\(\int x^4 (c+d x)^2 (a+b x^2)^{3/2} \, dx\) [1046]

Optimal result

Integrand size = 22, antiderivative size = 274 \[ \int x^4 (c+d x)^2 \left (a+b x^2\right )^{3/2} \, dx=-\frac {3 a^3 \left (2 b c^2-a d^2\right ) x \sqrt {a+b x^2}}{256 b^3}+\frac {a^2 \left (2 b c^2-a d^2\right ) x^3 \sqrt {a+b x^2}}{128 b^2}+\frac {a \left (2 b c^2-a d^2\right ) x^5 \sqrt {a+b x^2}}{32 b}+\frac {\left (2 b c^2-a d^2\right ) x^5 \left (a+b x^2\right )^{3/2}}{16 b}+\frac {2 a^2 c d \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}-\frac {4 a c d \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {2 c d \left (a+b x^2\right )^{9/2}}{9 b^3}+\frac {3 a^4 \left (2 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{7/2}} \] Output:

-3/256*a^3*(-a*d^2+2*b*c^2)*x*(b*x^2+a)^(1/2)/b^3+1/128*a^2*(-a*d^2+2*b*c^ 
2)*x^3*(b*x^2+a)^(1/2)/b^2+1/32*a*(-a*d^2+2*b*c^2)*x^5*(b*x^2+a)^(1/2)/b+1 
/16*(-a*d^2+2*b*c^2)*x^5*(b*x^2+a)^(3/2)/b+2/5*a^2*c*d*(b*x^2+a)^(5/2)/b^3 
+1/10*d^2*x^5*(b*x^2+a)^(5/2)/b-4/7*a*c*d*(b*x^2+a)^(7/2)/b^3+2/9*c*d*(b*x 
^2+a)^(9/2)/b^3+3/256*a^4*(-a*d^2+2*b*c^2)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/ 
2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 1.57 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.83 \[ \int x^4 (c+d x)^2 \left (a+b x^2\right )^{3/2} \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (a^4 d (4096 c+945 d x)+224 b^4 x^7 \left (45 c^2+80 c d x+36 d^2 x^2\right )+12 a^2 b^2 x^3 \left (105 c^2+128 c d x+42 d^2 x^2\right )-2 a^3 b x \left (945 c^2+1024 c d x+315 d^2 x^2\right )+16 a b^3 x^5 \left (945 c^2+1600 c d x+693 d^2 x^2\right )\right )+1890 a^5 d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}-\sqrt {a+b x^2}}\right )+3780 a^4 b c^2 \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{80640 b^{7/2}} \] Input:

Integrate[x^4*(c + d*x)^2*(a + b*x^2)^(3/2),x]
 

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(a^4*d*(4096*c + 945*d*x) + 224*b^4*x^7*(45*c^2 + 
 80*c*d*x + 36*d^2*x^2) + 12*a^2*b^2*x^3*(105*c^2 + 128*c*d*x + 42*d^2*x^2 
) - 2*a^3*b*x*(945*c^2 + 1024*c*d*x + 315*d^2*x^2) + 16*a*b^3*x^5*(945*c^2 
 + 1600*c*d*x + 693*d^2*x^2)) + 1890*a^5*d^2*ArcTanh[(Sqrt[b]*x)/(Sqrt[a] 
- Sqrt[a + b*x^2])] + 3780*a^4*b*c^2*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[ 
a + b*x^2])])/(80640*b^(7/2))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.97, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {541, 27, 533, 27, 533, 25, 27, 533, 27, 533, 27, 455, 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.

