3.100 \(\int (a+c x^2)^{5/2} (A+B x+C x^2) \, dx\)

Optimal. Leaf size=168 \[ \frac{5 a^3 (8 A c-a C) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 x \sqrt{a+c x^2} (8 A c-a C)}{128 c}+\frac{x \left (a+c x^2\right )^{5/2} (8 A c-a C)}{48 c}+\frac{5 a x \left (a+c x^2\right )^{3/2} (8 A c-a C)}{192 c}+\frac{B \left (a+c x^2\right )^{7/2}}{7 c}+\frac{C x \left (a+c x^2\right )^{7/2}}{8 c} \]

[Out]

(5*a^2*(8*A*c - a*C)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*(8*A*c - a*C)*x*(a + c*x^2)^(3/2))/(192*c) + ((8*A*c -
a*C)*x*(a + c*x^2)^(5/2))/(48*c) + (B*(a + c*x^2)^(7/2))/(7*c) + (C*x*(a + c*x^2)^(7/2))/(8*c) + (5*a^3*(8*A*c
- a*C)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.101682, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1815, 641, 195, 217, 206} \[ \frac{5 a^3 (8 A c-a C) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}+\frac{5 a^2 x \sqrt{a+c x^2} (8 A c-a C)}{128 c}+\frac{x \left (a+c x^2\right )^{5/2} (8 A c-a C)}{48 c}+\frac{5 a x \left (a+c x^2\right )^{3/2} (8 A c-a C)}{192 c}+\frac{B \left (a+c x^2\right )^{7/2}}{7 c}+\frac{C x \left (a+c x^2\right )^{7/2}}{8 c} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(5/2)*(A + B*x + C*x^2),x]

[Out]

(5*a^2*(8*A*c - a*C)*x*Sqrt[a + c*x^2])/(128*c) + (5*a*(8*A*c - a*C)*x*(a + c*x^2)^(3/2))/(192*c) + ((8*A*c -
a*C)*x*(a + c*x^2)^(5/2))/(48*c) + (B*(a + c*x^2)^(7/2))/(7*c) + (C*x*(a + c*x^2)^(7/2))/(8*c) + (5*a^3*(8*A*c
- a*C)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(128*c^(3/2))

Rule 1815

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[(e*x^(q - 1)*(a + b*x^2)^(p + 1))/(b*(q + 2*p + 1)), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x]
&& PolyQ[Pq, x] &&  !LeQ[p, -1]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \left (a+c x^2\right )^{5/2} \left (A+B x+C x^2\right ) \, dx &=\frac{C x \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\int (8 A c-a C+8 B c x) \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac{B \left (a+c x^2\right )^{7/2}}{7 c}+\frac{C x \left (a+c x^2\right )^{7/2}}{8 c}+\frac{(8 A c-a C) \int \left (a+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac{(8 A c-a C) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{B \left (a+c x^2\right )^{7/2}}{7 c}+\frac{C x \left (a+c x^2\right )^{7/2}}{8 c}+\frac{(5 a (8 A c-a C)) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c}\\ &=\frac{5 a (8 A c-a C) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{(8 A c-a C) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{B \left (a+c x^2\right )^{7/2}}{7 c}+\frac{C x \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 a^2 (8 A c-a C)\right ) \int \sqrt{a+c x^2} \, dx}{64 c}\\ &=\frac{5 a^2 (8 A c-a C) x \sqrt{a+c x^2}}{128 c}+\frac{5 a (8 A c-a C) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{(8 A c-a C) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{B \left (a+c x^2\right )^{7/2}}{7 c}+\frac{C x \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 a^3 (8 A c-a C)\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{128 c}\\ &=\frac{5 a^2 (8 A c-a C) x \sqrt{a+c x^2}}{128 c}+\frac{5 a (8 A c-a C) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{(8 A c-a C) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{B \left (a+c x^2\right )^{7/2}}{7 c}+\frac{C x \left (a+c x^2\right )^{7/2}}{8 c}+\frac{\left (5 a^3 (8 A c-a C)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{128 c}\\ &=\frac{5 a^2 (8 A c-a C) x \sqrt{a+c x^2}}{128 c}+\frac{5 a (8 A c-a C) x \left (a+c x^2\right )^{3/2}}{192 c}+\frac{(8 A c-a C) x \left (a+c x^2\right )^{5/2}}{48 c}+\frac{B \left (a+c x^2\right )^{7/2}}{7 c}+\frac{C x \left (a+c x^2\right )^{7/2}}{8 c}+\frac{5 a^3 (8 A c-a C) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{128 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.334406, size = 150, normalized size = 0.89 \[ \frac{\sqrt{a+c x^2} \left (\sqrt{c} \left (2 a^2 c x (924 A+x (576 B+413 C x))+3 a^3 (128 B+35 C x)+8 a c^2 x^3 (182 A+x (144 B+119 C x))+16 c^3 x^5 (28 A+3 x (8 B+7 C x))\right )-\frac{105 a^{5/2} (a C-8 A c) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}\right )}{2688 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(5/2)*(A + B*x + C*x^2),x]

