e) x^{10}}{11+m}+\frac {b^5 e x^{12}}{13+m}\right )}{a+b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(f*x)^m*Sqrt[(a + b*x^2)^2]*((a^5*d)/(1 + m) + (a^4*(5*b*d + a*e)*x^2)/(3 + m) + (5*a^3*b*(2*b*d + a*e)*x^4
)/(5 + m) + (10*a^2*b^2*(b*d + a*e)*x^6)/(7 + m) + (5*a*b^3*(b*d + 2*a*e)*x^8)/(9 + m) + (b^4*(b*d + 5*a*e)*x^
10)/(11 + m) + (b^5*e*x^12)/(13 + m)))/(a + b*x^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1098\) vs. \(2(323)=646\).
time = 0.04, size = 1099, normalized size = 2.75 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(e*x^2+d)*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

x*(b^5*e*m^6*x^12+36*b^5*e*m^5*x^12+5*a*b^4*e*m^6*x^10+b^5*d*m^6*x^10+505*b^5*e*m^4*x^12+190*a*b^4*e*m^5*x^10+
38*b^5*d*m^5*x^10+3480*b^5*e*m^3*x^12+10*a^2*b^3*e*m^6*x^8+5*a*b^4*d*m^6*x^8+2775*a*b^4*e*m^4*x^10+555*b^5*d*m
^4*x^10+12139*b^5*e*m^2*x^12+400*a^2*b^3*e*m^5*x^8+200*a*b^4*d*m^5*x^8+19700*a*b^4*e*m^3*x^10+3940*b^5*d*m^3*x
^10+19524*b^5*e*m*x^12+10*a^3*b^2*e*m^6*x^6+10*a^2*b^3*d*m^6*x^6+6130*a^2*b^3*e*m^4*x^8+3065*a*b^4*d*m^4*x^8+7
0195*a*b^4*e*m^2*x^10+14039*b^5*d*m^2*x^10+10395*b^5*e*x^12+420*a^3*b^2*e*m^5*x^6+420*a^2*b^3*d*m^5*x^6+45280*
a^2*b^3*e*m^3*x^8+22640*a*b^4*d*m^3*x^8+114510*a*b^4*e*m*x^10+22902*b^5*d*m*x^10+5*a^4*b*e*m^6*x^4+10*a^3*b^2*
d*m^6*x^4+6790*a^3*b^2*e*m^4*x^6+6790*a^2*b^3*d*m^4*x^6+166270*a^2*b^3*e*m^2*x^8+83135*a*b^4*d*m^2*x^8+61425*a
*b^4*e*x^10+12285*b^5*d*x^10+220*a^4*b*e*m^5*x^4+440*a^3*b^2*d*m^5*x^4+52920*a^3*b^2*e*m^3*x^6+52920*a^2*b^3*d
*m^3*x^6+276880*a^2*b^3*e*m*x^8+138440*a*b^4*d*m*x^8+a^5*e*m^6*x^2+5*a^4*b*d*m^6*x^2+3765*a^4*b*e*m^4*x^4+7530
*a^3*b^2*d*m^4*x^4+203350*a^3*b^2*e*m^2*x^6+203350*a^2*b^3*d*m^2*x^6+150150*a^2*b^3*e*x^8+75075*a*b^4*d*x^8+46
*a^5*e*m^5*x^2+230*a^4*b*d*m^5*x^2+31400*a^4*b*e*m^3*x^4+62800*a^3*b^2*d*m^3*x^4+349860*a^3*b^2*e*m*x^6+349860
*a^2*b^3*d*m*x^6+a^5*d*m^6+835*a^5*e*m^4*x^2+4175*a^4*b*d*m^4*x^2+129895*a^4*b*e*m^2*x^4+259790*a^3*b^2*d*m^2*
x^4+193050*a^3*b^2*e*x^6+193050*a^2*b^3*d*x^6+48*a^5*d*m^5+7540*a^5*e*m^3*x^2+37700*a^4*b*d*m^3*x^2+237180*a^4
*b*e*m*x^4+474360*a^3*b^2*d*m*x^4+925*a^5*d*m^4+34759*a^5*e*m^2*x^2+173795*a^4*b*d*m^2*x^2+135135*a^4*b*e*x^4+
270270*a^3*b^2*d*x^4+9120*a^5*d*m^3+73054*a^5*e*m*x^2+365270*a^4*b*d*m*x^2+48259*a^5*d*m^2+45045*a^5*e*x^2+225
225*a^4*b*d*x^2+129072*a^5*d*m+135135*a^5*d)*(f*x)^m*((b*x^2+a)^2)^(5/2)/(13+m)/(11+m)/(9+m)/(7+m)/(5+m)/(3+m)
/(1+m)/(b*x^2+a)^5

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Maxima [A]
