\(\int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{(e x)^{3/2}} \, dx\) [185]

Optimal result

Integrand size = 28, antiderivative size = 199 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{(e x)^{3/2}} \, dx=\frac {2 c^3 \sqrt {a x+b x^2}}{e \sqrt {e x}}+\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \left (a x+b x^2\right )^{3/2}}{3 b^3 (e x)^{3/2}}+\frac {2 d^2 (3 b c-2 a d) e \left (a x+b x^2\right )^{5/2}}{5 b^3 (e x)^{5/2}}+\frac {2 d^3 e^2 \left (a x+b x^2\right )^{7/2}}{7 b^3 (e x)^{7/2}}-\frac {2 \sqrt {a} c^3 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a x+b x^2}}{\sqrt {a} \sqrt {e x}}\right )}{e^{3/2}} \] Output:

2*c^3*(b*x^2+a*x)^(1/2)/e/(e*x)^(1/2)+2/3*d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)* 
(b*x^2+a*x)^(3/2)/b^3/(e*x)^(3/2)+2/5*d^2*(-2*a*d+3*b*c)*e*(b*x^2+a*x)^(5/ 
2)/b^3/(e*x)^(5/2)+2/7*d^3*e^2*(b*x^2+a*x)^(7/2)/b^3/(e*x)^(7/2)-2*a^(1/2) 
*c^3*arctanh(e^(1/2)*(b*x^2+a*x)^(1/2)/a^(1/2)/(e*x)^(1/2))/e^(3/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.80 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{(e x)^{3/2}} \, dx=\frac {2 (x (a+b x))^{3/2} \left (\sqrt {a+b x} \left (8 a^3 d^3-2 a^2 b d^2 (21 c+2 d x)+3 a b^2 d \left (35 c^2+7 c d x+d^2 x^2\right )+3 b^3 \left (35 c^3+35 c^2 d x+21 c d^2 x^2+5 d^3 x^3\right )\right )-105 \sqrt {a} b^3 c^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{105 b^3 (e x)^{3/2} (a+b x)^{3/2}} \] Input:

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

Output:

(2*(x*(a + b*x))^(3/2)*(Sqrt[a + b*x]*(8*a^3*d^3 - 2*a^2*b*d^2*(21*c + 2*d 
*x) + 3*a*b^2*d*(35*c^2 + 7*c*d*x + d^2*x^2) + 3*b^3*(35*c^3 + 35*c^2*d*x 
+ 21*c*d^2*x^2 + 5*d^3*x^3)) - 105*Sqrt[a]*b^3*c^3*ArcTanh[Sqrt[a + b*x]/S 
qrt[a]]))/(105*b^3*(e*x)^(3/2)*(a + b*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1262, 27, 2169, 27, 1221, 1131, 1136, 221}

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.

\(\displaystyle \int \frac {\sqrt {a x+b x^2} (c+d x)^3}{(e x)^{3/2}} \, dx\)

\(\Big \downarrow \) 1262

\(\displaystyle \frac {2 \int \frac {\sqrt {b x^2+a x} \left (7 b c^3 e^3+d^2 (21 b c-4 a d) x^2 e^3+21 b c^2 d x e^3\right )}{2 (e x)^{3/2}}dx}{7 b e^3}+\frac {2 d^3 \sqrt {e x} \left (a x+b x^2\right )^{3/2}}{7 b e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {b x^2+a x} \left (7 b c^3 e^3+d^2 (21 b c-4 a d) x^2 e^3+21 b c^2 d x e^3\right )}{(e x)^{3/2}}dx}{7 b e^3}+\frac {2 d^3 \sqrt {e x} \left (a x+b x^2\right )^{3/2}}{7 b e^2}\)

