\(\int x (c+d x)^3 (a+b x^2)^{5/2} \, dx\) [1139]

Optimal result

Integrand size = 20, antiderivative size = 254 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=-\frac {3 a^3 d \left (10 b c^2-a d^2\right ) x \sqrt {a+b x^2}}{256 b^2}-\frac {a^2 d \left (10 b c^2-a d^2\right ) x \left (a+b x^2\right )^{3/2}}{128 b^2}-\frac {a d \left (10 b c^2-a d^2\right ) x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac {c (c+d x)^2 \left (a+b x^2\right )^{7/2}}{30 b}+\frac {(c+d x)^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {\left (16 c \left (b c^2-10 a d^2\right )+7 d \left (2 b c^2-9 a d^2\right ) x\right ) \left (a+b x^2\right )^{7/2}}{1680 b^2}-\frac {3 a^4 d \left (10 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}} \] Output:

-3/256*a^3*d*(-a*d^2+10*b*c^2)*x*(b*x^2+a)^(1/2)/b^2-1/128*a^2*d*(-a*d^2+1 
0*b*c^2)*x*(b*x^2+a)^(3/2)/b^2-1/160*a*d*(-a*d^2+10*b*c^2)*x*(b*x^2+a)^(5/ 
2)/b^2+1/30*c*(d*x+c)^2*(b*x^2+a)^(7/2)/b+1/10*(d*x+c)^3*(b*x^2+a)^(7/2)/b 
+1/1680*(16*c*(-10*a*d^2+b*c^2)+7*d*(-9*a*d^2+2*b*c^2)*x)*(b*x^2+a)^(7/2)/ 
b^2-3/256*a^4*d*(-a*d^2+10*b*c^2)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/ 
2)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.93 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (-5 a^4 d^2 (512 c+63 d x)+10 a^3 b \left (384 c^3+315 c^2 d x+128 c d^2 x^2+21 d^3 x^3\right )+32 b^4 x^6 \left (120 c^3+315 c^2 d x+280 c d^2 x^2+84 d^3 x^3\right )+12 a^2 b^2 x^2 \left (960 c^3+2065 c^2 d x+1600 c d^2 x^2+434 d^3 x^3\right )+16 a b^3 x^4 \left (720 c^3+1785 c^2 d x+1520 c d^2 x^2+441 d^3 x^3\right )\right )-315 a^4 d \left (-10 b c^2+a d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{26880 b^{5/2}} \] Input:

Integrate[x*(c + d*x)^3*(a + b*x^2)^(5/2),x]
 

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(-5*a^4*d^2*(512*c + 63*d*x) + 10*a^3*b*(384*c^3 
+ 315*c^2*d*x + 128*c*d^2*x^2 + 21*d^3*x^3) + 32*b^4*x^6*(120*c^3 + 315*c^ 
2*d*x + 280*c*d^2*x^2 + 84*d^3*x^3) + 12*a^2*b^2*x^2*(960*c^3 + 2065*c^2*d 
*x + 1600*c*d^2*x^2 + 434*d^3*x^3) + 16*a*b^3*x^4*(720*c^3 + 1785*c^2*d*x 
+ 1520*c*d^2*x^2 + 441*d^3*x^3)) - 315*a^4*d*(-10*b*c^2 + a*d^2)*Log[-(Sqr 
t[b]*x) + Sqrt[a + b*x^2]])/(26880*b^(5/2))
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {541, 2340, 27, 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.

\(\displaystyle \int x \left (a+b x^2\right )^{5/2} (c+d x)^3 \, dx\)

