\(\int \frac {(A+B x) (b x+c x^2)^{5/2}}{x^{7/2}} \, dx\) [198]

Optimal result

Integrand size = 24, antiderivative size = 131 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx=\frac {2 A b^2 \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {2 A b \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}+\frac {2 A \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}-2 A b^{5/2} \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right ) \] Output:

2*A*b^2*(c*x^2+b*x)^(1/2)/x^(1/2)+2/3*A*b*(c*x^2+b*x)^(3/2)/x^(3/2)+2/5*A* 
(c*x^2+b*x)^(5/2)/x^(5/2)+2/7*B*(c*x^2+b*x)^(7/2)/c/x^(7/2)-2*A*b^(5/2)*ar 
ctanh((c*x^2+b*x)^(1/2)/b^(1/2)/x^(1/2))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.87 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx=\frac {2 \sqrt {x} \left ((b+c x) \left (15 b^3 B+3 c^3 x^2 (7 A+5 B x)+b c^2 x (77 A+45 B x)+b^2 c (161 A+45 B x)\right )-105 A b^{5/2} c \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{105 c \sqrt {x (b+c x)}} \] Input:

Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(7/2),x]
 

Output:

(2*Sqrt[x]*((b + c*x)*(15*b^3*B + 3*c^3*x^2*(7*A + 5*B*x) + b*c^2*x*(77*A 
+ 45*B*x) + b^2*c*(161*A + 45*B*x)) - 105*A*b^(5/2)*c*Sqrt[b + c*x]*ArcTan 
h[Sqrt[b + c*x]/Sqrt[b]]))/(105*c*Sqrt[x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1221, 1131, 1131, 1131, 1136, 220}

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.

\(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx\)

\(\Big \downarrow \) 1221

\(\displaystyle A \int \frac {\left (c x^2+b x\right )^{5/2}}{x^{7/2}}dx+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 1131

\(\displaystyle A \left (b \int \frac {\left (c x^2+b x\right )^{3/2}}{x^{5/2}}dx+\frac {2 \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}\right )+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 1131

\(\displaystyle A \left (b \left (b \int \frac {\sqrt {c x^2+b x}}{x^{3/2}}dx+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}\right )+\frac {2 \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}\right )+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 1131

\(\displaystyle A \left (b \left (b \left (b \int \frac {1}{\sqrt {x} \sqrt {c x^2+b x}}dx+\frac {2 \sqrt {b x+c x^2}}{\sqrt {x}}\right )+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}\right )+\frac {2 \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}\right )+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}\)

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 220

\(\displaystyle A \left (b \left (b \left (\frac {2 \sqrt {b x+c x^2}}{\sqrt {x}}-2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )\right )+\frac {2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}}\right )+\frac {2 \left (b x+c x^2\right )^{5/2}}{5 x^{5/2}}\right )+\frac {2 B \left (b x+c x^2\right )^{7/2}}{7 c x^{7/2}}\)

Input:

Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^(7/2),x]
 

Output:

(2*B*(b*x + c*x^2)^(7/2))/(7*c*x^(7/2)) + A*((2*(b*x + c*x^2)^(5/2))/(5*x^ 
(5/2)) + b*((2*(b*x + c*x^2)^(3/2))/(3*x^(3/2)) + b*((2*Sqrt[b*x + c*x^2]) 
/Sqrt[x] - 2*Sqrt[b]*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])))
 

Defintions of rubi rules used

rule 220

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

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]
 

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.15

method	result	size
default	\(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (-15 B \,c^{3} x^{3} \sqrt {c x +b}-21 A \,c^{3} x^{2} \sqrt {c x +b}-45 B b \,c^{2} x^{2} \sqrt {c x +b}+105 A \,b^{\frac {5}{2}} c \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-77 A b \,c^{2} x \sqrt {c x +b}-45 B \,b^{2} c x \sqrt {c x +b}-161 A \,b^{2} c \sqrt {c x +b}-15 B \,b^{3} \sqrt {c x +b}\right )}{105 \sqrt {x}\, \sqrt {c x +b}\, c}\)	\(151\)

Input:

int((B*x+A)*(c*x^2+b*x)^(5/2)/x^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-2/105*(x*(c*x+b))^(1/2)*(-15*B*c^3*x^3*(c*x+b)^(1/2)-21*A*c^3*x^2*(c*x+b) 
^(1/2)-45*B*b*c^2*x^2*(c*x+b)^(1/2)+105*A*b^(5/2)*c*arctanh((c*x+b)^(1/2)/ 
b^(1/2))-77*A*b*c^2*x*(c*x+b)^(1/2)-45*B*b^2*c*x*(c*x+b)^(1/2)-161*A*b^2*c 
*(c*x+b)^(1/2)-15*B*b^3*(c*x+b)^(1/2))/x^(1/2)/(c*x+b)^(1/2)/c
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.89 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx=\left [\frac {105 \, A b^{\frac {5}{2}} c x \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (15 \, B c^{3} x^{3} + 15 \, B b^{3} + 161 \, A b^{2} c + 3 \, {\left (15 \, B b c^{2} + 7 \, A c^{3}\right )} x^{2} + {\left (45 \, B b^{2} c + 77 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{105 \, c x}, \frac {2 \, {\left (105 \, A \sqrt {-b} b^{2} c x \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (15 \, B c^{3} x^{3} + 15 \, B b^{3} + 161 \, A b^{2} c + 3 \, {\left (15 \, B b c^{2} + 7 \, A c^{3}\right )} x^{2} + {\left (45 \, B b^{2} c + 77 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}\right )}}{105 \, c x}\right ] \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^(7/2),x, algorithm="fricas")
 