\(\Big \downarrow \) 2169

\(\displaystyle \frac {\frac {2 \int \frac {e^5 \left (35 b^2 c^3+d \left (105 b^2 c^2-42 a b d c+8 a^2 d^2\right ) x\right ) \sqrt {b x^2+a x}}{2 (e x)^{3/2}}dx}{5 b e^2}+\frac {2 d^2 e^2 \left (a x+b x^2\right )^{3/2} (21 b c-4 a d)}{5 b \sqrt {e x}}}{7 b e^3}+\frac {2 d^3 \sqrt {e x} \left (a x+b x^2\right )^{3/2}}{7 b e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e^3 \int \frac {\left (35 b^2 c^3+d \left (105 b^2 c^2-42 a b d c+8 a^2 d^2\right ) x\right ) \sqrt {b x^2+a x}}{(e x)^{3/2}}dx}{5 b}+\frac {2 d^2 e^2 \left (a x+b x^2\right )^{3/2} (21 b c-4 a d)}{5 b \sqrt {e x}}}{7 b e^3}+\frac {2 d^3 \sqrt {e x} \left (a x+b x^2\right )^{3/2}}{7 b e^2}\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {\frac {e^3 \left (35 b^2 c^3 \int \frac {\sqrt {b x^2+a x}}{(e x)^{3/2}}dx+\frac {2 d \left (a x+b x^2\right )^{3/2} \left (8 a^2 d^2-42 a b c d+105 b^2 c^2\right )}{3 b (e x)^{3/2}}\right )}{5 b}+\frac {2 d^2 e^2 \left (a x+b x^2\right )^{3/2} (21 b c-4 a d)}{5 b \sqrt {e x}}}{7 b e^3}+\frac {2 d^3 \sqrt {e x} \left (a x+b x^2\right )^{3/2}}{7 b e^2}\)

\(\Big \downarrow \) 1131

\(\displaystyle \frac {\frac {e^3 \left (35 b^2 c^3 \left (\frac {a \int \frac {1}{\sqrt {e x} \sqrt {b x^2+a x}}dx}{e}+\frac {2 \sqrt {a x+b x^2}}{e \sqrt {e x}}\right )+\frac {2 d \left (a x+b x^2\right )^{3/2} \left (8 a^2 d^2-42 a b c d+105 b^2 c^2\right )}{3 b (e x)^{3/2}}\right )}{5 b}+\frac {2 d^2 e^2 \left (a x+b x^2\right )^{3/2} (21 b c-4 a d)}{5 b \sqrt {e x}}}{7 b e^3}+\frac {2 d^3 \sqrt {e x} \left (a x+b x^2\right )^{3/2}}{7 b e^2}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {\frac {e^3 \left (35 b^2 c^3 \left (2 a \int \frac {1}{\frac {e \left (b x^2+a x\right )}{x}-a e}d\frac {\sqrt {b x^2+a x}}{\sqrt {e x}}+\frac {2 \sqrt {a x+b x^2}}{e \sqrt {e x}}\right )+\frac {2 d \left (a x+b x^2\right )^{3/2} \left (8 a^2 d^2-42 a b c d+105 b^2 c^2\right )}{3 b (e x)^{3/2}}\right )}{5 b}+\frac {2 d^2 e^2 \left (a x+b x^2\right )^{3/2} (21 b c-4 a d)}{5 b \sqrt {e x}}}{7 b e^3}+\frac {2 d^3 \sqrt {e x} \left (a x+b x^2\right )^{3/2}}{7 b e^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {e^3 \left (\frac {2 d \left (a x+b x^2\right )^{3/2} \left (8 a^2 d^2-42 a b c d+105 b^2 c^2\right )}{3 b (e x)^{3/2}}+35 b^2 c^3 \left (\frac {2 \sqrt {a x+b x^2}}{e \sqrt {e x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a x+b x^2}}{\sqrt {a} \sqrt {e x}}\right )}{e^{3/2}}\right )\right )}{5 b}+\frac {2 d^2 e^2 \left (a x+b x^2\right )^{3/2} (21 b c-4 a d)}{5 b \sqrt {e x}}}{7 b e^3}+\frac {2 d^3 \sqrt {e x} \left (a x+b x^2\right )^{3/2}}{7 b e^2}\)

Input:

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

Output:

(2*d^3*Sqrt[e*x]*(a*x + b*x^2)^(3/2))/(7*b*e^2) + ((2*d^2*(21*b*c - 4*a*d) 
*e^2*(a*x + b*x^2)^(3/2))/(5*b*Sqrt[e*x]) + (e^3*((2*d*(105*b^2*c^2 - 42*a 
*b*c*d + 8*a^2*d^2)*(a*x + b*x^2)^(3/2))/(3*b*(e*x)^(3/2)) + 35*b^2*c^3*(( 
2*Sqrt[a*x + b*x^2])/(e*Sqrt[e*x]) - (2*Sqrt[a]*ArcTanh[(Sqrt[e]*Sqrt[a*x 
+ b*x^2])/(Sqrt[a]*Sqrt[e*x])])/e^(3/2))))/(5*b))/(7*b*e^3)
 