\(\Big \downarrow \) 541

\(\displaystyle \frac {\int x \left (b x^2+a\right )^{5/2} \left (10 b c^3+30 b d^2 x^2 c+3 d \left (10 b c^2-a d^2\right ) x\right )dx}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {\int 3 b x \left (10 c \left (3 b c^2-2 a d^2\right )+9 d \left (10 b c^2-a d^2\right ) x\right ) \left (b x^2+a\right )^{5/2}dx}{9 b}+\frac {10}{3} c d^2 x^2 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \int x \left (10 c \left (3 b c^2-2 a d^2\right )+9 d \left (10 b c^2-a d^2\right ) x\right ) \left (b x^2+a\right )^{5/2}dx+\frac {10}{3} c d^2 x^2 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {1}{3} \left (\frac {9 d x \left (a+b x^2\right )^{7/2} \left (10 b c^2-a d^2\right )}{8 b}-\frac {\int \left (9 a d \left (10 b c^2-a d^2\right )-80 b c \left (3 b c^2-2 a d^2\right ) x\right ) \left (b x^2+a\right )^{5/2}dx}{8 b}\right )+\frac {10}{3} c d^2 x^2 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {1}{3} \left (\frac {9 d x \left (a+b x^2\right )^{7/2} \left (10 b c^2-a d^2\right )}{8 b}-\frac {9 a d \left (10 b c^2-a d^2\right ) \int \left (b x^2+a\right )^{5/2}dx-\frac {80}{7} c \left (a+b x^2\right )^{7/2} \left (3 b c^2-2 a d^2\right )}{8 b}\right )+\frac {10}{3} c d^2 x^2 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{3} \left (\frac {9 d x \left (a+b x^2\right )^{7/2} \left (10 b c^2-a d^2\right )}{8 b}-\frac {9 a d \left (10 b c^2-a d^2\right ) \left (\frac {5}{6} a \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {80}{7} c \left (a+b x^2\right )^{7/2} \left (3 b c^2-2 a d^2\right )}{8 b}\right )+\frac {10}{3} c d^2 x^2 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{3} \left (\frac {9 d x \left (a+b x^2\right )^{7/2} \left (10 b c^2-a d^2\right )}{8 b}-\frac {9 a d \left (10 b c^2-a d^2\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {80}{7} c \left (a+b x^2\right )^{7/2} \left (3 b c^2-2 a d^2\right )}{8 b}\right )+\frac {10}{3} c d^2 x^2 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{3} \left (\frac {9 d x \left (a+b x^2\right )^{7/2} \left (10 b c^2-a d^2\right )}{8 b}-\frac {9 a d \left (10 b c^2-a d^2\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {80}{7} c \left (a+b x^2\right )^{7/2} \left (3 b c^2-2 a d^2\right )}{8 b}\right )+\frac {10}{3} c d^2 x^2 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{3} \left (\frac {9 d x \left (a+b x^2\right )^{7/2} \left (10 b c^2-a d^2\right )}{8 b}-\frac {9 a d \left (10 b c^2-a d^2\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )-\frac {80}{7} c \left (a+b x^2\right )^{7/2} \left (3 b c^2-2 a d^2\right )}{8 b}\right )+\frac {10}{3} c d^2 x^2 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{3} \left (\frac {9 d x \left (a+b x^2\right )^{7/2} \left (10 b c^2-a d^2\right )}{8 b}-\frac {9 a d \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right ) \left (10 b c^2-a d^2\right )-\frac {80}{7} c \left (a+b x^2\right )^{7/2} \left (3 b c^2-2 a d^2\right )}{8 b}\right )+\frac {10}{3} c d^2 x^2 \left (a+b x^2\right )^{7/2}}{10 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{7/2}}{10 b}\)

Input:

Int[x*(c + d*x)^3*(a + b*x^2)^(5/2),x]
 

Output:

(d^3*x^3*(a + b*x^2)^(7/2))/(10*b) + ((10*c*d^2*x^2*(a + b*x^2)^(7/2))/3 + 
 ((9*d*(10*b*c^2 - a*d^2)*x*(a + b*x^2)^(7/2))/(8*b) - ((-80*c*(3*b*c^2 - 
2*a*d^2)*(a + b*x^2)^(7/2))/7 + 9*a*d*(10*b*c^2 - a*d^2)*((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))/3)/(10 
*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]
 

