\(\int x (c+d x) (a+b x^2)^{5/2} \, dx\) [1141]

Optimal result

Integrand size = 18, antiderivative size = 136 \[ \int x (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {5 a^3 d x \sqrt {a+b x^2}}{128 b}+\frac {5}{64} a^2 d x^3 \sqrt {a+b x^2}+\frac {5}{48} a d x^3 \left (a+b x^2\right )^{3/2}+\frac {1}{8} d x^3 \left (a+b x^2\right )^{5/2}+\frac {c \left (a+b x^2\right )^{7/2}}{7 b}-\frac {5 a^4 d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}} \] Output:

5/128*a^3*d*x*(b*x^2+a)^(1/2)/b+5/64*a^2*d*x^3*(b*x^2+a)^(1/2)+5/48*a*d*x^ 
3*(b*x^2+a)^(3/2)+1/8*d*x^3*(b*x^2+a)^(5/2)+1/7*c*(b*x^2+a)^(7/2)/b-5/128* 
a^4*d*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(3/2)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int x (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (48 b^3 x^6 (8 c+7 d x)+3 a^3 (128 c+35 d x)+8 a b^2 x^4 (144 c+119 d x)+2 a^2 b x^2 (576 c+413 d x)\right )+105 a^4 d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2688 b^{3/2}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(48*b^3*x^6*(8*c + 7*d*x) + 3*a^3*(128*c + 35*d*x 
) + 8*a*b^2*x^4*(144*c + 119*d*x) + 2*a^2*b*x^2*(576*c + 413*d*x)) + 105*a 
^4*d*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(2688*b^(3/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {533, 455, 211, 211, 211, 224, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \left (a+b x^2\right )^{5/2} (c+d x) \, dx\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac {\int (a d-8 b c x) \left (b x^2+a\right )^{5/2}dx}{8 b}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac {a d \int \left (b x^2+a\right )^{5/2}dx-\frac {8}{7} c \left (a+b x^2\right )^{7/2}}{8 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac {a d \left (\frac {5}{6} a \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {8}{7} c \left (a+b x^2\right )^{7/2}}{8 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac {a d \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {8}{7} c \left (a+b x^2\right )^{7/2}}{8 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac {a d \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {8}{7} c \left (a+b x^2\right )^{7/2}}{8 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac {a d \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {8}{7} c \left (a+b x^2\right )^{7/2}}{8 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d x \left (a+b x^2\right )^{7/2}}{8 b}-\frac {a d \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {8}{7} c \left (a+b x^2\right )^{7/2}}{8 b}\)

Input:

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

Output:

(d*x*(a + b*x^2)^(7/2))/(8*b) - ((-8*c*(a + b*x^2)^(7/2))/7 + a*d*((x*(a + 
 b*x^2)^(5/2))/6 + (5*a*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2 
])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/4))/6))/(8*b 
)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 219

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

rule 224

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 455

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

rule 533

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* 
p + 2))   Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], 
 x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer 
Q[2*p]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {c \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 b}+d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )\)	\(108\)
risch	\(\frac {\left (336 b^{3} d \,x^{7}+384 b^{3} c \,x^{6}+952 a \,b^{2} d \,x^{5}+1152 a \,b^{2} c \,x^{4}+826 a^{2} b d \,x^{3}+1152 a^{2} b c \,x^{2}+105 a^{3} d x +384 c \,a^{3}\right ) \sqrt {b \,x^{2}+a}}{2688 b}-\frac {5 a^{4} d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}\)	\(113\)

Input:

int(x*(d*x+c)*(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/7*c*(b*x^2+a)^(7/2)/b+d*(1/8*x*(b*x^2+a)^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a 
)^(5/2)+5/6*a*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^ 
(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.86 \[ \int x (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\left [\frac {105 \, a^{4} \sqrt {b} d \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (336 \, b^{4} d x^{7} + 384 \, b^{4} c x^{6} + 952 \, a b^{3} d x^{5} + 1152 \, a b^{3} c x^{4} + 826 \, a^{2} b^{2} d x^{3} + 1152 \, a^{2} b^{2} c x^{2} + 105 \, a^{3} b d x + 384 \, a^{3} b c\right )} \sqrt {b x^{2} + a}}{5376 \, b^{2}}, \frac {105 \, a^{4} \sqrt {-b} d \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (336 \, b^{4} d x^{7} + 384 \, b^{4} c x^{6} + 952 \, a b^{3} d x^{5} + 1152 \, a b^{3} c x^{4} + 826 \, a^{2} b^{2} d x^{3} + 1152 \, a^{2} b^{2} c x^{2} + 105 \, a^{3} b d x + 384 \, a^{3} b c\right )} \sqrt {b x^{2} + a}}{2688 \, b^{2}}\right ] \] Input:

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

Output:

[1/5376*(105*a^4*sqrt(b)*d*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) 
 + 2*(336*b^4*d*x^7 + 384*b^4*c*x^6 + 952*a*b^3*d*x^5 + 1152*a*b^3*c*x^4 + 
 826*a^2*b^2*d*x^3 + 1152*a^2*b^2*c*x^2 + 105*a^3*b*d*x + 384*a^3*b*c)*sqr 
t(b*x^2 + a))/b^2, 1/2688*(105*a^4*sqrt(-b)*d*arctan(sqrt(-b)*x/sqrt(b*x^2 
 + a)) + (336*b^4*d*x^7 + 384*b^4*c*x^6 + 952*a*b^3*d*x^5 + 1152*a*b^3*c*x 
^4 + 826*a^2*b^2*d*x^3 + 1152*a^2*b^2*c*x^2 + 105*a^3*b*d*x + 384*a^3*b*c) 
*sqrt(b*x^2 + a))/b^2]
 

Sympy [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.24 \[ \int x (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\begin {cases} - \frac {5 a^{4} d \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{128 b} + \sqrt {a + b x^{2}} \left (\frac {a^{3} c}{7 b} + \frac {5 a^{3} d x}{128 b} + \frac {3 a^{2} c x^{2}}{7} + \frac {59 a^{2} d x^{3}}{192} + \frac {3 a b c x^{4}}{7} + \frac {17 a b d x^{5}}{48} + \frac {b^{2} c x^{6}}{7} + \frac {b^{2} d x^{7}}{8}\right ) & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {c x^{2}}{2} + \frac {d x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-5*a**4*d*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sq 
rt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2), True))/(128*b) + sqrt(a + b*x**2 
)*(a**3*c/(7*b) + 5*a**3*d*x/(128*b) + 3*a**2*c*x**2/7 + 59*a**2*d*x**3/19 
2 + 3*a*b*c*x**4/7 + 17*a*b*d*x**5/48 + b**2*c*x**6/7 + b**2*d*x**7/8), Ne 
(b, 0)), (a**(5/2)*(c*x**2/2 + d*x**3/3), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.77 \[ \int x (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} d x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a d x}{48 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d x}{192 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} d x}{128 \, b} - \frac {5 \, a^{4} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} c}{7 \, b} \] Input:

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

Output:

1/8*(b*x^2 + a)^(7/2)*d*x/b - 1/48*(b*x^2 + a)^(5/2)*a*d*x/b - 5/192*(b*x^ 
2 + a)^(3/2)*a^2*d*x/b - 5/128*sqrt(b*x^2 + a)*a^3*d*x/b - 5/128*a^4*d*arc 
sinh(b*x/sqrt(a*b))/b^(3/2) + 1/7*(b*x^2 + a)^(7/2)*c/b
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.84 \[ \int x (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {5 \, a^{4} d \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {3}{2}}} + \frac {1}{2688} \, {\left (\frac {384 \, a^{3} c}{b} + {\left (\frac {105 \, a^{3} d}{b} + 2 \, {\left (576 \, a^{2} c + {\left (413 \, a^{2} d + 4 \, {\left (144 \, a b c + {\left (119 \, a b d + 6 \, {\left (7 \, b^{2} d x + 8 \, b^{2} c\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {b x^{2} + a} \] Input:

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

Output:

5/128*a^4*d*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) + 1/2688*(384*a 
^3*c/b + (105*a^3*d/b + 2*(576*a^2*c + (413*a^2*d + 4*(144*a*b*c + (119*a* 
b*d + 6*(7*b^2*d*x + 8*b^2*c)*x)*x)*x)*x)*x)*x)*sqrt(b*x^2 + a)
 

Mupad [F(-1)]

Timed out. \[ \int x (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\int x\,{\left (b\,x^2+a\right )}^{5/2}\,\left (c+d\,x\right ) \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.28 \[ \int x (c+d x) \left (a+b x^2\right )^{5/2} \, dx=\frac {384 \sqrt {b \,x^{2}+a}\, a^{3} b c +105 \sqrt {b \,x^{2}+a}\, a^{3} b d x +1152 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{2}+826 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{3}+1152 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{4}+952 \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{5}+384 \sqrt {b \,x^{2}+a}\, b^{4} c \,x^{6}+336 \sqrt {b \,x^{2}+a}\, b^{4} d \,x^{7}-105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d}{2688 b^{2}} \] Input:

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

Output:

(384*sqrt(a + b*x**2)*a**3*b*c + 105*sqrt(a + b*x**2)*a**3*b*d*x + 1152*sq 
rt(a + b*x**2)*a**2*b**2*c*x**2 + 826*sqrt(a + b*x**2)*a**2*b**2*d*x**3 + 
1152*sqrt(a + b*x**2)*a*b**3*c*x**4 + 952*sqrt(a + b*x**2)*a*b**3*d*x**5 + 
 384*sqrt(a + b*x**2)*b**4*c*x**6 + 336*sqrt(a + b*x**2)*b**4*d*x**7 - 105 
*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*d)/(2688*b**2)