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 221

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

rule 1131

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1)))   Int[(d + e*x)^(m + 1)*(a + 
 b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b 
*d*e + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne 
Q[m + 2*p + 1, 0] && IntegerQ[2*p]
 

rule 1136

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x 
_Symbol] :> Simp[2*e   Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + 
 b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 
- b*d*e + a*e^2, 0]
 

rule 1221

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 
)/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c 
*f - b*g))/(c*e*(m + 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && NeQ[m + 2*p + 2, 0]
 

rule 1262

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n + e*g^n*( 
m + p + n)*(d + e*x)^(n - 2)*(b*d - 2*a*e + (2*c*d - b*e)*x), x], x], x] /; 
 FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && 
IGtQ[n, 0] && NeQ[m + n + 2*p + 1, 0]
 

rule 2169

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - 
b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2 
, 0]
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.54

method	result	size
default	\(-\frac {2 \sqrt {x \left (b x +a \right )}\, \left (-15 b^{3} d^{3} x^{3} \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}-3 a \,b^{2} d^{3} x^{2} \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}-63 b^{3} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}+105 a \,b^{3} c^{3} e \,\operatorname {arctanh}\left (\frac {\sqrt {\left (b x +a \right ) e}}{\sqrt {a e}}\right )+4 a^{2} b \,d^{3} x \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}-21 a \,b^{2} c \,d^{2} x \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}-105 b^{3} c^{2} d x \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}-8 a^{3} d^{3} \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}+42 a^{2} b c \,d^{2} \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}-105 a \,b^{2} c^{2} d \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}-105 b^{3} c^{3} \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}\right )}{105 e \sqrt {e x}\, \sqrt {\left (b x +a \right ) e}\, b^{3} \sqrt {a e}}\)	\(306\)

Input:

int((d*x+c)^3*(b*x^2+a*x)^(1/2)/(e*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2/105*(x*(b*x+a))^(1/2)/e*(-15*b^3*d^3*x^3*((b*x+a)*e)^(1/2)*(a*e)^(1/2)- 
3*a*b^2*d^3*x^2*((b*x+a)*e)^(1/2)*(a*e)^(1/2)-63*b^3*c*d^2*x^2*((b*x+a)*e) 
^(1/2)*(a*e)^(1/2)+105*a*b^3*c^3*e*arctanh(((b*x+a)*e)^(1/2)/(a*e)^(1/2))+ 
4*a^2*b*d^3*x*((b*x+a)*e)^(1/2)*(a*e)^(1/2)-21*a*b^2*c*d^2*x*((b*x+a)*e)^( 
1/2)*(a*e)^(1/2)-105*b^3*c^2*d*x*((b*x+a)*e)^(1/2)*(a*e)^(1/2)-8*a^3*d^3*( 
(b*x+a)*e)^(1/2)*(a*e)^(1/2)+42*a^2*b*c*d^2*((b*x+a)*e)^(1/2)*(a*e)^(1/2)- 
105*a*b^2*c^2*d*((b*x+a)*e)^(1/2)*(a*e)^(1/2)-105*b^3*c^3*((b*x+a)*e)^(1/2 
)*(a*e)^(1/2))/(e*x)^(1/2)/((b*x+a)*e)^(1/2)/b^3/(a*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.87 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{(e x)^{3/2}} \, dx=\left [\frac {105 \, b^{3} c^{3} e x \sqrt {\frac {a}{e}} \log \left (-\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{2} + a x} \sqrt {e x} \sqrt {\frac {a}{e}}}{x^{2}}\right ) + 2 \, {\left (15 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} + 105 \, a b^{2} c^{2} d - 42 \, a^{2} b c d^{2} + 8 \, a^{3} d^{3} + 3 \, {\left (21 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + {\left (105 \, b^{3} c^{2} d + 21 \, a b^{2} c d^{2} - 4 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {e x}}{105 \, b^{3} e^{2} x}, \frac {2 \, {\left (105 \, b^{3} c^{3} e x \sqrt {-\frac {a}{e}} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {e x} \sqrt {-\frac {a}{e}}}{a x}\right ) + {\left (15 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} + 105 \, a b^{2} c^{2} d - 42 \, a^{2} b c d^{2} + 8 \, a^{3} d^{3} + 3 \, {\left (21 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + {\left (105 \, b^{3} c^{2} d + 21 \, a b^{2} c d^{2} - 4 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {e x}\right )}}{105 \, b^{3} e^{2} x}\right ] \] Input:

integrate((d*x+c)^3*(b*x^2+a*x)^(1/2)/(e*x)^(3/2),x, algorithm="fricas")
 