rule 541

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x 
] + Simp[1/(b*(m + n + 2*p + 1))   Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + 
 n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) 
*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt 
Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {c^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 b}+d^{3} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )+3 c \,d^{2} \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )+3 c^{2} d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )\)	\(270\)
risch	\(-\frac {\left (-2688 b^{4} d^{3} x^{9}-8960 b^{4} c \,d^{2} x^{8}-7056 a \,b^{3} d^{3} x^{7}-10080 b^{4} c^{2} d \,x^{7}-24320 a \,b^{3} c \,d^{2} x^{6}-3840 b^{4} c^{3} x^{6}-5208 a^{2} b^{2} d^{3} x^{5}-28560 a \,b^{3} c^{2} d \,x^{5}-19200 a^{2} b^{2} c \,d^{2} x^{4}-11520 a \,b^{3} c^{3} x^{4}-210 a^{3} b \,d^{3} x^{3}-24780 a^{2} b^{2} c^{2} d \,x^{3}-1280 a^{3} b c \,d^{2} x^{2}-11520 a^{2} b^{2} c^{3} x^{2}+315 a^{4} d^{3} x -3150 a^{3} b \,c^{2} d x +2560 a^{4} c \,d^{2}-3840 a^{3} b \,c^{3}\right ) \sqrt {b \,x^{2}+a}}{26880 b^{2}}+\frac {3 a^{4} d \left (a \,d^{2}-10 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {5}{2}}}\)	\(271\)

Input:

int(x*(d*x+c)^3*(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/7*c^3*(b*x^2+a)^(7/2)/b+d^3*(1/10*x^3*(b*x^2+a)^(7/2)/b-3/10*a/b*(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 
)))))))+3*c*d^2*(1/9*x^2*(b*x^2+a)^(7/2)/b-2/63*a/b^2*(b*x^2+a)^(7/2))+3*c 
^2*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 = 0.14 (sec) , antiderivative size = 583, normalized size of antiderivative = 2.30 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\left [-\frac {315 \, {\left (10 \, a^{4} b c^{2} d - a^{5} d^{3}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (2688 \, b^{5} d^{3} x^{9} + 8960 \, b^{5} c d^{2} x^{8} + 3840 \, a^{3} b^{2} c^{3} - 2560 \, a^{4} b c d^{2} + 1008 \, {\left (10 \, b^{5} c^{2} d + 7 \, a b^{4} d^{3}\right )} x^{7} + 1280 \, {\left (3 \, b^{5} c^{3} + 19 \, a b^{4} c d^{2}\right )} x^{6} + 168 \, {\left (170 \, a b^{4} c^{2} d + 31 \, a^{2} b^{3} d^{3}\right )} x^{5} + 3840 \, {\left (3 \, a b^{4} c^{3} + 5 \, a^{2} b^{3} c d^{2}\right )} x^{4} + 210 \, {\left (118 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} d^{3}\right )} x^{3} + 1280 \, {\left (9 \, a^{2} b^{3} c^{3} + a^{3} b^{2} c d^{2}\right )} x^{2} + 315 \, {\left (10 \, a^{3} b^{2} c^{2} d - a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{53760 \, b^{3}}, \frac {315 \, {\left (10 \, a^{4} b c^{2} d - a^{5} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (2688 \, b^{5} d^{3} x^{9} + 8960 \, b^{5} c d^{2} x^{8} + 3840 \, a^{3} b^{2} c^{3} - 2560 \, a^{4} b c d^{2} + 1008 \, {\left (10 \, b^{5} c^{2} d + 7 \, a b^{4} d^{3}\right )} x^{7} + 1280 \, {\left (3 \, b^{5} c^{3} + 19 \, a b^{4} c d^{2}\right )} x^{6} + 168 \, {\left (170 \, a b^{4} c^{2} d + 31 \, a^{2} b^{3} d^{3}\right )} x^{5} + 3840 \, {\left (3 \, a b^{4} c^{3} + 5 \, a^{2} b^{3} c d^{2}\right )} x^{4} + 210 \, {\left (118 \, a^{2} b^{3} c^{2} d + a^{3} b^{2} d^{3}\right )} x^{3} + 1280 \, {\left (9 \, a^{2} b^{3} c^{3} + a^{3} b^{2} c d^{2}\right )} x^{2} + 315 \, {\left (10 \, a^{3} b^{2} c^{2} d - a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{26880 \, b^{3}}\right ] \] Input:

integrate(x*(d*x+c)^3*(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[-1/53760*(315*(10*a^4*b*c^2*d - a^5*d^3)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b* 
x^2 + a)*sqrt(b)*x - a) - 2*(2688*b^5*d^3*x^9 + 8960*b^5*c*d^2*x^8 + 3840* 
a^3*b^2*c^3 - 2560*a^4*b*c*d^2 + 1008*(10*b^5*c^2*d + 7*a*b^4*d^3)*x^7 + 1 
280*(3*b^5*c^3 + 19*a*b^4*c*d^2)*x^6 + 168*(170*a*b^4*c^2*d + 31*a^2*b^3*d 
^3)*x^5 + 3840*(3*a*b^4*c^3 + 5*a^2*b^3*c*d^2)*x^4 + 210*(118*a^2*b^3*c^2* 
d + a^3*b^2*d^3)*x^3 + 1280*(9*a^2*b^3*c^3 + a^3*b^2*c*d^2)*x^2 + 315*(10* 
a^3*b^2*c^2*d - a^4*b*d^3)*x)*sqrt(b*x^2 + a))/b^3, 1/26880*(315*(10*a^4*b 
*c^2*d - a^5*d^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (2688*b^5* 
d^3*x^9 + 8960*b^5*c*d^2*x^8 + 3840*a^3*b^2*c^3 - 2560*a^4*b*c*d^2 + 1008* 
(10*b^5*c^2*d + 7*a*b^4*d^3)*x^7 + 1280*(3*b^5*c^3 + 19*a*b^4*c*d^2)*x^6 + 
 168*(170*a*b^4*c^2*d + 31*a^2*b^3*d^3)*x^5 + 3840*(3*a*b^4*c^3 + 5*a^2*b^ 
3*c*d^2)*x^4 + 210*(118*a^2*b^3*c^2*d + a^3*b^2*d^3)*x^3 + 1280*(9*a^2*b^3 
*c^3 + a^3*b^2*c*d^2)*x^2 + 315*(10*a^3*b^2*c^2*d - a^4*b*d^3)*x)*sqrt(b*x 
^2 + a))/b^3]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (235) = 470\).

Time = 0.66 (sec) , antiderivative size = 745, normalized size of antiderivative = 2.93 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

integrate(x*(d*x+c)**3*(b*x**2+a)**(5/2),x)
 

Output:

Piecewise((-a*(3*a**3*c**2*d - 3*a*(a**3*d**3 + 9*a**2*b*c**2*d - 5*a*(3*a 
**2*b*d**3 + 9*a*b**2*c**2*d - 7*a*(21*a*b**2*d**3/10 + 3*b**3*c**2*d)/(8* 
b))/(6*b))/(4*b))*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt( 
b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2), True))/(2*b) + sqrt(a + b*x**2)*(b* 
*2*c*d**2*x**8/3 + b**2*d**3*x**9/10 + x**7*(21*a*b**2*d**3/10 + 3*b**3*c* 
*2*d)/(8*b) + x**6*(19*a*b**2*c*d**2/3 + b**3*c**3)/(7*b) + x**5*(3*a**2*b 
*d**3 + 9*a*b**2*c**2*d - 7*a*(21*a*b**2*d**3/10 + 3*b**3*c**2*d)/(8*b))/( 
6*b) + x**4*(9*a**2*b*c*d**2 + 3*a*b**2*c**3 - 6*a*(19*a*b**2*c*d**2/3 + b 
**3*c**3)/(7*b))/(5*b) + x**3*(a**3*d**3 + 9*a**2*b*c**2*d - 5*a*(3*a**2*b 
*d**3 + 9*a*b**2*c**2*d - 7*a*(21*a*b**2*d**3/10 + 3*b**3*c**2*d)/(8*b))/( 
6*b))/(4*b) + x**2*(3*a**3*c*d**2 + 3*a**2*b*c**3 - 4*a*(9*a**2*b*c*d**2 + 
 3*a*b**2*c**3 - 6*a*(19*a*b**2*c*d**2/3 + b**3*c**3)/(7*b))/(5*b))/(3*b) 
+ x*(3*a**3*c**2*d - 3*a*(a**3*d**3 + 9*a**2*b*c**2*d - 5*a*(3*a**2*b*d**3 
 + 9*a*b**2*c**2*d - 7*a*(21*a*b**2*d**3/10 + 3*b**3*c**2*d)/(8*b))/(6*b)) 
/(4*b))/(2*b) + (a**3*c**3 - 2*a*(3*a**3*c*d**2 + 3*a**2*b*c**3 - 4*a*(9*a 
**2*b*c*d**2 + 3*a*b**2*c**3 - 6*a*(19*a*b**2*c*d**2/3 + b**3*c**3)/(7*b)) 
/(5*b))/(3*b))/b), Ne(b, 0)), (a**(5/2)*(c**3*x**2/2 + c**2*d*x**3 + 3*c*d 
**2*x**4/4 + d**3*x**5/5), True))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.12 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} d^{3} x^{3}}{10 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} c d^{2} x^{2}}{3 \, b} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} c^{2} d x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a c^{2} d x}{16 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} c^{2} d x}{64 \, b} - \frac {15 \, \sqrt {b x^{2} + a} a^{3} c^{2} d x}{128 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a d^{3} x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} d^{3} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} d^{3} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{4} d^{3} x}{256 \, b^{2}} - \frac {15 \, a^{4} c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} + \frac {3 \, a^{5} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} c^{3}}{7 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a c d^{2}}{21 \, b^{2}} \] Input:

integrate(x*(d*x+c)^3*(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

1/10*(b*x^2 + a)^(7/2)*d^3*x^3/b + 1/3*(b*x^2 + a)^(7/2)*c*d^2*x^2/b + 3/8 
*(b*x^2 + a)^(7/2)*c^2*d*x/b - 1/16*(b*x^2 + a)^(5/2)*a*c^2*d*x/b - 5/64*( 
b*x^2 + a)^(3/2)*a^2*c^2*d*x/b - 15/128*sqrt(b*x^2 + a)*a^3*c^2*d*x/b - 3/ 
80*(b*x^2 + a)^(7/2)*a*d^3*x/b^2 + 1/160*(b*x^2 + a)^(5/2)*a^2*d^3*x/b^2 + 
 1/128*(b*x^2 + a)^(3/2)*a^3*d^3*x/b^2 + 3/256*sqrt(b*x^2 + a)*a^4*d^3*x/b 
^2 - 15/128*a^4*c^2*d*arcsinh(b*x/sqrt(a*b))/b^(3/2) + 3/256*a^5*d^3*arcsi 
nh(b*x/sqrt(a*b))/b^(5/2) + 1/7*(b*x^2 + a)^(7/2)*c^3/b - 2/21*(b*x^2 + a) 
^(7/2)*a*c*d^2/b^2
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.25 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\frac {1}{26880} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (3 \, b^{2} d^{3} x + 10 \, b^{2} c d^{2}\right )} x + \frac {9 \, {\left (10 \, b^{10} c^{2} d + 7 \, a b^{9} d^{3}\right )}}{b^{8}}\right )} x + \frac {80 \, {\left (3 \, b^{10} c^{3} + 19 \, a b^{9} c d^{2}\right )}}{b^{8}}\right )} x + \frac {21 \, {\left (170 \, a b^{9} c^{2} d + 31 \, a^{2} b^{8} d^{3}\right )}}{b^{8}}\right )} x + \frac {480 \, {\left (3 \, a b^{9} c^{3} + 5 \, a^{2} b^{8} c d^{2}\right )}}{b^{8}}\right )} x + \frac {105 \, {\left (118 \, a^{2} b^{8} c^{2} d + a^{3} b^{7} d^{3}\right )}}{b^{8}}\right )} x + \frac {640 \, {\left (9 \, a^{2} b^{8} c^{3} + a^{3} b^{7} c d^{2}\right )}}{b^{8}}\right )} x + \frac {315 \, {\left (10 \, a^{3} b^{7} c^{2} d - a^{4} b^{6} d^{3}\right )}}{b^{8}}\right )} x + \frac {1280 \, {\left (3 \, a^{3} b^{7} c^{3} - 2 \, a^{4} b^{6} c d^{2}\right )}}{b^{8}}\right )} + \frac {3 \, {\left (10 \, a^{4} b c^{2} d - a^{5} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {5}{2}}} \] Input:

integrate(x*(d*x+c)^3*(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

1/26880*sqrt(b*x^2 + a)*((2*((4*((2*(7*(8*(3*b^2*d^3*x + 10*b^2*c*d^2)*x + 
 9*(10*b^10*c^2*d + 7*a*b^9*d^3)/b^8)*x + 80*(3*b^10*c^3 + 19*a*b^9*c*d^2) 
/b^8)*x + 21*(170*a*b^9*c^2*d + 31*a^2*b^8*d^3)/b^8)*x + 480*(3*a*b^9*c^3 
+ 5*a^2*b^8*c*d^2)/b^8)*x + 105*(118*a^2*b^8*c^2*d + a^3*b^7*d^3)/b^8)*x + 
 640*(9*a^2*b^8*c^3 + a^3*b^7*c*d^2)/b^8)*x + 315*(10*a^3*b^7*c^2*d - a^4* 
b^6*d^3)/b^8)*x + 1280*(3*a^3*b^7*c^3 - 2*a^4*b^6*c*d^2)/b^8) + 3/256*(10* 
a^4*b*c^2*d - a^5*d^3)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int x (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\int x\,{\left (b\,x^2+a\right )}^{5/2}\,{\left (c+d\,x\right )}^3 \,d x \] Input:

int(x*(a + b*x^2)^(5/2)*(c + d*x)^3,x)
 

Output:

int(x*(a + b*x^2)^(5/2)*(c + d*x)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.72 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\frac {-2560 \sqrt {b \,x^{2}+a}\, a^{4} b c \,d^{2}-315 \sqrt {b \,x^{2}+a}\, a^{4} b \,d^{3} x +3840 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c^{3}+3150 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c^{2} d x +1280 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c \,d^{2} x^{2}+210 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d^{3} x^{3}+11520 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c^{3} x^{2}+24780 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c^{2} d \,x^{3}+19200 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c \,d^{2} x^{4}+5208 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d^{3} x^{5}+11520 \sqrt {b \,x^{2}+a}\, a \,b^{4} c^{3} x^{4}+28560 \sqrt {b \,x^{2}+a}\, a \,b^{4} c^{2} d \,x^{5}+24320 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,d^{2} x^{6}+7056 \sqrt {b \,x^{2}+a}\, a \,b^{4} d^{3} x^{7}+3840 \sqrt {b \,x^{2}+a}\, b^{5} c^{3} x^{6}+10080 \sqrt {b \,x^{2}+a}\, b^{5} c^{2} d \,x^{7}+8960 \sqrt {b \,x^{2}+a}\, b^{5} c \,d^{2} x^{8}+2688 \sqrt {b \,x^{2}+a}\, b^{5} d^{3} x^{9}+315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5} d^{3}-3150 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b \,c^{2} d}{26880 b^{3}} \] Input:

int(x*(d*x+c)^3*(b*x^2+a)^(5/2),x)
 

Output:

( - 2560*sqrt(a + b*x**2)*a**4*b*c*d**2 - 315*sqrt(a + b*x**2)*a**4*b*d**3 
*x + 3840*sqrt(a + b*x**2)*a**3*b**2*c**3 + 3150*sqrt(a + b*x**2)*a**3*b** 
2*c**2*d*x + 1280*sqrt(a + b*x**2)*a**3*b**2*c*d**2*x**2 + 210*sqrt(a + b* 
x**2)*a**3*b**2*d**3*x**3 + 11520*sqrt(a + b*x**2)*a**2*b**3*c**3*x**2 + 2 
4780*sqrt(a + b*x**2)*a**2*b**3*c**2*d*x**3 + 19200*sqrt(a + b*x**2)*a**2* 
b**3*c*d**2*x**4 + 5208*sqrt(a + b*x**2)*a**2*b**3*d**3*x**5 + 11520*sqrt( 
a + b*x**2)*a*b**4*c**3*x**4 + 28560*sqrt(a + b*x**2)*a*b**4*c**2*d*x**5 + 
 24320*sqrt(a + b*x**2)*a*b**4*c*d**2*x**6 + 7056*sqrt(a + b*x**2)*a*b**4* 
d**3*x**7 + 3840*sqrt(a + b*x**2)*b**5*c**3*x**6 + 10080*sqrt(a + b*x**2)* 
b**5*c**2*d*x**7 + 8960*sqrt(a + b*x**2)*b**5*c*d**2*x**8 + 2688*sqrt(a + 
b*x**2)*b**5*d**3*x**9 + 315*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sq 
rt(a))*a**5*d**3 - 3150*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a) 
)*a**4*b*c**2*d)/(26880*b**3)