\(\int \frac {(c+d x) (a x^2+b x^3)^{5/2}}{x^{11}} \, dx\) [271]

Optimal result

Integrand size = 24, antiderivative size = 206 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{11}} \, dx=-\frac {b (3 b c+22 a d) \sqrt {a x^2+b x^3}}{48 x^4}-\frac {b^2 (3 b c+118 a d) \sqrt {a x^2+b x^3}}{192 a x^3}+\frac {b^3 (3 b c-10 a d) \sqrt {a x^2+b x^3}}{128 a^2 x^2}-\frac {(b c+2 a d) \left (a x^2+b x^3\right )^{3/2}}{8 x^7}-\frac {c \left (a x^2+b x^3\right )^{5/2}}{5 x^{10}}-\frac {b^4 (3 b c-10 a d) \text {arctanh}\left (\frac {\sqrt {a x^2+b x^3}}{\sqrt {a} x}\right )}{128 a^{5/2}} \] Output:

-1/48*b*(22*a*d+3*b*c)*(b*x^3+a*x^2)^(1/2)/x^4-1/192*b^2*(118*a*d+3*b*c)*( 
b*x^3+a*x^2)^(1/2)/a/x^3+1/128*b^3*(-10*a*d+3*b*c)*(b*x^3+a*x^2)^(1/2)/a^2 
/x^2-1/8*(2*a*d+b*c)*(b*x^3+a*x^2)^(3/2)/x^7-1/5*c*(b*x^3+a*x^2)^(5/2)/x^1 
0-1/128*b^4*(-10*a*d+3*b*c)*arctanh((b*x^3+a*x^2)^(1/2)/a^(1/2)/x)/a^(5/2)
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.75 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{11}} \, dx=-\frac {\sqrt {x^2 (a+b x)} \left (\sqrt {a} \sqrt {a+b x} \left (-45 b^4 c x^4+30 a b^3 x^3 (c+5 d x)+96 a^4 (4 c+5 d x)+16 a^3 b x (63 c+85 d x)+4 a^2 b^2 x^2 (186 c+295 d x)\right )+15 b^4 (3 b c-10 a d) x^5 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{1920 a^{5/2} x^6 \sqrt {a+b x}} \] Input:

Integrate[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/x^11,x]
 

Output:

-1/1920*(Sqrt[x^2*(a + b*x)]*(Sqrt[a]*Sqrt[a + b*x]*(-45*b^4*c*x^4 + 30*a* 
b^3*x^3*(c + 5*d*x) + 96*a^4*(4*c + 5*d*x) + 16*a^3*b*x*(63*c + 85*d*x) + 
4*a^2*b^2*x^2*(186*c + 295*d*x)) + 15*b^4*(3*b*c - 10*a*d)*x^5*ArcTanh[Sqr 
t[a + b*x]/Sqrt[a]]))/(a^(5/2)*x^6*Sqrt[a + b*x])
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1944, 1926, 1926, 1926, 1931, 1914, 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.

\(\displaystyle \int \frac {\left (a x^2+b x^3\right )^{5/2} (c+d x)}{x^{11}} \, dx\)

\(\Big \downarrow \) 1944

\(\displaystyle -\frac {(3 b c-10 a d) \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^{10}}dx}{10 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{5 a x^{12}}\)

\(\Big \downarrow \) 1926

\(\displaystyle -\frac {(3 b c-10 a d) \left (\frac {5}{8} b \int \frac {\left (b x^3+a x^2\right )^{3/2}}{x^7}dx-\frac {\left (a x^2+b x^3\right )^{5/2}}{4 x^9}\right )}{10 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{5 a x^{12}}\)

\(\Big \downarrow \) 1926

\(\displaystyle -\frac {(3 b c-10 a d) \left (\frac {5}{8} b \left (\frac {1}{2} b \int \frac {\sqrt {b x^3+a x^2}}{x^4}dx-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{4 x^9}\right )}{10 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{5 a x^{12}}\)