time = 0.31, size = 494, normalized size = 1.24 \begin {gather*} \frac {{\left ({\left (m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945\right )} b^{5} f^{m} x^{11} + 5 \, {\left (m^{5} + 27 \, m^{4} + 262 \, m^{3} + 1122 \, m^{2} + 2041 \, m + 1155\right )} a b^{4} f^{m} x^{9} + 10 \, {\left (m^{5} + 29 \, m^{4} + 302 \, m^{3} + 1366 \, m^{2} + 2577 \, m + 1485\right )} a^{2} b^{3} f^{m} x^{7} + 10 \, {\left (m^{5} + 31 \, m^{4} + 350 \, m^{3} + 1730 \, m^{2} + 3489 \, m + 2079\right )} a^{3} b^{2} f^{m} x^{5} + 5 \, {\left (m^{5} + 33 \, m^{4} + 406 \, m^{3} + 2262 \, m^{2} + 5353 \, m + 3465\right )} a^{4} b f^{m} x^{3} + {\left (m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395\right )} a^{5} f^{m} x\right )} d x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} + \frac {{\left ({\left (m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395\right )} b^{5} f^{m} x^{13} + 5 \, {\left (m^{5} + 37 \, m^{4} + 518 \, m^{3} + 3422 \, m^{2} + 10617 \, m + 12285\right )} a b^{4} f^{m} x^{11} + 10 \, {\left (m^{5} + 39 \, m^{4} + 574 \, m^{3} + 3954 \, m^{2} + 12673 \, m + 15015\right )} a^{2} b^{3} f^{m} x^{9} + 10 \, {\left (m^{5} + 41 \, m^{4} + 638 \, m^{3} + 4654 \, m^{2} + 15681 \, m + 19305\right )} a^{3} b^{2} f^{m} x^{7} + 5 \, {\left (m^{5} + 43 \, m^{4} + 710 \, m^{3} + 5570 \, m^{2} + 20409 \, m + 27027\right )} a^{4} b f^{m} x^{5} + {\left (m^{5} + 45 \, m^{4} + 790 \, m^{3} + 6750 \, m^{2} + 28009 \, m + 45045\right )} a^{5} f^{m} x^{3}\right )} e^{\left (m \log \left (x\right ) + 1\right )}}{m^{6} + 48 \, m^{5} + 925 \, m^{4} + 9120 \, m^{3} + 48259 \, m^{2} + 129072 \, m + 135135} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(e*x^2+d)*(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*f^m*x^11 + 5*(m^5 + 27*m^4 + 262*m^3 + 1122*m^2 + 2041*
m + 1155)*a*b^4*f^m*x^9 + 10*(m^5 + 29*m^4 + 302*m^3 + 1366*m^2 + 2577*m + 1485)*a^2*b^3*f^m*x^7 + 10*(m^5 + 3
1*m^4 + 350*m^3 + 1730*m^2 + 3489*m + 2079)*a^3*b^2*f^m*x^5 + 5*(m^5 + 33*m^4 + 406*m^3 + 2262*m^2 + 5353*m +
3465)*a^4*b*f^m*x^3 + (m^5 + 35*m^4 + 470*m^3 + 3010*m^2 + 9129*m + 10395)*a^5*f^m*x)*d*x^m/(m^6 + 36*m^5 + 50
5*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395) + ((m^5 + 35*m^4 + 470*m^3 + 3010*m^2 + 9129*m + 10395)*b^5*f^
m*x^13 + 5*(m^5 + 37*m^4 + 518*m^3 + 3422*m^2 + 10617*m + 12285)*a*b^4*f^m*x^11 + 10*(m^5 + 39*m^4 + 574*m^3 +
3954*m^2 + 12673*m + 15015)*a^2*b^3*f^m*x^9 + 10*(m^5 + 41*m^4 + 638*m^3 + 4654*m^2 + 15681*m + 19305)*a^3*b^
2*f^m*x^7 + 5*(m^5 + 43*m^4 + 710*m^3 + 5570*m^2 + 20409*m + 27027)*a^4*b*f^m*x^5 + (m^5 + 45*m^4 + 790*m^3 +
6750*m^2 + 28009*m + 45045)*a^5*f^m*x^3)*e^(m*log(x) + 1)/(m^6 + 48*m^5 + 925*m^4 + 9120*m^3 + 48259*m^2 + 129
072*m + 135135)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (329) = 658\).