Output:

[1/105*(105*b^3*c^3*e*x*sqrt(a/e)*log(-(b*x^2 + 2*a*x - 2*sqrt(b*x^2 + a*x 
)*sqrt(e*x)*sqrt(a/e))/x^2) + 2*(15*b^3*d^3*x^3 + 105*b^3*c^3 + 105*a*b^2* 
c^2*d - 42*a^2*b*c*d^2 + 8*a^3*d^3 + 3*(21*b^3*c*d^2 + a*b^2*d^3)*x^2 + (1 
05*b^3*c^2*d + 21*a*b^2*c*d^2 - 4*a^2*b*d^3)*x)*sqrt(b*x^2 + a*x)*sqrt(e*x 
))/(b^3*e^2*x), 2/105*(105*b^3*c^3*e*x*sqrt(-a/e)*arctan(sqrt(b*x^2 + a*x) 
*sqrt(e*x)*sqrt(-a/e)/(a*x)) + (15*b^3*d^3*x^3 + 105*b^3*c^3 + 105*a*b^2*c 
^2*d - 42*a^2*b*c*d^2 + 8*a^3*d^3 + 3*(21*b^3*c*d^2 + a*b^2*d^3)*x^2 + (10 
5*b^3*c^2*d + 21*a*b^2*c*d^2 - 4*a^2*b*d^3)*x)*sqrt(b*x^2 + a*x)*sqrt(e*x) 
)/(b^3*e^2*x)]
 

Sympy [F]

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

integrate((d*x+c)**3*(b*x**2+a*x)**(1/2)/(e*x)**(3/2),x)
 

Output:

Integral(sqrt(x*(a + b*x))*(c + d*x)**3/(e*x)**(3/2), x)
 

Maxima [F]

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

integrate((d*x+c)^3*(b*x^2+a*x)^(1/2)/(e*x)^(3/2),x, algorithm="maxima")
 

Output:

c^3*integrate(sqrt(b*x + a)/x, x)/e^(3/2) + 2/105*(105*b^3*c^2*d*x^3 + 105 
*a*b^2*c^2*d*x^2 + (15*b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 - 4*a^2*b*d^3*x + 8*a 
^3*d^3)*x^2 + 21*(3*b^3*c*d^2*x^3 + a*b^2*c*d^2*x^2 - 2*a^2*b*c*d^2*x)*x)* 
sqrt(b*x + a)/(b^3*e^(3/2)*x^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (165) = 330\).

Time = 0.22 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.71 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{(e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {\frac {105 \, a c^{3} e^{4} {\left | e \right |} \arctan \left (\frac {\sqrt {b e x + a e}}{\sqrt {-a e}}\right )}{\sqrt {-a e}} + \frac {105 \, \sqrt {b e x + a e} b^{21} c^{3} e^{3} {\left | e \right |} + 105 \, {\left (b e x + a e\right )}^{\frac {3}{2}} b^{20} c^{2} d e^{2} {\left | e \right |} - 105 \, {\left (b e x + a e\right )}^{\frac {3}{2}} a b^{19} c d^{2} e^{2} {\left | e \right |} + 35 \, {\left (b e x + a e\right )}^{\frac {3}{2}} a^{2} b^{18} d^{3} e^{2} {\left | e \right |} + 63 \, {\left (b e x + a e\right )}^{\frac {5}{2}} b^{19} c d^{2} e {\left | e \right |} - 42 \, {\left (b e x + a e\right )}^{\frac {5}{2}} a b^{18} d^{3} e {\left | e \right |} + 15 \, {\left (b e x + a e\right )}^{\frac {7}{2}} b^{18} d^{3} {\left | e \right |}}{b^{21}}}{e^{5}} - \frac {105 \, a b^{3} c^{3} e {\left | e \right |} \arctan \left (\frac {\sqrt {a e}}{\sqrt {-a e}}\right ) + 105 \, \sqrt {a e} \sqrt {-a e} b^{3} c^{3} {\left | e \right |} + 105 \, \sqrt {a e} \sqrt {-a e} a b^{2} c^{2} d {\left | e \right |} - 42 \, \sqrt {a e} \sqrt {-a e} a^{2} b c d^{2} {\left | e \right |} + 8 \, \sqrt {a e} \sqrt {-a e} a^{3} d^{3} {\left | e \right |}}{\sqrt {-a e} b^{3} e^{2}}\right )}}{105 \, e} \] Input:

integrate((d*x+c)^3*(b*x^2+a*x)^(1/2)/(e*x)^(3/2),x, algorithm="giac")
 

