\(\int \frac {(c+d x)^2 (a+b x^2)^{3/2}}{x^7} \, dx\) [1057]

Optimal result

Integrand size = 22, antiderivative size = 154 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=\frac {\left (b c^2-6 a d^2\right ) \sqrt {a+b x^2}}{24 x^4}+\frac {5 b \left (b c^2-6 a d^2\right ) \sqrt {a+b x^2}}{48 a x^2}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac {2 c d \left (a+b x^2\right )^{5/2}}{5 a x^5}+\frac {b^2 \left (b c^2-6 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \] Output:

1/24*(-6*a*d^2+b*c^2)*(b*x^2+a)^(1/2)/x^4+5/48*b*(-6*a*d^2+b*c^2)*(b*x^2+a 
)^(1/2)/a/x^2-1/6*c^2*(b*x^2+a)^(5/2)/a/x^6-2/5*c*d*(b*x^2+a)^(5/2)/a/x^5+ 
1/16*b^2*(-6*a*d^2+b*c^2)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)
 

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=-\frac {\sqrt {a+b x^2} \left (3 b^2 c x^4 (5 c+32 d x)+a^2 \left (40 c^2+96 c d x+60 d^2 x^2\right )+2 a b x^2 \left (35 c^2+96 c d x+75 d^2 x^2\right )\right )}{240 a x^6}+\frac {b^2 \left (-b c^2+6 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}} \] Input:

Integrate[((c + d*x)^2*(a + b*x^2)^(3/2))/x^7,x]
 

Output:

-1/240*(Sqrt[a + b*x^2]*(3*b^2*c*x^4*(5*c + 32*d*x) + a^2*(40*c^2 + 96*c*d 
*x + 60*d^2*x^2) + 2*a*b*x^2*(35*c^2 + 96*c*d*x + 75*d^2*x^2)))/(a*x^6) + 
(b^2*(-(b*c^2) + 6*a*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/ 
(8*a^(3/2))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {540, 25, 534, 243, 51, 51, 73, 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.

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

\(\Big \downarrow \) 540

\(\displaystyle -\frac {\int -\frac {\left (12 a c d-\left (b c^2-6 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (12 a c d-\left (b c^2-6 a d^2\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {-\left (b c^2-6 a d^2\right ) \int \frac {\left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {12 c d \left (a+b x^2\right )^{5/2}}{5 x^5}}{6 a}-\frac {c^2 \left (a+b x^2\right )^{5/2}}{6 a x^6}\)

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[((c + d*x)^2*(a + b*x^2)^(3/2))/x^7,x]
 

Output:

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

Defintions of rubi rules used

rule 25

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

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.92

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (96 b^{2} c d \,x^{5}+150 a b \,d^{2} x^{4}+15 b^{2} c^{2} x^{4}+192 a b c d \,x^{3}+60 a^{2} d^{2} x^{2}+70 a b \,c^{2} x^{2}+96 a^{2} c d x +40 a^{2} c^{2}\right )}{240 x^{6} a}-\frac {\left (6 a \,d^{2}-b \,c^{2}\right ) b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{16 a^{\frac {3}{2}}}\)	\(141\)
default	\(c^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+d^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )-\frac {2 c d \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}\)	\(253\)

Input:

int((d*x+c)^2*(b*x^2+a)^(3/2)/x^7,x,method=_RETURNVERBOSE)
 

Output:

-1/240*(b*x^2+a)^(1/2)*(96*b^2*c*d*x^5+150*a*b*d^2*x^4+15*b^2*c^2*x^4+192* 
a*b*c*d*x^3+60*a^2*d^2*x^2+70*a*b*c^2*x^2+96*a^2*c*d*x+40*a^2*c^2)/x^6/a-1 
/16*(6*a*d^2-b*c^2)*b^2/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.01 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=\left [-\frac {15 \, {\left (b^{3} c^{2} - 6 \, a b^{2} d^{2}\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (96 \, a b^{2} c d x^{5} + 192 \, a^{2} b c d x^{3} + 96 \, a^{3} c d x + 40 \, a^{3} c^{2} + 15 \, {\left (a b^{2} c^{2} + 10 \, a^{2} b d^{2}\right )} x^{4} + 10 \, {\left (7 \, a^{2} b c^{2} + 6 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{480 \, a^{2} x^{6}}, -\frac {15 \, {\left (b^{3} c^{2} - 6 \, a b^{2} d^{2}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (96 \, a b^{2} c d x^{5} + 192 \, a^{2} b c d x^{3} + 96 \, a^{3} c d x + 40 \, a^{3} c^{2} + 15 \, {\left (a b^{2} c^{2} + 10 \, a^{2} b d^{2}\right )} x^{4} + 10 \, {\left (7 \, a^{2} b c^{2} + 6 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{240 \, a^{2} x^{6}}\right ] \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)/x^7,x, algorithm="fricas")
 

Output:

[-1/480*(15*(b^3*c^2 - 6*a*b^2*d^2)*sqrt(a)*x^6*log(-(b*x^2 - 2*sqrt(b*x^2 
 + a)*sqrt(a) + 2*a)/x^2) + 2*(96*a*b^2*c*d*x^5 + 192*a^2*b*c*d*x^3 + 96*a 
^3*c*d*x + 40*a^3*c^2 + 15*(a*b^2*c^2 + 10*a^2*b*d^2)*x^4 + 10*(7*a^2*b*c^ 
2 + 6*a^3*d^2)*x^2)*sqrt(b*x^2 + a))/(a^2*x^6), -1/240*(15*(b^3*c^2 - 6*a* 
b^2*d^2)*sqrt(-a)*x^6*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (96*a*b^2*c*d*x 
^5 + 192*a^2*b*c*d*x^3 + 96*a^3*c*d*x + 40*a^3*c^2 + 15*(a*b^2*c^2 + 10*a^ 
2*b*d^2)*x^4 + 10*(7*a^2*b*c^2 + 6*a^3*d^2)*x^2)*sqrt(b*x^2 + a))/(a^2*x^6 
)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (139) = 278\).