time = 0.37, size = 866, normalized size = 2.16 \begin {gather*} \frac {{\left ({\left (b^{5} d m^{6} + 38 \, b^{5} d m^{5} + 555 \, b^{5} d m^{4} + 3940 \, b^{5} d m^{3} + 14039 \, b^{5} d m^{2} + 22902 \, b^{5} d m + 12285 \, b^{5} d\right )} x^{11} + 5 \, {\left (a b^{4} d m^{6} + 40 \, a b^{4} d m^{5} + 613 \, a b^{4} d m^{4} + 4528 \, a b^{4} d m^{3} + 16627 \, a b^{4} d m^{2} + 27688 \, a b^{4} d m + 15015 \, a b^{4} d\right )} x^{9} + 10 \, {\left (a^{2} b^{3} d m^{6} + 42 \, a^{2} b^{3} d m^{5} + 679 \, a^{2} b^{3} d m^{4} + 5292 \, a^{2} b^{3} d m^{3} + 20335 \, a^{2} b^{3} d m^{2} + 34986 \, a^{2} b^{3} d m + 19305 \, a^{2} b^{3} d\right )} x^{7} + 10 \, {\left (a^{3} b^{2} d m^{6} + 44 \, a^{3} b^{2} d m^{5} + 753 \, a^{3} b^{2} d m^{4} + 6280 \, a^{3} b^{2} d m^{3} + 25979 \, a^{3} b^{2} d m^{2} + 47436 \, a^{3} b^{2} d m + 27027 \, a^{3} b^{2} d\right )} x^{5} + 5 \, {\left (a^{4} b d m^{6} + 46 \, a^{4} b d m^{5} + 835 \, a^{4} b d m^{4} + 7540 \, a^{4} b d m^{3} + 34759 \, a^{4} b d m^{2} + 73054 \, a^{4} b d m + 45045 \, a^{4} b d\right )} x^{3} + {\left (a^{5} d m^{6} + 48 \, a^{5} d m^{5} + 925 \, a^{5} d m^{4} + 9120 \, a^{5} d m^{3} + 48259 \, a^{5} d m^{2} + 129072 \, a^{5} d m + 135135 \, a^{5} d\right )} x + {\left ({\left (b^{5} m^{6} + 36 \, b^{5} m^{5} + 505 \, b^{5} m^{4} + 3480 \, b^{5} m^{3} + 12139 \, b^{5} m^{2} + 19524 \, b^{5} m + 10395 \, b^{5}\right )} x^{13} + 5 \, {\left (a b^{4} m^{6} + 38 \, a b^{4} m^{5} + 555 \, a b^{4} m^{4} + 3940 \, a b^{4} m^{3} + 14039 \, a b^{4} m^{2} + 22902 \, a b^{4} m + 12285 \, a b^{4}\right )} x^{11} + 10 \, {\left (a^{2} b^{3} m^{6} + 40 \, a^{2} b^{3} m^{5} + 613 \, a^{2} b^{3} m^{4} + 4528 \, a^{2} b^{3} m^{3} + 16627 \, a^{2} b^{3} m^{2} + 27688 \, a^{2} b^{3} m + 15015 \, a^{2} b^{3}\right )} x^{9} + 10 \, {\left (a^{3} b^{2} m^{6} + 42 \, a^{3} b^{2} m^{5} + 679 \, a^{3} b^{2} m^{4} + 5292 \, a^{3} b^{2} m^{3} + 20335 \, a^{3} b^{2} m^{2} + 34986 \, a^{3} b^{2} m + 19305 \, a^{3} b^{2}\right )} x^{7} + 5 \, {\left (a^{4} b m^{6} + 44 \, a^{4} b m^{5} + 753 \, a^{4} b m^{4} + 6280 \, a^{4} b m^{3} + 25979 \, a^{4} b m^{2} + 47436 \, a^{4} b m + 27027 \, a^{4} b\right )} x^{5} + {\left (a^{5} m^{6} + 46 \, a^{5} m^{5} + 835 \, a^{5} m^{4} + 7540 \, a^{5} m^{3} + 34759 \, a^{5} m^{2} + 73054 \, a^{5} m + 45045 \, a^{5}\right )} x^{3}\right )} e\right )} \left (f x\right )^{m}}{m^{7} + 49 \, m^{6} + 973 \, m^{5} + 10045 \, m^{4} + 57379 \, m^{3} + 177331 \, m^{2} + 264207 \, m + 135135} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^5*d*m^6 + 38*b^5*d*m^5 + 555*b^5*d*m^4 + 3940*b^5*d*m^3 + 14039*b^5*d*m^2 + 22902*b^5*d*m + 12285*b^5*d)*x
^11 + 5*(a*b^4*d*m^6 + 40*a*b^4*d*m^5 + 613*a*b^4*d*m^4 + 4528*a*b^4*d*m^3 + 16627*a*b^4*d*m^2 + 27688*a*b^4*d
*m + 15015*a*b^4*d)*x^9 + 10*(a^2*b^3*d*m^6 + 42*a^2*b^3*d*m^5 + 679*a^2*b^3*d*m^4 + 5292*a^2*b^3*d*m^3 + 2033
5*a^2*b^3*d*m^2 + 34986*a^2*b^3*d*m + 19305*a^2*b^3*d)*x^7 + 10*(a^3*b^2*d*m^6 + 44*a^3*b^2*d*m^5 + 753*a^3*b^
2*d*m^4 + 6280*a^3*b^2*d*m^3 + 25979*a^3*b^2*d*m^2 + 47436*a^3*b^2*d*m + 27027*a^3*b^2*d)*x^5 + 5*(a^4*b*d*m^6
+ 46*a^4*b*d*m^5 + 835*a^4*b*d*m^4 + 7540*a^4*b*d*m^3 + 34759*a^4*b*d*m^2 + 73054*a^4*b*d*m + 45045*a^4*b*d)*
x^3 + (a^5*d*m^6 + 48*a^5*d*m^5 + 925*a^5*d*m^4 + 9120*a^5*d*m^3 + 48259*a^5*d*m^2 + 129072*a^5*d*m + 135135*a
^5*d)*x + ((b^5*m^6 + 36*b^5*m^5 + 505*b^5*m^4 + 3480*b^5*m^3 + 12139*b^5*m^2 + 19524*b^5*m + 10395*b^5)*x^13
+ 5*(a*b^4*m^6 + 38*a*b^4*m^5 + 555*a*b^4*m^4 + 3940*a*b^4*m^3 + 14039*a*b^4*m^2 + 22902*a*b^4*m + 12285*a*b^4
)*x^11 + 10*(a^2*b^3*m^6 + 40*a^2*b^3*m^5 + 613*a^2*b^3*m^4 + 4528*a^2*b^3*m^3 + 16627*a^2*b^3*m^2 + 27688*a^2
*b^3*m + 15015*a^2*b^3)*x^9 + 10*(a^3*b^2*m^6 + 42*a^3*b^2*m^5 + 679*a^3*b^2*m^4 + 5292*a^3*b^2*m^3 + 20335*a^
3*b^2*m^2 + 34986*a^3*b^2*m + 19305*a^3*b^2)*x^7 + 5*(a^4*b*m^6 + 44*a^4*b*m^5 + 753*a^4*b*m^4 + 6280*a^4*b*m^
3 + 25979*a^4*b*m^2 + 47436*a^4*b*m + 27027*a^4*b)*x^5 + (a^5*m^6 + 46*a^5*m^5 + 835*a^5*m^4 + 7540*a^5*m^3 +
34759*a^5*m^2 + 73054*a^5*m + 45045*a^5)*x^3)*e)*(f*x)^m/(m^7 + 49*m^6 + 973*m^5 + 10045*m^4 + 57379*m^3 + 177
331*m^2 + 264207*m + 135135)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (f x\right )^{m} \left (d + e x^{2}\right ) \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2213 vs. \(2 (329) = 658\).