\(\Big \downarrow \) 1926

\(\displaystyle -\frac {(3 b c-10 a d) \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \int \frac {1}{x \sqrt {b x^3+a x^2}}dx-\frac {\sqrt {a x^2+b x^3}}{2 x^3}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{4 x^9}\right )}{10 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{5 a x^{12}}\)

\(\Big \downarrow \) 1931

\(\displaystyle -\frac {(3 b c-10 a d) \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \left (-\frac {b \int \frac {1}{\sqrt {b x^3+a x^2}}dx}{2 a}-\frac {\sqrt {a x^2+b x^3}}{a x^2}\right )-\frac {\sqrt {a x^2+b x^3}}{2 x^3}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{4 x^9}\right )}{10 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{5 a x^{12}}\)

\(\Big \downarrow \) 1914

\(\displaystyle -\frac {(3 b c-10 a d) \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \left (\frac {b \int \frac {1}{1-\frac {a x^2}{b x^3+a x^2}}d\frac {x}{\sqrt {b x^3+a x^2}}}{a}-\frac {\sqrt {a x^2+b x^3}}{a x^2}\right )-\frac {\sqrt {a x^2+b x^3}}{2 x^3}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{4 x^9}\right )}{10 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{5 a x^{12}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{3/2}}-\frac {\sqrt {a x^2+b x^3}}{a x^2}\right )-\frac {\sqrt {a x^2+b x^3}}{2 x^3}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{3 x^6}\right )-\frac {\left (a x^2+b x^3\right )^{5/2}}{4 x^9}\right ) (3 b c-10 a d)}{10 a}-\frac {c \left (a x^2+b x^3\right )^{7/2}}{5 a x^{12}}\)

Input:

Int[((c + d*x)*(a*x^2 + b*x^3)^(5/2))/x^11,x]
 

Output:

-1/5*(c*(a*x^2 + b*x^3)^(7/2))/(a*x^12) - ((3*b*c - 10*a*d)*(-1/4*(a*x^2 + 
 b*x^3)^(5/2)/x^9 + (5*b*(-1/3*(a*x^2 + b*x^3)^(3/2)/x^6 + (b*(-1/2*Sqrt[a 
*x^2 + b*x^3]/x^3 + (b*(-(Sqrt[a*x^2 + b*x^3]/(a*x^2)) + (b*ArcTanh[(Sqrt[ 
a]*x)/Sqrt[a*x^2 + b*x^3]])/a^(3/2)))/4))/2))/8))/(10*a)
 

Defintions of rubi rules used

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 1914

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[2/(2 - n) 
Subst[Int[1/(1 - a*x^2), x], x, x/Sqrt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, 
n}, x] && NeQ[n, 2]
 

rule 1926

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] 
 :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + j*p + 1))), x] - Simp[b*p 
*((n - j)/(c^n*(m + j*p + 1)))   Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), 
 x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (Integer 
sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
 

rule 1931

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p 
+ 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1)))   I 
nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] 
 &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[ 
m + j*p + 1, 0]
 

rule 1944

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.)), x_Symbol] :> Simp[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j 
+ b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] + Simp[(a*d*(m + j*p + 1) - b 
*c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1))   Int[(e*x)^(m + n)*(a*x^ 
j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j 
+ n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m + j*p, -1 
] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, (-n)*p - 1])) && ( 
GtQ[e, 0] || IntegersQ[j, n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1 
, 0]
 