Output:

2/105*((105*a*c^3*e^4*abs(e)*arctan(sqrt(b*e*x + a*e)/sqrt(-a*e))/sqrt(-a* 
e) + (105*sqrt(b*e*x + a*e)*b^21*c^3*e^3*abs(e) + 105*(b*e*x + a*e)^(3/2)* 
b^20*c^2*d*e^2*abs(e) - 105*(b*e*x + a*e)^(3/2)*a*b^19*c*d^2*e^2*abs(e) + 
35*(b*e*x + a*e)^(3/2)*a^2*b^18*d^3*e^2*abs(e) + 63*(b*e*x + a*e)^(5/2)*b^ 
19*c*d^2*e*abs(e) - 42*(b*e*x + a*e)^(5/2)*a*b^18*d^3*e*abs(e) + 15*(b*e*x 
 + a*e)^(7/2)*b^18*d^3*abs(e))/b^21)/e^5 - (105*a*b^3*c^3*e*abs(e)*arctan( 
sqrt(a*e)/sqrt(-a*e)) + 105*sqrt(a*e)*sqrt(-a*e)*b^3*c^3*abs(e) + 105*sqrt 
(a*e)*sqrt(-a*e)*a*b^2*c^2*d*abs(e) - 42*sqrt(a*e)*sqrt(-a*e)*a^2*b*c*d^2* 
abs(e) + 8*sqrt(a*e)*sqrt(-a*e)*a^3*d^3*abs(e))/(sqrt(-a*e)*b^3*e^2))/e
 

Mupad [F(-1)]

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

int(((a*x + b*x^2)^(1/2)*(c + d*x)^3)/(e*x)^(3/2),x)
 

Output:

int(((a*x + b*x^2)^(1/2)*(c + d*x)^3)/(e*x)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^3 \sqrt {a x+b x^2}}{(e x)^{3/2}} \, dx=\frac {\sqrt {e}\, \left (16 \sqrt {b x +a}\, a^{3} d^{3}-84 \sqrt {b x +a}\, a^{2} b c \,d^{2}-8 \sqrt {b x +a}\, a^{2} b \,d^{3} x +210 \sqrt {b x +a}\, a \,b^{2} c^{2} d +42 \sqrt {b x +a}\, a \,b^{2} c \,d^{2} x +6 \sqrt {b x +a}\, a \,b^{2} d^{3} x^{2}+210 \sqrt {b x +a}\, b^{3} c^{3}+210 \sqrt {b x +a}\, b^{3} c^{2} d x +126 \sqrt {b x +a}\, b^{3} c \,d^{2} x^{2}+30 \sqrt {b x +a}\, b^{3} d^{3} x^{3}+105 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{3} c^{3}-105 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{3} c^{3}\right )}{105 b^{3} e^{2}} \] Input:

int((d*x+c)^3*(b*x^2+a*x)^(1/2)/(e*x)^(3/2),x)
 

Output:

(sqrt(e)*(16*sqrt(a + b*x)*a**3*d**3 - 84*sqrt(a + b*x)*a**2*b*c*d**2 - 8* 
sqrt(a + b*x)*a**2*b*d**3*x + 210*sqrt(a + b*x)*a*b**2*c**2*d + 42*sqrt(a 
+ b*x)*a*b**2*c*d**2*x + 6*sqrt(a + b*x)*a*b**2*d**3*x**2 + 210*sqrt(a + b 
*x)*b**3*c**3 + 210*sqrt(a + b*x)*b**3*c**2*d*x + 126*sqrt(a + b*x)*b**3*c 
*d**2*x**2 + 30*sqrt(a + b*x)*b**3*d**3*x**3 + 105*sqrt(a)*log(sqrt(a + b* 
x) - sqrt(a))*b**3*c**3 - 105*sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*b**3*c* 
*3))/(105*b**3*e**2)