time = 3.65, size = 2213, normalized size = 5.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((f*x)^m*b^5*m^6*x^13*e*sgn(b*x^2 + a) + 36*(f*x)^m*b^5*m^5*x^13*e*sgn(b*x^2 + a) + (f*x)^m*b^5*d*m^6*x^11*sgn
(b*x^2 + a) + 5*(f*x)^m*a*b^4*m^6*x^11*e*sgn(b*x^2 + a) + 505*(f*x)^m*b^5*m^4*x^13*e*sgn(b*x^2 + a) + 38*(f*x)
^m*b^5*d*m^5*x^11*sgn(b*x^2 + a) + 190*(f*x)^m*a*b^4*m^5*x^11*e*sgn(b*x^2 + a) + 3480*(f*x)^m*b^5*m^3*x^13*e*s
gn(b*x^2 + a) + 5*(f*x)^m*a*b^4*d*m^6*x^9*sgn(b*x^2 + a) + 555*(f*x)^m*b^5*d*m^4*x^11*sgn(b*x^2 + a) + 10*(f*x
)^m*a^2*b^3*m^6*x^9*e*sgn(b*x^2 + a) + 2775*(f*x)^m*a*b^4*m^4*x^11*e*sgn(b*x^2 + a) + 12139*(f*x)^m*b^5*m^2*x^
13*e*sgn(b*x^2 + a) + 200*(f*x)^m*a*b^4*d*m^5*x^9*sgn(b*x^2 + a) + 3940*(f*x)^m*b^5*d*m^3*x^11*sgn(b*x^2 + a)
+ 400*(f*x)^m*a^2*b^3*m^5*x^9*e*sgn(b*x^2 + a) + 19700*(f*x)^m*a*b^4*m^3*x^11*e*sgn(b*x^2 + a) + 19524*(f*x)^m
*b^5*m*x^13*e*sgn(b*x^2 + a) + 10*(f*x)^m*a^2*b^3*d*m^6*x^7*sgn(b*x^2 + a) + 3065*(f*x)^m*a*b^4*d*m^4*x^9*sgn(
b*x^2 + a) + 14039*(f*x)^m*b^5*d*m^2*x^11*sgn(b*x^2 + a) + 10*(f*x)^m*a^3*b^2*m^6*x^7*e*sgn(b*x^2 + a) + 6130*
(f*x)^m*a^2*b^3*m^4*x^9*e*sgn(b*x^2 + a) + 70195*(f*x)^m*a*b^4*m^2*x^11*e*sgn(b*x^2 + a) + 10395*(f*x)^m*b^5*x
^13*e*sgn(b*x^2 + a) + 420*(f*x)^m*a^2*b^3*d*m^5*x^7*sgn(b*x^2 + a) + 22640*(f*x)^m*a*b^4*d*m^3*x^9*sgn(b*x^2
+ a) + 22902*(f*x)^m*b^5*d*m*x^11*sgn(b*x^2 + a) + 420*(f*x)^m*a^3*b^2*m^5*x^7*e*sgn(b*x^2 + a) + 45280*(f*x)^
m*a^2*b^3*m^3*x^9*e*sgn(b*x^2 + a) + 114510*(f*x)^m*a*b^4*m*x^11*e*sgn(b*x^2 + a) + 10*(f*x)^m*a^3*b^2*d*m^6*x
^5*sgn(b*x^2 + a) + 6790*(f*x)^m*a^2*b^3*d*m^4*x^7*sgn(b*x^2 + a) + 83135*(f*x)^m*a*b^4*d*m^2*x^9*sgn(b*x^2 +
a) + 12285*(f*x)^m*b^5*d*x^11*sgn(b*x^2 + a) + 5*(f*x)^m*a^4*b*m^6*x^5*e*sgn(b*x^2 + a) + 6790*(f*x)^m*a^3*b^2
*m^4*x^7*e*sgn(b*x^2 + a) + 166270*(f*x)^m*a^2*b^3*m^2*x^9*e*sgn(b*x^2 + a) + 61425*(f*x)^m*a*b^4*x^11*e*sgn(b
*x^2 + a) + 440*(f*x)^m*a^3*b^2*d*m^5*x^5*sgn(b*x^2 + a) + 52920*(f*x)^m*a^2*b^3*d*m^3*x^7*sgn(b*x^2 + a) + 13
8440*(f*x)^m*a*b^4*d*m*x^9*sgn(b*x^2 + a) + 220*(f*x)^m*a^4*b*m^5*x^5*e*sgn(b*x^2 + a) + 52920*(f*x)^m*a^3*b^2