[Out]

(Sqrt[a + c*x^2]*(Sqrt[c]*(3*a^3*(128*B + 35*C*x) + 16*c^3*x^5*(28*A + 3*x*(8*B + 7*C*x)) + 8*a*c^2*x^3*(182*A
+ x*(144*B + 119*C*x)) + 2*a^2*c*x*(924*A + x*(576*B + 413*C*x))) - (105*a^(5/2)*(-8*A*c + a*C)*ArcSinh[(Sqrt
[c]*x)/Sqrt[a]])/Sqrt[1 + (c*x^2)/a]))/(2688*c^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.05, size = 181, normalized size = 1.1 \begin{align*}{\frac{Cx}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Cax}{48\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}Cx}{192\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,C{a}^{3}x}{128\,c}\sqrt{c{x}^{2}+a}}-{\frac{5\,C{a}^{4}}{128}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{B}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Ax}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,aAx}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}Ax}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,A{a}^{3}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(5/2)*(C*x^2+B*x+A),x)

[Out]

1/8*C*x*(c*x^2+a)^(7/2)/c-1/48*C*a/c*x*(c*x^2+a)^(5/2)-5/192*C*a^2/c*x*(c*x^2+a)^(3/2)-5/128*C*a^3/c*x*(c*x^2+
a)^(1/2)-5/128*C*a^4/c^(3/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))+1/7*B*(c*x^2+a)^(7/2)/c+1/6*A*x*(c*x^2+a)^(5/2)+5/2
4*A*a*x*(c*x^2+a)^(3/2)+5/16*A*a^2*x*(c*x^2+a)^(1/2)+5/16*A*a^3/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)*(C*x^2+B*x+A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.16134, size = 809, normalized size = 4.82 \begin{align*} \left [-\frac{105 \,{\left (C a^{4} - 8 \, A a^{3} c\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (336 \, C c^{4} x^{7} + 384 \, B c^{4} x^{6} + 1152 \, B a c^{3} x^{4} + 1152 \, B a^{2} c^{2} x^{2} + 56 \,{\left (17 \, C a c^{3} + 8 \, A c^{4}\right )} x^{5} + 384 \, B a^{3} c + 14 \,{\left (59 \, C a^{2} c^{2} + 104 \, A a c^{3}\right )} x^{3} + 21 \,{\left (5 \, C a^{3} c + 88 \, A a^{2} c^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{5376 \, c^{2}}, \frac{105 \,{\left (C a^{4} - 8 \, A a^{3} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (336 \, C c^{4} x^{7} + 384 \, B c^{4} x^{6} + 1152 \, B a c^{3} x^{4} + 1152 \, B a^{2} c^{2} x^{2} + 56 \,{\left (17 \, C a c^{3} + 8 \, A c^{4}\right )} x^{5} + 384 \, B a^{3} c + 14 \,{\left (59 \, C a^{2} c^{2} + 104 \, A a c^{3}\right )} x^{3} + 21 \,{\left (5 \, C a^{3} c + 88 \, A a^{2} c^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{2688 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)*(C*x^2+B*x+A),x, algorithm="fricas")

[Out]

