3.3.40 \(\int \frac {(A+B x^2) (b x^2+c x^4)^{3/2}}{x^{11/2}} \, dx\) [240]

Optimal. Leaf size=200 \[ \frac {4 (3 b B+7 A c) \sqrt {b x^2+c x^4}}{21 \sqrt {x}}+\frac {2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac {4 b^{3/4} (3 b B+7 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{c} \sqrt {b x^2+c x^4}} \]

[Out]

2/21*(7*A*c+3*B*b)*(c*x^4+b*x^2)^(3/2)/b/x^(5/2)-2/3*A*(c*x^4+b*x^2)^(5/2)/b/x^(13/2)+4/21*(7*A*c+3*B*b)*(c*x^
4+b*x^2)^(1/2)/x^(1/2)+4/21*b^(3/4)*(7*A*c+3*B*b)*x*(cos(2*arctan(c^(1/4)*x^(1/2)/b^(1/4)))^2)^(1/2)/cos(2*arc
tan(c^(1/4)*x^(1/2)/b^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/b^(1/4))),1/2*2^(1/2))*(b^(1/2)+x*c^(1/2)
)*((c*x^2+b)/(b^(1/2)+x*c^(1/2))^2)^(1/2)/c^(1/4)/(c*x^4+b*x^2)^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2063, 2046, 2057, 335, 226} \begin {gather*} \frac {4 b^{3/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (7 A c+3 b B) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{c} \sqrt {b x^2+c x^4}}+\frac {4 \sqrt {b x^2+c x^4} (7 A c+3 b B)}{21 \sqrt {x}}+\frac {2 \left (b x^2+c x^4\right )^{3/2} (7 A c+3 b B)}{21 b x^{5/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(11/2),x]

[Out]

(4*(3*b*B + 7*A*c)*Sqrt[b*x^2 + c*x^4])/(21*Sqrt[x]) + (2*(3*b*B + 7*A*c)*(b*x^2 + c*x^4)^(3/2))/(21*b*x^(5/2)
) - (2*A*(b*x^2 + c*x^4)^(5/2))/(3*b*x^(13/2)) + (4*b^(3/4)*(3*b*B + 7*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b +
c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(21*c^(1/4)*Sqrt[b*x^2 +
c*x^4])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2046

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b
*x^n)^p/(c*(m + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 2063

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j + b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] + Dist[(a*d*(m + j*p + 1
) - b*c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1)), Int[(e*x)^(m + n)*(a*x^j + b*x^(j + n))^p, x], x] /; F
reeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m
+ j*p, -1] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, (-n)*p - 1])) && (GtQ[e, 0] || IntegersQ[j,
n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1, 0]

Rubi steps

\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{11/2}} \, dx &=-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}-\frac {\left (2 \left (-\frac {3 b B}{2}-\frac {7 A c}{2}\right )\right ) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{7/2}} \, dx}{3 b}\\ &=\frac {2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac {1}{7} (2 (3 b B+7 A c)) \int \frac {\sqrt {b x^2+c x^4}}{x^{3/2}} \, dx\\ &=\frac {4 (3 b B+7 A c) \sqrt {b x^2+c x^4}}{21 \sqrt {x}}+\frac {2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac {1}{21} (4 b (3 b B+7 A c)) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {4 (3 b B+7 A c) \sqrt {b x^2+c x^4}}{21 \sqrt {x}}+\frac {2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac {\left (4 b (3 b B+7 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{21 \sqrt {b x^2+c x^4}}\\ &=\frac {4 (3 b B+7 A c) \sqrt {b x^2+c x^4}}{21 \sqrt {x}}+\frac {2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac {\left (8 b (3 b B+7 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{21 \sqrt {b x^2+c x^4}}\\ &=\frac {4 (3 b B+7 A c) \sqrt {b x^2+c x^4}}{21 \sqrt {x}}+\frac {2 (3 b B+7 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b x^{5/2}}-\frac {2 A \left (b x^2+c x^4\right )^{5/2}}{3 b x^{13/2}}+\frac {4 b^{3/4} (3 b B+7 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{c} \sqrt {b x^2+c x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.05, size = 101, normalized size = 0.50 \begin {gather*} -\frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (A \left (b+c x^2\right )^2 \sqrt {1+\frac {c x^2}{b}}-b (3 b B+7 A c) x^2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{b}\right )\right )}{3 b x^{5/2} \sqrt {1+\frac {c x^2}{b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(11/2),x]

[Out]

(-2*Sqrt[x^2*(b + c*x^2)]*(A*(b + c*x^2)^2*Sqrt[1 + (c*x^2)/b] - b*(3*b*B + 7*A*c)*x^2*Hypergeometric2F1[-3/2,
1/4, 5/4, -((c*x^2)/b)]))/(3*b*x^(5/2)*Sqrt[1 + (c*x^2)/b])

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Maple [A]
time = 0.39, size = 260, normalized size = 1.30

method	result	size
risch	\(-\frac {2 \left (-3 B c \,x^{4}-7 A c \,x^{2}-9 b B \,x^{2}+7 A b \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{21 x^{\frac {5}{2}}}+\frac {4 b \left (7 A c +3 B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{21 c \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\)	\(200\)
default	\(\frac {2 \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}} \left (14 A \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b c}\, b c x +6 B \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b c}\, b^{2} x +3 B \,c^{3} x^{6}+7 A \,c^{3} x^{4}+12 B b \,c^{2} x^{4}+9 B \,b^{2} c \,x^{2}-7 A \,b^{2} c \right )}{21 x^{\frac {9}{2}} \left (c \,x^{2}+b \right )^{2} c}\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(11/2),x,method=_RETURNVERBOSE)

[Out]

2/21*(c*x^4+b*x^2)^(3/2)/x^(9/2)/(c*x^2+b)^2*(14*A*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*
c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*EllipticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*
2^(1/2))*(-b*c)^(1/2)*b*c*x+6*B*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1
/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*EllipticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)^(1/
2)*b^2*x+3*B*c^3*x^6+7*A*c^3*x^4+12*B*b*c^2*x^4+9*B*b^2*c*x^2-7*A*b^2*c)/c

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^(11/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.63, size = 86, normalized size = 0.43 \begin {gather*} \frac {2 \, {\left (4 \, {\left (3 \, B b^{2} + 7 \, A b c\right )} \sqrt {c} x^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (3 \, B c^{2} x^{4} - 7 \, A b c + {\left (9 \, B b c + 7 \, A c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{21 \, c x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(11/2),x, algorithm="fricas")

[Out]

2/21*(4*(3*B*b^2 + 7*A*b*c)*sqrt(c)*x^3*weierstrassPInverse(-4*b/c, 0, x) + (3*B*c^2*x^4 - 7*A*b*c + (9*B*b*c
+ 7*A*c^2)*x^2)*sqrt(c*x^4 + b*x^2)*sqrt(x))/(c*x^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{\frac {11}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(c*x**4+b*x**2)**(3/2)/x**(11/2),x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)*(A + B*x**2)/x**(11/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^(3/2)/x^(11/2),x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)/x^(11/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(11/2),x)

[Out]

int(((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(11/2), x)

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