\(\int x (c+d x)^3 (a+b x^2)^{3/2} \, dx\) [1063]

Optimal result

Integrand size = 20, antiderivative size = 218 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=-\frac {3 a^2 d \left (8 b c^2-a d^2\right ) x \sqrt {a+b x^2}}{128 b^2}-\frac {a d \left (8 b c^2-a d^2\right ) x \left (a+b x^2\right )^{3/2}}{64 b^2}+\frac {3 c (c+d x)^2 \left (a+b x^2\right )^{5/2}}{56 b}+\frac {(c+d x)^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {\left (12 c \left (b c^2-8 a d^2\right )+5 d \left (2 b c^2-7 a d^2\right ) x\right ) \left (a+b x^2\right )^{5/2}}{560 b^2}-\frac {3 a^3 d \left (8 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \] Output:

-3/128*a^2*d*(-a*d^2+8*b*c^2)*x*(b*x^2+a)^(1/2)/b^2-1/64*a*d*(-a*d^2+8*b*c 
^2)*x*(b*x^2+a)^(3/2)/b^2+3/56*c*(d*x+c)^2*(b*x^2+a)^(5/2)/b+1/8*(d*x+c)^3 
*(b*x^2+a)^(5/2)/b+1/560*(12*c*(-8*a*d^2+b*c^2)+5*d*(-7*a*d^2+2*b*c^2)*x)* 
(b*x^2+a)^(5/2)/b^2-3/128*a^3*d*(-a*d^2+8*b*c^2)*arctanh(b^(1/2)*x/(b*x^2+ 
a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.90 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (-3 a^3 d^2 (256 c+35 d x)+16 b^3 x^4 \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )+2 a^2 b \left (448 c^3+420 c^2 d x+192 c d^2 x^2+35 d^3 x^3\right )+8 a b^2 x^2 \left (224 c^3+490 c^2 d x+384 c d^2 x^2+105 d^3 x^3\right )\right )-105 a^3 d \left (-8 b c^2+a d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{4480 b^{5/2}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(-3*a^3*d^2*(256*c + 35*d*x) + 16*b^3*x^4*(56*c^3 
 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x^3) + 2*a^2*b*(448*c^3 + 420*c^2* 
d*x + 192*c*d^2*x^2 + 35*d^3*x^3) + 8*a*b^2*x^2*(224*c^3 + 490*c^2*d*x + 3 
84*c*d^2*x^2 + 105*d^3*x^3)) - 105*a^3*d*(-8*b*c^2 + a*d^2)*Log[-(Sqrt[b]* 
x) + Sqrt[a + b*x^2]])/(4480*b^(5/2))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.99, 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, 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.

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

\(\Big \downarrow \) 541

\(\displaystyle \frac {\int x \left (b x^2+a\right )^{3/2} \left (8 b c^3+24 b d^2 x^2 c+3 d \left (8 b c^2-a d^2\right ) x\right )dx}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {\int b x \left (8 c \left (7 b c^2-6 a d^2\right )+21 d \left (8 b c^2-a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b}+\frac {24}{7} c d^2 x^2 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} \int x \left (8 c \left (7 b c^2-6 a d^2\right )+21 d \left (8 b c^2-a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}dx+\frac {24}{7} c d^2 x^2 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 d x \left (a+b x^2\right )^{5/2} \left (8 b c^2-a d^2\right )}{2 b}-\frac {\int 3 \left (7 a d \left (8 b c^2-a d^2\right )-16 b c \left (7 b c^2-6 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{6 b}\right )+\frac {24}{7} c d^2 x^2 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 d x \left (a+b x^2\right )^{5/2} \left (8 b c^2-a d^2\right )}{2 b}-\frac {\int \left (7 a d \left (8 b c^2-a d^2\right )-16 b c \left (7 b c^2-6 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{2 b}\right )+\frac {24}{7} c d^2 x^2 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 d x \left (a+b x^2\right )^{5/2} \left (8 b c^2-a d^2\right )}{2 b}-\frac {7 a d \left (8 b c^2-a d^2\right ) \int \left (b x^2+a\right )^{3/2}dx-\frac {16}{5} c \left (a+b x^2\right )^{5/2} \left (7 b c^2-6 a d^2\right )}{2 b}\right )+\frac {24}{7} c d^2 x^2 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 d x \left (a+b x^2\right )^{5/2} \left (8 b c^2-a d^2\right )}{2 b}-\frac {7 a d \left (8 b c^2-a d^2\right ) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {16}{5} c \left (a+b x^2\right )^{5/2} \left (7 b c^2-6 a d^2\right )}{2 b}\right )+\frac {24}{7} c d^2 x^2 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 d x \left (a+b x^2\right )^{5/2} \left (8 b c^2-a d^2\right )}{2 b}-\frac {7 a d \left (8 b c^2-a d^2\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {16}{5} c \left (a+b x^2\right )^{5/2} \left (7 b c^2-6 a d^2\right )}{2 b}\right )+\frac {24}{7} c d^2 x^2 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 d x \left (a+b x^2\right )^{5/2} \left (8 b c^2-a d^2\right )}{2 b}-\frac {7 a d \left (8 b c^2-a d^2\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {16}{5} c \left (a+b x^2\right )^{5/2} \left (7 b c^2-6 a d^2\right )}{2 b}\right )+\frac {24}{7} c d^2 x^2 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 d x \left (a+b x^2\right )^{5/2} \left (8 b c^2-a d^2\right )}{2 b}-\frac {7 a d \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) \left (8 b c^2-a d^2\right )-\frac {16}{5} c \left (a+b x^2\right )^{5/2} \left (7 b c^2-6 a d^2\right )}{2 b}\right )+\frac {24}{7} c d^2 x^2 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {d^3 x^3 \left (a+b x^2\right )^{5/2}}{8 b}\)