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.76

method	result	size
risch	\(-\frac {\left (150 x^{4} a \,b^{3} d -45 x^{4} b^{4} c +1180 a^{2} b^{2} d \,x^{3}+30 a \,b^{3} c \,x^{3}+1360 a^{3} b d \,x^{2}+744 a^{2} b^{2} c \,x^{2}+480 a^{4} d x +1008 a^{3} b c x +384 c \,a^{4}\right ) \sqrt {x^{2} \left (b x +a \right )}}{1920 x^{6} a^{2}}+\frac {\left (10 a d -3 b c \right ) b^{4} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {x^{2} \left (b x +a \right )}}{128 a^{\frac {5}{2}} x \sqrt {b x +a}}\)	\(156\)
pseudoelliptic	\(-\frac {55 \left (b^{9} x^{10} \left (a d -\frac {13 b c}{20}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )-\frac {52 \sqrt {b x +a}\, \left (\frac {80 x^{4} b^{4} \left (\frac {22 d x}{13}+c \right ) a^{\frac {11}{2}}}{99}-\frac {320 x^{3} b^{3} \left (\frac {5 d x}{3}+c \right ) a^{\frac {13}{2}}}{429}-\frac {343168 x^{2} \left (\frac {3090 d x}{2681}+c \right ) b^{2} a^{\frac {15}{2}}}{1287}-\frac {587776 x \left (\frac {185 d x}{164}+c \right ) b \,a^{\frac {17}{2}}}{1287}+\left (-\frac {286720 d x}{1287}-\frac {28672 c}{143}\right ) a^{\frac {19}{2}}+\left (\frac {35 x^{3} b^{3} \left (\frac {30 d x}{13}+c \right ) a^{\frac {3}{2}}}{24}-\frac {7 x^{2} \left (\frac {25 d x}{13}+c \right ) b^{2} a^{\frac {5}{2}}}{6}+b x \left (\frac {70 d x}{39}+c \right ) a^{\frac {7}{2}}+\left (-\frac {20 d x}{13}-\frac {8 c}{9}\right ) a^{\frac {9}{2}}-\frac {35 \sqrt {a}\, b^{4} c \,x^{4}}{16}\right ) x^{5} b^{5}\right )}{175}\right )}{32768 a^{\frac {15}{2}} x^{10}}\)	\(201\)
default	\(-\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}} \left (150 \left (b x +a \right )^{\frac {9}{2}} a^{\frac {7}{2}} d -45 \left (b x +a \right )^{\frac {9}{2}} a^{\frac {5}{2}} b c +580 \left (b x +a \right )^{\frac {7}{2}} a^{\frac {9}{2}} d +210 \left (b x +a \right )^{\frac {7}{2}} a^{\frac {7}{2}} b c -150 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{3} b^{5} d \,x^{5}+45 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{2} b^{6} c \,x^{5}-1280 \left (b x +a \right )^{\frac {5}{2}} a^{\frac {11}{2}} d +384 \left (b x +a \right )^{\frac {5}{2}} a^{\frac {9}{2}} b c +700 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {13}{2}} d -210 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {11}{2}} b c -150 \sqrt {b x +a}\, a^{\frac {15}{2}} d +45 \sqrt {b x +a}\, a^{\frac {13}{2}} b c \right )}{1920 b \,x^{10} \left (b x +a \right )^{\frac {5}{2}} a^{\frac {9}{2}}}\)	\(216\)

Input:

int((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^11,x,method=_RETURNVERBOSE)
 

Output:

-1/1920*(150*a*b^3*d*x^4-45*b^4*c*x^4+1180*a^2*b^2*d*x^3+30*a*b^3*c*x^3+13 
60*a^3*b*d*x^2+744*a^2*b^2*c*x^2+480*a^4*d*x+1008*a^3*b*c*x+384*a^4*c)/x^6 
/a^2*(x^2*(b*x+a))^(1/2)+1/128*(10*a*d-3*b*c)*b^4/a^(5/2)*arctanh((b*x+a)^ 
(1/2)/a^(1/2))*(x^2*(b*x+a))^(1/2)/x/(b*x+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.67 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{11}} \, dx=\left [-\frac {15 \, {\left (3 \, b^{5} c - 10 \, a b^{4} d\right )} \sqrt {a} x^{6} \log \left (\frac {b x^{2} + 2 \, a x + 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, {\left (384 \, a^{5} c - 15 \, {\left (3 \, a b^{4} c - 10 \, a^{2} b^{3} d\right )} x^{4} + 10 \, {\left (3 \, a^{2} b^{3} c + 118 \, a^{3} b^{2} d\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{2} c + 170 \, a^{4} b d\right )} x^{2} + 48 \, {\left (21 \, a^{4} b c + 10 \, a^{5} d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{3840 \, a^{3} x^{6}}, \frac {15 \, {\left (3 \, b^{5} c - 10 \, a b^{4} d\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{b x^{2} + a x}\right ) - {\left (384 \, a^{5} c - 15 \, {\left (3 \, a b^{4} c - 10 \, a^{2} b^{3} d\right )} x^{4} + 10 \, {\left (3 \, a^{2} b^{3} c + 118 \, a^{3} b^{2} d\right )} x^{3} + 8 \, {\left (93 \, a^{3} b^{2} c + 170 \, a^{4} b d\right )} x^{2} + 48 \, {\left (21 \, a^{4} b c + 10 \, a^{5} d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{1920 \, a^{3} x^{6}}\right ] \] Input:

integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^11,x, algorithm="fricas")
 

Output:

[-1/3840*(15*(3*b^5*c - 10*a*b^4*d)*sqrt(a)*x^6*log((b*x^2 + 2*a*x + 2*sqr 
t(b*x^3 + a*x^2)*sqrt(a))/x^2) + 2*(384*a^5*c - 15*(3*a*b^4*c - 10*a^2*b^3 
*d)*x^4 + 10*(3*a^2*b^3*c + 118*a^3*b^2*d)*x^3 + 8*(93*a^3*b^2*c + 170*a^4 
*b*d)*x^2 + 48*(21*a^4*b*c + 10*a^5*d)*x)*sqrt(b*x^3 + a*x^2))/(a^3*x^6), 
1/1920*(15*(3*b^5*c - 10*a*b^4*d)*sqrt(-a)*x^6*arctan(sqrt(b*x^3 + a*x^2)* 
sqrt(-a)/(b*x^2 + a*x)) - (384*a^5*c - 15*(3*a*b^4*c - 10*a^2*b^3*d)*x^4 + 
 10*(3*a^2*b^3*c + 118*a^3*b^2*d)*x^3 + 8*(93*a^3*b^2*c + 170*a^4*b*d)*x^2 
 + 48*(21*a^4*b*c + 10*a^5*d)*x)*sqrt(b*x^3 + a*x^2))/(a^3*x^6)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{11}} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {5}{2}} \left (c + d x\right )}{x^{11}}\, dx \] Input:

integrate((d*x+c)*(b*x**3+a*x**2)**(5/2)/x**11,x)
 

Output:

Integral((x**2*(a + b*x))**(5/2)*(c + d*x)/x**11, x)
 

Maxima [F]

\[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{11}} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {5}{2}} {\left (d x + c\right )}}{x^{11}} \,d x } \] Input:

integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^11,x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a*x^2)^(5/2)*(d*x + c)/x^11, x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{11}} \, dx=\frac {1}{1920} \, b^{5} {\left (\frac {15 \, {\left (3 \, b c \mathrm {sgn}\left (x\right ) - 10 \, a d \mathrm {sgn}\left (x\right )\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} b} + \frac {45 \, {\left (b x + a\right )}^{\frac {9}{2}} b c \mathrm {sgn}\left (x\right ) - 210 \, {\left (b x + a\right )}^{\frac {7}{2}} a b c \mathrm {sgn}\left (x\right ) - 384 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b c \mathrm {sgn}\left (x\right ) + 210 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b c \mathrm {sgn}\left (x\right ) - 45 \, \sqrt {b x + a} a^{4} b c \mathrm {sgn}\left (x\right ) - 150 \, {\left (b x + a\right )}^{\frac {9}{2}} a d \mathrm {sgn}\left (x\right ) - 580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} d \mathrm {sgn}\left (x\right ) + 1280 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} d \mathrm {sgn}\left (x\right ) - 700 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} d \mathrm {sgn}\left (x\right ) + 150 \, \sqrt {b x + a} a^{5} d \mathrm {sgn}\left (x\right )}{a^{2} b^{6} x^{5}}\right )} \] Input:

integrate((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^11,x, algorithm="giac")
 