[-1/5376*(105*(C*a^4 - 8*A*a^3*c)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(336*C*c^4*x^7 +
384*B*c^4*x^6 + 1152*B*a*c^3*x^4 + 1152*B*a^2*c^2*x^2 + 56*(17*C*a*c^3 + 8*A*c^4)*x^5 + 384*B*a^3*c + 14*(59*
C*a^2*c^2 + 104*A*a*c^3)*x^3 + 21*(5*C*a^3*c + 88*A*a^2*c^2)*x)*sqrt(c*x^2 + a))/c^2, 1/2688*(105*(C*a^4 - 8*A
*a^3*c)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (336*C*c^4*x^7 + 384*B*c^4*x^6 + 1152*B*a*c^3*x^4 + 1152
*B*a^2*c^2*x^2 + 56*(17*C*a*c^3 + 8*A*c^4)*x^5 + 384*B*a^3*c + 14*(59*C*a^2*c^2 + 104*A*a*c^3)*x^3 + 21*(5*C*a
^3*c + 88*A*a^2*c^2)*x)*sqrt(c*x^2 + a))/c^2]

________________________________________________________________________________________

Sympy [A]  time = 30.5604, size = 510, normalized size = 3.04 \begin{align*} \frac{A a^{\frac{5}{2}} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 A a^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{35 A a^{\frac{3}{2}} c x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 A \sqrt{a} c^{2} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{5 A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 \sqrt{c}} + \frac{A c^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B a^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + 2 B a c \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + B c^{2} \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \frac{5 C a^{\frac{7}{2}} x}{128 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{133 C a^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{127 C a^{\frac{3}{2}} c x^{5}}{192 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{23 C \sqrt{a} c^{2} x^{7}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{5 C a^{4} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{128 c^{\frac{3}{2}}} + \frac{C c^{3} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(5/2)*(C*x**2+B*x+A),x)

[Out]

A*a**(5/2)*x*sqrt(1 + c*x**2/a)/2 + 3*A*a**(5/2)*x/(16*sqrt(1 + c*x**2/a)) + 35*A*a**(3/2)*c*x**3/(48*sqrt(1 +
c*x**2/a)) + 17*A*sqrt(a)*c**2*x**5/(24*sqrt(1 + c*x**2/a)) + 5*A*a**3*asinh(sqrt(c)*x/sqrt(a))/(16*sqrt(c))
+ A*c**3*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a)) + B*a**2*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/
2)/(3*c), True)) + 2*B*a*c*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x*
*4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + B*c**2*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**
3) - 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c,
0)), (sqrt(a)*x**6/6, True)) + 5*C*a**(7/2)*x/(128*c*sqrt(1 + c*x**2/a)) + 133*C*a**(5/2)*x**3/(384*sqrt(1 +
c*x**2/a)) + 127*C*a**(3/2)*c*x**5/(192*sqrt(1 + c*x**2/a)) + 23*C*sqrt(a)*c**2*x**7/(48*sqrt(1 + c*x**2/a)) -
5*C*a**4*asinh(sqrt(c)*x/sqrt(a))/(128*c**(3/2)) + C*c**3*x**9/(8*sqrt(a)*sqrt(1 + c*x**2/a))

________________________________________________________________________________________

Giac [A]  time = 1.1761, size = 227, normalized size = 1.35 \begin{align*} \frac{1}{2688} \,{\left (\frac{384 \, B a^{3}}{c} +{\left (2 \,{\left (576 \, B a^{2} +{\left (4 \,{\left (144 \, B a c +{\left (6 \,{\left (7 \, C c^{2} x + 8 \, B c^{2}\right )} x + \frac{7 \,{\left (17 \, C a c^{7} + 8 \, A c^{8}\right )}}{c^{6}}\right )} x\right )} x + \frac{7 \,{\left (59 \, C a^{2} c^{6} + 104 \, A a c^{7}\right )}}{c^{6}}\right )} x\right )} x + \frac{21 \,{\left (5 \, C a^{3} c^{5} + 88 \, A a^{2} c^{6}\right )}}{c^{6}}\right )} x\right )} \sqrt{c x^{2} + a} + \frac{5 \,{\left (C a^{4} - 8 \, A a^{3} c\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{128 \, c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)*(C*x^2+B*x+A),x, algorithm="giac")

[Out]

1/2688*(384*B*a^3/c + (2*(576*B*a^2 + (4*(144*B*a*c + (6*(7*C*c^2*x + 8*B*c^2)*x + 7*(17*C*a*c^7 + 8*A*c^8)/c^
6)*x)*x + 7*(59*C*a^2*c^6 + 104*A*a*c^7)/c^6)*x)*x + 21*(5*C*a^3*c^5 + 88*A*a^2*c^6)/c^6)*x)*sqrt(c*x^2 + a) +
5/128*(C*a^4 - 8*A*a^3*c)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)