*m^3*x^7*e*sgn(b*x^2 + a) + 276880*(f*x)^m*a^2*b^3*m*x^9*e*sgn(b*x^2 + a) + 5*(f*x)^m*a^4*b*d*m^6*x^3*sgn(b*x^
2 + a) + 7530*(f*x)^m*a^3*b^2*d*m^4*x^5*sgn(b*x^2 + a) + 203350*(f*x)^m*a^2*b^3*d*m^2*x^7*sgn(b*x^2 + a) + 750
75*(f*x)^m*a*b^4*d*x^9*sgn(b*x^2 + a) + (f*x)^m*a^5*m^6*x^3*e*sgn(b*x^2 + a) + 3765*(f*x)^m*a^4*b*m^4*x^5*e*sg
n(b*x^2 + a) + 203350*(f*x)^m*a^3*b^2*m^2*x^7*e*sgn(b*x^2 + a) + 150150*(f*x)^m*a^2*b^3*x^9*e*sgn(b*x^2 + a) +
230*(f*x)^m*a^4*b*d*m^5*x^3*sgn(b*x^2 + a) + 62800*(f*x)^m*a^3*b^2*d*m^3*x^5*sgn(b*x^2 + a) + 349860*(f*x)^m*
a^2*b^3*d*m*x^7*sgn(b*x^2 + a) + 46*(f*x)^m*a^5*m^5*x^3*e*sgn(b*x^2 + a) + 31400*(f*x)^m*a^4*b*m^3*x^5*e*sgn(b
*x^2 + a) + 349860*(f*x)^m*a^3*b^2*m*x^7*e*sgn(b*x^2 + a) + (f*x)^m*a^5*d*m^6*x*sgn(b*x^2 + a) + 4175*(f*x)^m*
a^4*b*d*m^4*x^3*sgn(b*x^2 + a) + 259790*(f*x)^m*a^3*b^2*d*m^2*x^5*sgn(b*x^2 + a) + 193050*(f*x)^m*a^2*b^3*d*x^
7*sgn(b*x^2 + a) + 835*(f*x)^m*a^5*m^4*x^3*e*sgn(b*x^2 + a) + 129895*(f*x)^m*a^4*b*m^2*x^5*e*sgn(b*x^2 + a) +
193050*(f*x)^m*a^3*b^2*x^7*e*sgn(b*x^2 + a) + 48*(f*x)^m*a^5*d*m^5*x*sgn(b*x^2 + a) + 37700*(f*x)^m*a^4*b*d*m^
3*x^3*sgn(b*x^2 + a) + 474360*(f*x)^m*a^3*b^2*d*m*x^5*sgn(b*x^2 + a) + 7540*(f*x)^m*a^5*m^3*x^3*e*sgn(b*x^2 +
a) + 237180*(f*x)^m*a^4*b*m*x^5*e*sgn(b*x^2 + a) + 925*(f*x)^m*a^5*d*m^4*x*sgn(b*x^2 + a) + 173795*(f*x)^m*a^4
*b*d*m^2*x^3*sgn(b*x^2 + a) + 270270*(f*x)^m*a^3*b^2*d*x^5*sgn(b*x^2 + a) + 34759*(f*x)^m*a^5*m^2*x^3*e*sgn(b*
x^2 + a) + 135135*(f*x)^m*a^4*b*x^5*e*sgn(b*x^2 + a) + 9120*(f*x)^m*a^5*d*m^3*x*sgn(b*x^2 + a) + 365270*(f*x)^
m*a^4*b*d*m*x^3*sgn(b*x^2 + a) + 73054*(f*x)^m*a^5*m*x^3*e*sgn(b*x^2 + a) + 48259*(f*x)^m*a^5*d*m^2*x*sgn(b*x^
2 + a) + 225225*(f*x)^m*a^4*b*d*x^3*sgn(b*x^2 + a) + 45045*(f*x)^m*a^5*x^3*e*sgn(b*x^2 + a) + 129072*(f*x)^m*a
^5*d*m*x*sgn(b*x^2 + a) + 135135*(f*x)^m*a^5*d*x*sgn(b*x^2 + a))/(m^7 + 49*m^6 + 973*m^5 + 10045*m^4 + 57379*m
^3 + 177331*m^2 + 264207*m + 135135)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (f\,x\right )}^m\,\left (e\,x^2+d\right )\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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