Output:

1/1920*b^5*(15*(3*b*c*sgn(x) - 10*a*d*sgn(x))*arctan(sqrt(b*x + a)/sqrt(-a 
))/(sqrt(-a)*a^2*b) + (45*(b*x + a)^(9/2)*b*c*sgn(x) - 210*(b*x + a)^(7/2) 
*a*b*c*sgn(x) - 384*(b*x + a)^(5/2)*a^2*b*c*sgn(x) + 210*(b*x + a)^(3/2)*a 
^3*b*c*sgn(x) - 45*sqrt(b*x + a)*a^4*b*c*sgn(x) - 150*(b*x + a)^(9/2)*a*d* 
sgn(x) - 580*(b*x + a)^(7/2)*a^2*d*sgn(x) + 1280*(b*x + a)^(5/2)*a^3*d*sgn 
(x) - 700*(b*x + a)^(3/2)*a^4*d*sgn(x) + 150*sqrt(b*x + a)*a^5*d*sgn(x))/( 
a^2*b^6*x^5))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{11}} \, dx=\int \frac {{\left (b\,x^3+a\,x^2\right )}^{5/2}\,\left (c+d\,x\right )}{x^{11}} \,d x \] Input:

int(((a*x^2 + b*x^3)^(5/2)*(c + d*x))/x^11,x)
 

Output:

int(((a*x^2 + b*x^3)^(5/2)*(c + d*x))/x^11, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.17 \[ \int \frac {(c+d x) \left (a x^2+b x^3\right )^{5/2}}{x^{11}} \, dx=\frac {-768 \sqrt {b x +a}\, a^{5} c -960 \sqrt {b x +a}\, a^{5} d x -2016 \sqrt {b x +a}\, a^{4} b c x -2720 \sqrt {b x +a}\, a^{4} b d \,x^{2}-1488 \sqrt {b x +a}\, a^{3} b^{2} c \,x^{2}-2360 \sqrt {b x +a}\, a^{3} b^{2} d \,x^{3}-60 \sqrt {b x +a}\, a^{2} b^{3} c \,x^{3}-300 \sqrt {b x +a}\, a^{2} b^{3} d \,x^{4}+90 \sqrt {b x +a}\, a \,b^{4} c \,x^{4}-150 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a \,b^{4} d \,x^{5}+45 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{5} c \,x^{5}+150 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a \,b^{4} d \,x^{5}-45 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{5} c \,x^{5}}{3840 a^{3} x^{5}} \] Input:

int((d*x+c)*(b*x^3+a*x^2)^(5/2)/x^11,x)
 

Output:

( - 768*sqrt(a + b*x)*a**5*c - 960*sqrt(a + b*x)*a**5*d*x - 2016*sqrt(a + 
b*x)*a**4*b*c*x - 2720*sqrt(a + b*x)*a**4*b*d*x**2 - 1488*sqrt(a + b*x)*a* 
*3*b**2*c*x**2 - 2360*sqrt(a + b*x)*a**3*b**2*d*x**3 - 60*sqrt(a + b*x)*a* 
*2*b**3*c*x**3 - 300*sqrt(a + b*x)*a**2*b**3*d*x**4 + 90*sqrt(a + b*x)*a*b 
**4*c*x**4 - 150*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*a*b**4*d*x**5 + 45*s 
qrt(a)*log(sqrt(a + b*x) - sqrt(a))*b**5*c*x**5 + 150*sqrt(a)*log(sqrt(a + 
 b*x) + sqrt(a))*a*b**4*d*x**5 - 45*sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*b 
**5*c*x**5)/(3840*a**3*x**5)