Input:

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

Output:

(d^3*x^3*(a + b*x^2)^(5/2))/(8*b) + ((24*c*d^2*x^2*(a + b*x^2)^(5/2))/7 + 
((7*d*(8*b*c^2 - a*d^2)*x*(a + b*x^2)^(5/2))/(2*b) - ((-16*c*(7*b*c^2 - 6* 
a*d^2)*(a + b*x^2)^(5/2))/5 + 7*a*d*(8*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)
 

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 = 0.45 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.98

method	result	size
risch	\(-\frac {\left (-560 b^{3} d^{3} x^{7}-1920 b^{3} c \,d^{2} x^{6}-840 a \,b^{2} d^{3} x^{5}-2240 b^{3} c^{2} d \,x^{5}-3072 a \,b^{2} c \,d^{2} x^{4}-896 b^{3} c^{3} x^{4}-70 a^{2} b \,d^{3} x^{3}-3920 a \,b^{2} c^{2} d \,x^{3}-384 a^{2} b c \,d^{2} x^{2}-1792 a \,b^{2} c^{3} x^{2}+105 a^{3} d^{3} x -840 a^{2} b \,c^{2} d x +768 a^{3} c \,d^{2}-896 a^{2} b \,c^{3}\right ) \sqrt {b \,x^{2}+a}}{4480 b^{2}}+\frac {3 a^{3} d \left (a \,d^{2}-8 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {5}{2}}}\)	\(213\)
default	\(\frac {c^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+d^{3} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+3 c \,d^{2} \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+3 c^{2} d \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )\)	\(238\)

Input:

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

Output:

-1/4480/b^2*(-560*b^3*d^3*x^7-1920*b^3*c*d^2*x^6-840*a*b^2*d^3*x^5-2240*b^ 
3*c^2*d*x^5-3072*a*b^2*c*d^2*x^4-896*b^3*c^3*x^4-70*a^2*b*d^3*x^3-3920*a*b 
^2*c^2*d*x^3-384*a^2*b*c*d^2*x^2-1792*a*b^2*c^3*x^2+105*a^3*d^3*x-840*a^2* 
b*c^2*d*x+768*a^3*c*d^2-896*a^2*b*c^3)*(b*x^2+a)^(1/2)+3/128*a^3*d*(a*d^2- 
8*b*c^2)/b^(5/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.15 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\left [-\frac {105 \, {\left (8 \, a^{3} b c^{2} d - a^{4} d^{3}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (560 \, b^{4} d^{3} x^{7} + 1920 \, b^{4} c d^{2} x^{6} + 896 \, a^{2} b^{2} c^{3} - 768 \, a^{3} b c d^{2} + 280 \, {\left (8 \, b^{4} c^{2} d + 3 \, a b^{3} d^{3}\right )} x^{5} + 128 \, {\left (7 \, b^{4} c^{3} + 24 \, a b^{3} c d^{2}\right )} x^{4} + 70 \, {\left (56 \, a b^{3} c^{2} d + a^{2} b^{2} d^{3}\right )} x^{3} + 128 \, {\left (14 \, a b^{3} c^{3} + 3 \, a^{2} b^{2} c d^{2}\right )} x^{2} + 105 \, {\left (8 \, a^{2} b^{2} c^{2} d - a^{3} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{8960 \, b^{3}}, \frac {105 \, {\left (8 \, a^{3} b c^{2} d - a^{4} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (560 \, b^{4} d^{3} x^{7} + 1920 \, b^{4} c d^{2} x^{6} + 896 \, a^{2} b^{2} c^{3} - 768 \, a^{3} b c d^{2} + 280 \, {\left (8 \, b^{4} c^{2} d + 3 \, a b^{3} d^{3}\right )} x^{5} + 128 \, {\left (7 \, b^{4} c^{3} + 24 \, a b^{3} c d^{2}\right )} x^{4} + 70 \, {\left (56 \, a b^{3} c^{2} d + a^{2} b^{2} d^{3}\right )} x^{3} + 128 \, {\left (14 \, a b^{3} c^{3} + 3 \, a^{2} b^{2} c d^{2}\right )} x^{2} + 105 \, {\left (8 \, a^{2} b^{2} c^{2} d - a^{3} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{4480 \, b^{3}}\right ] \] Input:

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

Output:

[-1/8960*(105*(8*a^3*b*c^2*d - a^4*d^3)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^ 
2 + a)*sqrt(b)*x - a) - 2*(560*b^4*d^3*x^7 + 1920*b^4*c*d^2*x^6 + 896*a^2* 
b^2*c^3 - 768*a^3*b*c*d^2 + 280*(8*b^4*c^2*d + 3*a*b^3*d^3)*x^5 + 128*(7*b 
^4*c^3 + 24*a*b^3*c*d^2)*x^4 + 70*(56*a*b^3*c^2*d + a^2*b^2*d^3)*x^3 + 128 
*(14*a*b^3*c^3 + 3*a^2*b^2*c*d^2)*x^2 + 105*(8*a^2*b^2*c^2*d - a^3*b*d^3)* 
x)*sqrt(b*x^2 + a))/b^3, 1/4480*(105*(8*a^3*b*c^2*d - a^4*d^3)*sqrt(-b)*ar 
ctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (560*b^4*d^3*x^7 + 1920*b^4*c*d^2*x^6 + 
 896*a^2*b^2*c^3 - 768*a^3*b*c*d^2 + 280*(8*b^4*c^2*d + 3*a*b^3*d^3)*x^5 + 
 128*(7*b^4*c^3 + 24*a*b^3*c*d^2)*x^4 + 70*(56*a*b^3*c^2*d + a^2*b^2*d^3)* 
x^3 + 128*(14*a*b^3*c^3 + 3*a^2*b^2*c*d^2)*x^2 + 105*(8*a^2*b^2*c^2*d - a^ 
3*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. 464 vs. \(2 (204) = 408\).