Output:

[1/105*(105*A*b^(5/2)*c*x*log(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2 + b*x)*sqrt(b 
)*sqrt(x))/x^2) + 2*(15*B*c^3*x^3 + 15*B*b^3 + 161*A*b^2*c + 3*(15*B*b*c^2 
 + 7*A*c^3)*x^2 + (45*B*b^2*c + 77*A*b*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(x))/ 
(c*x), 2/105*(105*A*sqrt(-b)*b^2*c*x*arctan(sqrt(c*x^2 + b*x)*sqrt(-b)/(b* 
sqrt(x))) + (15*B*c^3*x^3 + 15*B*b^3 + 161*A*b^2*c + 3*(15*B*b*c^2 + 7*A*c 
^3)*x^2 + (45*B*b^2*c + 77*A*b*c^2)*x)*sqrt(c*x^2 + b*x)*sqrt(x))/(c*x)]
 

Sympy [F]

\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{\frac {7}{2}}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**(7/2),x)
 

Output:

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**(7/2), x)
 

Maxima [F]

\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{x^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^(7/2),x, algorithm="maxima")
 

Output:

A*b^2*integrate(sqrt(c*x + b)/x, x) + 2/105*(35*(B*b^2*c + 2*A*b*c^2)*x^3 
+ (15*B*c^3*x^3 + 3*B*b*c^2*x^2 - 4*B*b^2*c*x + 8*B*b^3)*x^2 + 35*(B*b^3 + 
 2*A*b^2*c)*x^2 + 7*(3*(2*B*b*c^2 + A*c^3)*x^3 + (2*B*b^2*c + A*b*c^2)*x^2 
 - 2*(2*B*b^3 + A*b^2*c)*x)*x)*sqrt(c*x + b)/(c*x^2)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.67 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx=\frac {2 \, A b^{3} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + \frac {2 \, {\left (15 \, {\left (c x + b\right )}^{\frac {7}{2}} B c^{6} + 21 \, {\left (c x + b\right )}^{\frac {5}{2}} A c^{7} + 35 \, {\left (c x + b\right )}^{\frac {3}{2}} A b c^{7} + 105 \, \sqrt {c x + b} A b^{2} c^{7}\right )}}{105 \, c^{7}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^(7/2),x, algorithm="giac")
 

Output:

2*A*b^3*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) + 2/105*(15*(c*x + b)^(7/2 
)*B*c^6 + 21*(c*x + b)^(5/2)*A*c^7 + 35*(c*x + b)^(3/2)*A*b*c^7 + 105*sqrt 
(c*x + b)*A*b^2*c^7)/c^7
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{x^{7/2}} \,d x \] Input:

int(((b*x + c*x^2)^(5/2)*(A + B*x))/x^(7/2),x)
 

Output:

int(((b*x + c*x^2)^(5/2)*(A + B*x))/x^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{7/2}} \, dx=\frac {322 \sqrt {c x +b}\, a \,b^{2} c +154 \sqrt {c x +b}\, a b \,c^{2} x +42 \sqrt {c x +b}\, a \,c^{3} x^{2}+30 \sqrt {c x +b}\, b^{4}+90 \sqrt {c x +b}\, b^{3} c x +90 \sqrt {c x +b}\, b^{2} c^{2} x^{2}+30 \sqrt {c x +b}\, b \,c^{3} x^{3}+105 \sqrt {b}\, \mathrm {log}\left (\sqrt {c x +b}-\sqrt {b}\right ) a \,b^{2} c -105 \sqrt {b}\, \mathrm {log}\left (\sqrt {c x +b}+\sqrt {b}\right ) a \,b^{2} c}{105 c} \] Input:

int((B*x+A)*(c*x^2+b*x)^(5/2)/x^(7/2),x)
 

Output:

(322*sqrt(b + c*x)*a*b**2*c + 154*sqrt(b + c*x)*a*b*c**2*x + 42*sqrt(b + c 
*x)*a*c**3*x**2 + 30*sqrt(b + c*x)*b**4 + 90*sqrt(b + c*x)*b**3*c*x + 90*s 
qrt(b + c*x)*b**2*c**2*x**2 + 30*sqrt(b + c*x)*b*c**3*x**3 + 105*sqrt(b)*l 
og(sqrt(b + c*x) - sqrt(b))*a*b**2*c - 105*sqrt(b)*log(sqrt(b + c*x) + sqr 
t(b))*a*b**2*c)/(105*c)