\(\displaystyle \int x^4 \left (a+b x^2\right )^{3/2} (c+d x)^2 \, dx\)

\(\Big \downarrow \) 541

\(\displaystyle \frac {\int 5 x^4 \left (2 b c^2+4 b d x c-a d^2\right ) \left (b x^2+a\right )^{3/2}dx}{10 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x^4 \left (2 b c^2+4 b d x c-a d^2\right ) \left (b x^2+a\right )^{3/2}dx}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}-\frac {\int b x^3 \left (16 a c d-9 \left (2 b c^2-a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{9 b}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}-\frac {1}{9} \int x^3 \left (16 a c d-9 \left (2 b c^2-a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {1}{9} \left (\frac {\int -a x^2 \left (27 \left (2 b c^2-a d^2\right )+128 b c d x\right ) \left (b x^2+a\right )^{3/2}dx}{8 b}+\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {\int a x^2 \left (27 \left (2 b c^2-a d^2\right )+128 b c d x\right ) \left (b x^2+a\right )^{3/2}dx}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \int x^2 \left (27 \left (2 b c^2-a d^2\right )+128 b c d x\right ) \left (b x^2+a\right )^{3/2}dx}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \left (\frac {128}{7} c d x^2 \left (a+b x^2\right )^{5/2}-\frac {\int b x \left (256 a c d-189 \left (2 b c^2-a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b}\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \left (\frac {128}{7} c d x^2 \left (a+b x^2\right )^{5/2}-\frac {1}{7} \int x \left (256 a c d-189 \left (2 b c^2-a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}dx\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \left (\frac {1}{7} \left (\frac {\int -3 a \left (63 \left (2 b c^2-a d^2\right )+512 b c d x\right ) \left (b x^2+a\right )^{3/2}dx}{6 b}+\frac {63 x \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{2 b}\right )+\frac {128}{7} c d x^2 \left (a+b x^2\right )^{5/2}\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \left (\frac {1}{7} \left (\frac {63 x \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{2 b}-\frac {a \int \left (63 \left (2 b c^2-a d^2\right )+512 b c d x\right ) \left (b x^2+a\right )^{3/2}dx}{2 b}\right )+\frac {128}{7} c d x^2 \left (a+b x^2\right )^{5/2}\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \left (\frac {1}{7} \left (\frac {63 x \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{2 b}-\frac {a \left (63 \left (2 b c^2-a d^2\right ) \int \left (b x^2+a\right )^{3/2}dx+\frac {512}{5} c d \left (a+b x^2\right )^{5/2}\right )}{2 b}\right )+\frac {128}{7} c d x^2 \left (a+b x^2\right )^{5/2}\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \left (\frac {1}{7} \left (\frac {63 x \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{2 b}-\frac {a \left (63 \left (2 b c^2-a d^2\right ) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {512}{5} c d \left (a+b x^2\right )^{5/2}\right )}{2 b}\right )+\frac {128}{7} c d x^2 \left (a+b x^2\right )^{5/2}\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \left (\frac {1}{7} \left (\frac {63 x \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{2 b}-\frac {a \left (63 \left (2 b c^2-a d^2\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {512}{5} c d \left (a+b x^2\right )^{5/2}\right )}{2 b}\right )+\frac {128}{7} c d x^2 \left (a+b x^2\right )^{5/2}\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \left (\frac {1}{7} \left (\frac {63 x \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{2 b}-\frac {a \left (63 \left (2 b c^2-a d^2\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {512}{5} c d \left (a+b x^2\right )^{5/2}\right )}{2 b}\right )+\frac {128}{7} c d x^2 \left (a+b x^2\right )^{5/2}\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{8 b}-\frac {a \left (\frac {1}{7} \left (\frac {63 x \left (a+b x^2\right )^{5/2} \left (2 b c^2-a d^2\right )}{2 b}-\frac {a \left (63 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) \left (2 b c^2-a d^2\right )+\frac {512}{5} c d \left (a+b x^2\right )^{5/2}\right )}{2 b}\right )+\frac {128}{7} c d x^2 \left (a+b x^2\right )^{5/2}\right )}{8 b}\right )+\frac {4}{9} c d x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {d^2 x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

Input:

Int[x^4*(c + d*x)^2*(a + b*x^2)^(3/2),x]
 

Output:

(d^2*x^5*(a + b*x^2)^(5/2))/(10*b) + ((4*c*d*x^4*(a + b*x^2)^(5/2))/9 + (( 
9*(2*b*c^2 - a*d^2)*x^3*(a + b*x^2)^(5/2))/(8*b) - (a*((128*c*d*x^2*(a + b 
*x^2)^(5/2))/7 + ((63*(2*b*c^2 - a*d^2)*x*(a + b*x^2)^(5/2))/(2*b) - (a*(( 
512*c*d*(a + b*x^2)^(5/2))/5 + 63*(2*b*c^2 - a*d^2)*((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)))/(2*b))/7))/(8*b))/9)/(2*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.77

method	result	size
risch	\(\frac {\left (8064 b^{4} d^{2} x^{9}+17920 d \,b^{4} c \,x^{8}+11088 a \,b^{3} d^{2} x^{7}+10080 b^{4} c^{2} x^{7}+25600 a c d \,b^{3} x^{6}+504 a^{2} b^{2} d^{2} x^{5}+15120 a \,b^{3} c^{2} x^{5}+1536 a^{2} b^{2} c d \,x^{4}-630 a^{3} b \,d^{2} x^{3}+1260 a^{2} b^{2} c^{2} x^{3}-2048 a^{3} d c b \,x^{2}+945 a^{4} d^{2} x -1890 a^{3} b \,c^{2} x +4096 a^{4} d c \right ) \sqrt {b \,x^{2}+a}}{80640 b^{3}}-\frac {3 a^{4} \left (a \,d^{2}-2 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {7}{2}}}\)	\(210\)
default	\(c^{2} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+d^{2} \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )+2 c d \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )\)	\(289\)

Input:

int(x^4*(d*x+c)^2*(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/80640/b^3*(8064*b^4*d^2*x^9+17920*b^4*c*d*x^8+11088*a*b^3*d^2*x^7+10080* 
b^4*c^2*x^7+25600*a*b^3*c*d*x^6+504*a^2*b^2*d^2*x^5+15120*a*b^3*c^2*x^5+15 
36*a^2*b^2*c*d*x^4-630*a^3*b*d^2*x^3+1260*a^2*b^2*c^2*x^3-2048*a^3*b*c*d*x 
^2+945*a^4*d^2*x-1890*a^3*b*c^2*x+4096*a^4*c*d)*(b*x^2+a)^(1/2)-3/256*a^4* 
(a*d^2-2*b*c^2)/b^(7/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.68 \[ \int x^4 (c+d x)^2 \left (a+b x^2\right )^{3/2} \, dx=\left [-\frac {945 \, {\left (2 \, a^{4} b c^{2} - a^{5} d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8064 \, b^{5} d^{2} x^{9} + 17920 \, b^{5} c d x^{8} + 25600 \, a b^{4} c d x^{6} + 1536 \, a^{2} b^{3} c d x^{4} - 2048 \, a^{3} b^{2} c d x^{2} + 1008 \, {\left (10 \, b^{5} c^{2} + 11 \, a b^{4} d^{2}\right )} x^{7} + 4096 \, a^{4} b c d + 504 \, {\left (30 \, a b^{4} c^{2} + a^{2} b^{3} d^{2}\right )} x^{5} + 630 \, {\left (2 \, a^{2} b^{3} c^{2} - a^{3} b^{2} d^{2}\right )} x^{3} - 945 \, {\left (2 \, a^{3} b^{2} c^{2} - a^{4} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{161280 \, b^{4}}, -\frac {945 \, {\left (2 \, a^{4} b c^{2} - a^{5} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8064 \, b^{5} d^{2} x^{9} + 17920 \, b^{5} c d x^{8} + 25600 \, a b^{4} c d x^{6} + 1536 \, a^{2} b^{3} c d x^{4} - 2048 \, a^{3} b^{2} c d x^{2} + 1008 \, {\left (10 \, b^{5} c^{2} + 11 \, a b^{4} d^{2}\right )} x^{7} + 4096 \, a^{4} b c d + 504 \, {\left (30 \, a b^{4} c^{2} + a^{2} b^{3} d^{2}\right )} x^{5} + 630 \, {\left (2 \, a^{2} b^{3} c^{2} - a^{3} b^{2} d^{2}\right )} x^{3} - 945 \, {\left (2 \, a^{3} b^{2} c^{2} - a^{4} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{80640 \, b^{4}}\right ] \] Input:

integrate(x^4*(d*x+c)^2*(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

[-1/161280*(945*(2*a^4*b*c^2 - a^5*d^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^ 
2 + a)*sqrt(b)*x - a) - 2*(8064*b^5*d^2*x^9 + 17920*b^5*c*d*x^8 + 25600*a* 
b^4*c*d*x^6 + 1536*a^2*b^3*c*d*x^4 - 2048*a^3*b^2*c*d*x^2 + 1008*(10*b^5*c 
^2 + 11*a*b^4*d^2)*x^7 + 4096*a^4*b*c*d + 504*(30*a*b^4*c^2 + a^2*b^3*d^2) 
*x^5 + 630*(2*a^2*b^3*c^2 - a^3*b^2*d^2)*x^3 - 945*(2*a^3*b^2*c^2 - a^4*b* 
d^2)*x)*sqrt(b*x^2 + a))/b^4, -1/80640*(945*(2*a^4*b*c^2 - a^5*d^2)*sqrt(- 
b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (8064*b^5*d^2*x^9 + 17920*b^5*c*d* 
x^8 + 25600*a*b^4*c*d*x^6 + 1536*a^2*b^3*c*d*x^4 - 2048*a^3*b^2*c*d*x^2 + 
1008*(10*b^5*c^2 + 11*a*b^4*d^2)*x^7 + 4096*a^4*b*c*d + 504*(30*a*b^4*c^2 
+ a^2*b^3*d^2)*x^5 + 630*(2*a^2*b^3*c^2 - a^3*b^2*d^2)*x^3 - 945*(2*a^3*b^ 
2*c^2 - a^4*b*d^2)*x)*sqrt(b*x^2 + a))/b^4]
 

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.52 \[ \int x^4 (c+d x)^2 \left (a+b x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 a^{2} \left (a^{2} c^{2} - \frac {5 a \left (a^{2} d^{2} + 2 a b c^{2} - \frac {7 a \left (\frac {11 a b d^{2}}{10} + b^{2} c^{2}\right )}{8 b}\right )}{6 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b^{2}} + \sqrt {a + b x^{2}} \cdot \left (\frac {16 a^{4} c d}{315 b^{3}} - \frac {8 a^{3} c d x^{2}}{315 b^{2}} + \frac {2 a^{2} c d x^{4}}{105 b} + \frac {20 a c d x^{6}}{63} - \frac {3 a x \left (a^{2} c^{2} - \frac {5 a \left (a^{2} d^{2} + 2 a b c^{2} - \frac {7 a \left (\frac {11 a b d^{2}}{10} + b^{2} c^{2}\right )}{8 b}\right )}{6 b}\right )}{8 b^{2}} + \frac {2 b c d x^{8}}{9} + \frac {b d^{2} x^{9}}{10} + \frac {x^{7} \cdot \left (\frac {11 a b d^{2}}{10} + b^{2} c^{2}\right )}{8 b} + \frac {x^{5} \left (a^{2} d^{2} + 2 a b c^{2} - \frac {7 a \left (\frac {11 a b d^{2}}{10} + b^{2} c^{2}\right )}{8 b}\right )}{6 b} + \frac {x^{3} \left (a^{2} c^{2} - \frac {5 a \left (a^{2} d^{2} + 2 a b c^{2} - \frac {7 a \left (\frac {11 a b d^{2}}{10} + b^{2} c^{2}\right )}{8 b}\right )}{6 b}\right )}{4 b}\right ) & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {c^{2} x^{5}}{5} + \frac {c d x^{6}}{3} + \frac {d^{2} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \] Input:

integrate(x**4*(d*x+c)**2*(b*x**2+a)**(3/2),x)
 

Output:

Piecewise((3*a**2*(a**2*c**2 - 5*a*(a**2*d**2 + 2*a*b*c**2 - 7*a*(11*a*b*d 
**2/10 + b**2*c**2)/(8*b))/(6*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))/(8*b**2) + s 
qrt(a + b*x**2)*(16*a**4*c*d/(315*b**3) - 8*a**3*c*d*x**2/(315*b**2) + 2*a 
**2*c*d*x**4/(105*b) + 20*a*c*d*x**6/63 - 3*a*x*(a**2*c**2 - 5*a*(a**2*d** 
2 + 2*a*b*c**2 - 7*a*(11*a*b*d**2/10 + b**2*c**2)/(8*b))/(6*b))/(8*b**2) + 
 2*b*c*d*x**8/9 + b*d**2*x**9/10 + x**7*(11*a*b*d**2/10 + b**2*c**2)/(8*b) 
 + x**5*(a**2*d**2 + 2*a*b*c**2 - 7*a*(11*a*b*d**2/10 + b**2*c**2)/(8*b))/ 
(6*b) + x**3*(a**2*c**2 - 5*a*(a**2*d**2 + 2*a*b*c**2 - 7*a*(11*a*b*d**2/1 
0 + b**2*c**2)/(8*b))/(6*b))/(4*b)), Ne(b, 0)), (a**(3/2)*(c**2*x**5/5 + c 
*d*x**6/3 + d**2*x**7/7), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.04 \[ \int x^4 (c+d x)^2 \left (a+b x^2\right )^{3/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d^{2} x^{5}}{10 \, b} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c d x^{4}}{9 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2} x^{3}}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a d^{2} x^{3}}{16 \, b^{2}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a c d x^{2}}{63 \, b^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a c^{2} x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} c^{2} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{3} c^{2} x}{128 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} d^{2} x}{32 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} d^{2} x}{128 \, b^{3}} - \frac {3 \, \sqrt {b x^{2} + a} a^{4} d^{2} x}{256 \, b^{3}} + \frac {3 \, a^{4} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} - \frac {3 \, a^{5} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {7}{2}}} + \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} c d}{315 \, b^{3}} \] Input:

integrate(x^4*(d*x+c)^2*(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

1/10*(b*x^2 + a)^(5/2)*d^2*x^5/b + 2/9*(b*x^2 + a)^(5/2)*c*d*x^4/b + 1/8*( 
b*x^2 + a)^(5/2)*c^2*x^3/b - 1/16*(b*x^2 + a)^(5/2)*a*d^2*x^3/b^2 - 8/63*( 
b*x^2 + a)^(5/2)*a*c*d*x^2/b^2 - 1/16*(b*x^2 + a)^(5/2)*a*c^2*x/b^2 + 1/64 
*(b*x^2 + a)^(3/2)*a^2*c^2*x/b^2 + 3/128*sqrt(b*x^2 + a)*a^3*c^2*x/b^2 + 1 
/32*(b*x^2 + a)^(5/2)*a^2*d^2*x/b^3 - 1/128*(b*x^2 + a)^(3/2)*a^3*d^2*x/b^ 
3 - 3/256*sqrt(b*x^2 + a)*a^4*d^2*x/b^3 + 3/128*a^4*c^2*arcsinh(b*x/sqrt(a 
*b))/b^(5/2) - 3/256*a^5*d^2*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 16/315*(b*x^ 
2 + a)^(5/2)*a^2*c*d/b^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.86 \[ \int x^4 (c+d x)^2 \left (a+b x^2\right )^{3/2} \, dx=\frac {1}{80640} \, {\left (\frac {4096 \, a^{4} c d}{b^{3}} - {\left (2 \, {\left (\frac {1024 \, a^{3} c d}{b^{2}} - {\left (4 \, {\left (\frac {192 \, a^{2} c d}{b} + {\left (2 \, {\left (1600 \, a c d + 7 \, {\left (8 \, {\left (9 \, b d^{2} x + 20 \, b c d\right )} x + \frac {9 \, {\left (10 \, b^{9} c^{2} + 11 \, a b^{8} d^{2}\right )}}{b^{8}}\right )} x\right )} x + \frac {63 \, {\left (30 \, a b^{8} c^{2} + a^{2} b^{7} d^{2}\right )}}{b^{8}}\right )} x\right )} x + \frac {315 \, {\left (2 \, a^{2} b^{7} c^{2} - a^{3} b^{6} d^{2}\right )}}{b^{8}}\right )} x\right )} x + \frac {945 \, {\left (2 \, a^{3} b^{6} c^{2} - a^{4} b^{5} d^{2}\right )}}{b^{8}}\right )} x\right )} \sqrt {b x^{2} + a} - \frac {3 \, {\left (2 \, a^{4} b c^{2} - a^{5} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {7}{2}}} \] Input:

integrate(x^4*(d*x+c)^2*(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

1/80640*(4096*a^4*c*d/b^3 - (2*(1024*a^3*c*d/b^2 - (4*(192*a^2*c*d/b + (2* 
(1600*a*c*d + 7*(8*(9*b*d^2*x + 20*b*c*d)*x + 9*(10*b^9*c^2 + 11*a*b^8*d^2 
)/b^8)*x)*x + 63*(30*a*b^8*c^2 + a^2*b^7*d^2)/b^8)*x)*x + 315*(2*a^2*b^7*c 
^2 - a^3*b^6*d^2)/b^8)*x)*x + 945*(2*a^3*b^6*c^2 - a^4*b^5*d^2)/b^8)*x)*sq 
rt(b*x^2 + a) - 3/256*(2*a^4*b*c^2 - a^5*d^2)*log(abs(-sqrt(b)*x + sqrt(b* 
x^2 + a)))/b^(7/2)
 

Mupad [F(-1)]

Timed out. \[ \int x^4 (c+d x)^2 \left (a+b x^2\right )^{3/2} \, dx=\int x^4\,{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2 \,d x \] Input:

int(x^4*(a + b*x^2)^(3/2)*(c + d*x)^2,x)
 

Output:

int(x^4*(a + b*x^2)^(3/2)*(c + d*x)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.25 \[ \int x^4 (c+d x)^2 \left (a+b x^2\right )^{3/2} \, dx=\frac {4096 \sqrt {b \,x^{2}+a}\, a^{4} b c d +945 \sqrt {b \,x^{2}+a}\, a^{4} b \,d^{2} x -1890 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c^{2} x -2048 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c d \,x^{2}-630 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d^{2} x^{3}+1260 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c^{2} x^{3}+1536 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c d \,x^{4}+504 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d^{2} x^{5}+15120 \sqrt {b \,x^{2}+a}\, a \,b^{4} c^{2} x^{5}+25600 \sqrt {b \,x^{2}+a}\, a \,b^{4} c d \,x^{6}+11088 \sqrt {b \,x^{2}+a}\, a \,b^{4} d^{2} x^{7}+10080 \sqrt {b \,x^{2}+a}\, b^{5} c^{2} x^{7}+17920 \sqrt {b \,x^{2}+a}\, b^{5} c d \,x^{8}+8064 \sqrt {b \,x^{2}+a}\, b^{5} d^{2} x^{9}-945 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5} d^{2}+1890 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b \,c^{2}}{80640 b^{4}} \] Input:

int(x^4*(d*x+c)^2*(b*x^2+a)^(3/2),x)
 

Output:

(4096*sqrt(a + b*x**2)*a**4*b*c*d + 945*sqrt(a + b*x**2)*a**4*b*d**2*x - 1 
890*sqrt(a + b*x**2)*a**3*b**2*c**2*x - 2048*sqrt(a + b*x**2)*a**3*b**2*c* 
d*x**2 - 630*sqrt(a + b*x**2)*a**3*b**2*d**2*x**3 + 1260*sqrt(a + b*x**2)* 
a**2*b**3*c**2*x**3 + 1536*sqrt(a + b*x**2)*a**2*b**3*c*d*x**4 + 504*sqrt( 
a + b*x**2)*a**2*b**3*d**2*x**5 + 15120*sqrt(a + b*x**2)*a*b**4*c**2*x**5 
+ 25600*sqrt(a + b*x**2)*a*b**4*c*d*x**6 + 11088*sqrt(a + b*x**2)*a*b**4*d 
**2*x**7 + 10080*sqrt(a + b*x**2)*b**5*c**2*x**7 + 17920*sqrt(a + b*x**2)* 
b**5*c*d*x**8 + 8064*sqrt(a + b*x**2)*b**5*d**2*x**9 - 945*sqrt(b)*log((sq 
rt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**5*d**2 + 1890*sqrt(b)*log((sqrt(a 
+ b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b*c**2)/(80640*b**4)