Time = 0.60 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.13 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\begin {cases} - \frac {a \left (3 a^{2} c^{2} d - \frac {3 a \left (a^{2} d^{3} + 6 a b c^{2} d - \frac {5 a \left (\frac {9 a b d^{3}}{8} + 3 b^{2} c^{2} d\right )}{6 b}\right )}{4 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2 b} + \sqrt {a + b x^{2}} \cdot \left (\frac {3 b c d^{2} x^{6}}{7} + \frac {b d^{3} x^{7}}{8} + \frac {x^{5} \cdot \left (\frac {9 a b d^{3}}{8} + 3 b^{2} c^{2} d\right )}{6 b} + \frac {x^{4} \cdot \left (\frac {24 a b c d^{2}}{7} + b^{2} c^{3}\right )}{5 b} + \frac {x^{3} \left (a^{2} d^{3} + 6 a b c^{2} d - \frac {5 a \left (\frac {9 a b d^{3}}{8} + 3 b^{2} c^{2} d\right )}{6 b}\right )}{4 b} + \frac {x^{2} \cdot \left (3 a^{2} c d^{2} + 2 a b c^{3} - \frac {4 a \left (\frac {24 a b c d^{2}}{7} + b^{2} c^{3}\right )}{5 b}\right )}{3 b} + \frac {x \left (3 a^{2} c^{2} d - \frac {3 a \left (a^{2} d^{3} + 6 a b c^{2} d - \frac {5 a \left (\frac {9 a b d^{3}}{8} + 3 b^{2} c^{2} d\right )}{6 b}\right )}{4 b}\right )}{2 b} + \frac {a^{2} c^{3} - \frac {2 a \left (3 a^{2} c d^{2} + 2 a b c^{3} - \frac {4 a \left (\frac {24 a b c d^{2}}{7} + b^{2} c^{3}\right )}{5 b}\right )}{3 b}}{b}\right ) & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {c^{3} x^{2}}{2} + c^{2} d x^{3} + \frac {3 c d^{2} x^{4}}{4} + \frac {d^{3} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a*(3*a**2*c**2*d - 3*a*(a**2*d**3 + 6*a*b*c**2*d - 5*a*(9*a*b* 
d**3/8 + 3*b**2*c**2*d)/(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)*(3*b*c*d**2*x**6/7 + b*d**3*x**7/8 + x**5*(9*a*b*d**3/8 + 
 3*b**2*c**2*d)/(6*b) + x**4*(24*a*b*c*d**2/7 + b**2*c**3)/(5*b) + x**3*(a 
**2*d**3 + 6*a*b*c**2*d - 5*a*(9*a*b*d**3/8 + 3*b**2*c**2*d)/(6*b))/(4*b) 
+ x**2*(3*a**2*c*d**2 + 2*a*b*c**3 - 4*a*(24*a*b*c*d**2/7 + b**2*c**3)/(5* 
b))/(3*b) + x*(3*a**2*c**2*d - 3*a*(a**2*d**3 + 6*a*b*c**2*d - 5*a*(9*a*b* 
d**3/8 + 3*b**2*c**2*d)/(6*b))/(4*b))/(2*b) + (a**2*c**3 - 2*a*(3*a**2*c*d 
**2 + 2*a*b*c**3 - 4*a*(24*a*b*c*d**2/7 + b**2*c**3)/(5*b))/(3*b))/b), Ne( 
b, 0)), (a**(3/2)*(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 = 0.04 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.11 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d^{3} x^{3}}{8 \, b} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c d^{2} x^{2}}{7 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2} d x}{2 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a c^{2} d x}{8 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a^{2} c^{2} d x}{16 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a d^{3} x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} d^{3} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{3} d^{3} x}{128 \, b^{2}} - \frac {3 \, a^{3} c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} + \frac {3 \, a^{4} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{3}}{5 \, b} - \frac {6 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a c d^{2}}{35 \, b^{2}} \] Input:

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

Output:

1/8*(b*x^2 + a)^(5/2)*d^3*x^3/b + 3/7*(b*x^2 + a)^(5/2)*c*d^2*x^2/b + 1/2* 
(b*x^2 + a)^(5/2)*c^2*d*x/b - 1/8*(b*x^2 + a)^(3/2)*a*c^2*d*x/b - 3/16*sqr 
t(b*x^2 + a)*a^2*c^2*d*x/b - 1/16*(b*x^2 + a)^(5/2)*a*d^3*x/b^2 + 1/64*(b* 
x^2 + a)^(3/2)*a^2*d^3*x/b^2 + 3/128*sqrt(b*x^2 + a)*a^3*d^3*x/b^2 - 3/16* 
a^3*c^2*d*arcsinh(b*x/sqrt(a*b))/b^(3/2) + 3/128*a^4*d^3*arcsinh(b*x/sqrt( 
a*b))/b^(5/2) + 1/5*(b*x^2 + a)^(5/2)*c^3/b - 6/35*(b*x^2 + a)^(5/2)*a*c*d 
^2/b^2
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.15 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\frac {1}{4480} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, b d^{3} x + 24 \, b c d^{2}\right )} x + \frac {7 \, {\left (8 \, b^{7} c^{2} d + 3 \, a b^{6} d^{3}\right )}}{b^{6}}\right )} x + \frac {16 \, {\left (7 \, b^{7} c^{3} + 24 \, a b^{6} c d^{2}\right )}}{b^{6}}\right )} x + \frac {35 \, {\left (56 \, a b^{6} c^{2} d + a^{2} b^{5} d^{3}\right )}}{b^{6}}\right )} x + \frac {64 \, {\left (14 \, a b^{6} c^{3} + 3 \, a^{2} b^{5} c d^{2}\right )}}{b^{6}}\right )} x + \frac {105 \, {\left (8 \, a^{2} b^{5} c^{2} d - a^{3} b^{4} d^{3}\right )}}{b^{6}}\right )} x + \frac {128 \, {\left (7 \, a^{2} b^{5} c^{3} - 6 \, a^{3} b^{4} c d^{2}\right )}}{b^{6}}\right )} + \frac {3 \, {\left (8 \, a^{3} b c^{2} d - a^{4} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} \] Input:

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

Output:

1/4480*sqrt(b*x^2 + a)*((2*((4*(5*(2*(7*b*d^3*x + 24*b*c*d^2)*x + 7*(8*b^7 
*c^2*d + 3*a*b^6*d^3)/b^6)*x + 16*(7*b^7*c^3 + 24*a*b^6*c*d^2)/b^6)*x + 35 
*(56*a*b^6*c^2*d + a^2*b^5*d^3)/b^6)*x + 64*(14*a*b^6*c^3 + 3*a^2*b^5*c*d^ 
2)/b^6)*x + 105*(8*a^2*b^5*c^2*d - a^3*b^4*d^3)/b^6)*x + 128*(7*a^2*b^5*c^ 
3 - 6*a^3*b^4*c*d^2)/b^6) + 3/128*(8*a^3*b*c^2*d - a^4*d^3)*log(abs(-sqrt( 
b)*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.59 \[ \int x (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\frac {-768 \sqrt {b \,x^{2}+a}\, a^{3} b c \,d^{2}-105 \sqrt {b \,x^{2}+a}\, a^{3} b \,d^{3} x +896 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{3}+840 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{2} d x +384 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,d^{2} x^{2}+70 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d^{3} x^{3}+1792 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{3} x^{2}+3920 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} d \,x^{3}+3072 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,d^{2} x^{4}+840 \sqrt {b \,x^{2}+a}\, a \,b^{3} d^{3} x^{5}+896 \sqrt {b \,x^{2}+a}\, b^{4} c^{3} x^{4}+2240 \sqrt {b \,x^{2}+a}\, b^{4} c^{2} d \,x^{5}+1920 \sqrt {b \,x^{2}+a}\, b^{4} c \,d^{2} x^{6}+560 \sqrt {b \,x^{2}+a}\, b^{4} d^{3} x^{7}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} d^{3}-840 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b \,c^{2} d}{4480 b^{3}} \] Input:

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

Output:

( - 768*sqrt(a + b*x**2)*a**3*b*c*d**2 - 105*sqrt(a + b*x**2)*a**3*b*d**3* 
x + 896*sqrt(a + b*x**2)*a**2*b**2*c**3 + 840*sqrt(a + b*x**2)*a**2*b**2*c 
**2*d*x + 384*sqrt(a + b*x**2)*a**2*b**2*c*d**2*x**2 + 70*sqrt(a + b*x**2) 
*a**2*b**2*d**3*x**3 + 1792*sqrt(a + b*x**2)*a*b**3*c**3*x**2 + 3920*sqrt( 
a + b*x**2)*a*b**3*c**2*d*x**3 + 3072*sqrt(a + b*x**2)*a*b**3*c*d**2*x**4 
+ 840*sqrt(a + b*x**2)*a*b**3*d**3*x**5 + 896*sqrt(a + b*x**2)*b**4*c**3*x 
**4 + 2240*sqrt(a + b*x**2)*b**4*c**2*d*x**5 + 1920*sqrt(a + b*x**2)*b**4* 
c*d**2*x**6 + 560*sqrt(a + b*x**2)*b**4*d**3*x**7 + 105*sqrt(b)*log((sqrt( 
a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*d**3 - 840*sqrt(b)*log((sqrt(a + b* 
x**2) + sqrt(b)*x)/sqrt(a))*a**3*b*c**2*d)/(4480*b**3)