Time = 11.95 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.29 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=- \frac {a^{2} c^{2}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {a^{2} d^{2}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 a \sqrt {b} c^{2}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 a \sqrt {b} c d \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {3 a \sqrt {b} d^{2}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 b^{\frac {3}{2}} c^{2}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {4 b^{\frac {3}{2}} c d \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{2}} - \frac {b^{\frac {3}{2}} d^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {b^{\frac {3}{2}} d^{2}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {b^{\frac {5}{2}} c^{2}}{16 a x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 b^{\frac {5}{2}} c d \sqrt {\frac {a}{b x^{2}} + 1}}{5 a} - \frac {3 b^{2} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} + \frac {b^{3} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {3}{2}}} \] Input:

integrate((d*x+c)**2*(b*x**2+a)**(3/2)/x**7,x)
 

Output:

-a**2*c**2/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - a**2*d**2/(4*sqrt(b)*x* 
*5*sqrt(a/(b*x**2) + 1)) - 11*a*sqrt(b)*c**2/(24*x**5*sqrt(a/(b*x**2) + 1) 
) - 2*a*sqrt(b)*c*d*sqrt(a/(b*x**2) + 1)/(5*x**4) - 3*a*sqrt(b)*d**2/(8*x* 
*3*sqrt(a/(b*x**2) + 1)) - 17*b**(3/2)*c**2/(48*x**3*sqrt(a/(b*x**2) + 1)) 
 - 4*b**(3/2)*c*d*sqrt(a/(b*x**2) + 1)/(5*x**2) - b**(3/2)*d**2*sqrt(a/(b* 
x**2) + 1)/(2*x) - b**(3/2)*d**2/(8*x*sqrt(a/(b*x**2) + 1)) - b**(5/2)*c** 
2/(16*a*x*sqrt(a/(b*x**2) + 1)) - 2*b**(5/2)*c*d*sqrt(a/(b*x**2) + 1)/(5*a 
) - 3*b**2*d**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*sqrt(a)) + b**3*c**2*asinh(s 
qrt(a)/(sqrt(b)*x))/(16*a**(3/2))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.63 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=\frac {b^{3} c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} - \frac {3 \, b^{2} d^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} c^{2}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} b^{3} c^{2}}{16 \, a^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} d^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} b^{2} d^{2}}{8 \, a} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} c^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b d^{2}}{8 \, a^{2} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} b c^{2}}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} d^{2}}{4 \, a x^{4}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} c d}{5 \, a x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} c^{2}}{6 \, a x^{6}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)/x^7,x, algorithm="maxima")
 

Output:

1/16*b^3*c^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 3/8*b^2*d^2*arcsinh(a 
/(sqrt(a*b)*abs(x)))/sqrt(a) - 1/48*(b*x^2 + a)^(3/2)*b^3*c^2/a^3 - 1/16*s 
qrt(b*x^2 + a)*b^3*c^2/a^2 + 1/8*(b*x^2 + a)^(3/2)*b^2*d^2/a^2 + 3/8*sqrt( 
b*x^2 + a)*b^2*d^2/a + 1/48*(b*x^2 + a)^(5/2)*b^2*c^2/(a^3*x^2) - 1/8*(b*x 
^2 + a)^(5/2)*b*d^2/(a^2*x^2) + 1/24*(b*x^2 + a)^(5/2)*b*c^2/(a^2*x^4) - 1 
/4*(b*x^2 + a)^(5/2)*d^2/(a*x^4) - 2/5*(b*x^2 + a)^(5/2)*c*d/(a*x^5) - 1/6 
*(b*x^2 + a)^(5/2)*c^2/(a*x^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (130) = 260\).

Time = 0.14 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.81 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=-\frac {{\left (b^{3} c^{2} - 6 \, a b^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a} + \frac {15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{11} b^{3} c^{2} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{11} a b^{2} d^{2} + 480 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a b^{\frac {5}{2}} c d + 235 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} a b^{3} c^{2} - 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} a^{2} b^{2} d^{2} - 480 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {5}{2}} c d + 390 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} a^{2} b^{3} c^{2} + 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} a^{3} b^{2} d^{2} + 960 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} c d + 390 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a^{3} b^{3} c^{2} + 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} a^{4} b^{2} d^{2} - 960 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} c d + 235 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{4} b^{3} c^{2} - 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a^{5} b^{2} d^{2} + 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} c d + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{5} b^{3} c^{2} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{6} b^{2} d^{2} - 96 \, a^{6} b^{\frac {5}{2}} c d}{120 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{6} a} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(3/2)/x^7,x, algorithm="giac")
 

Output:

-1/8*(b^3*c^2 - 6*a*b^2*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a 
))/(sqrt(-a)*a) + 1/120*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^11*b^3*c^2 + 150 
*(sqrt(b)*x - sqrt(b*x^2 + a))^11*a*b^2*d^2 + 480*(sqrt(b)*x - sqrt(b*x^2 
+ a))^10*a*b^(5/2)*c*d + 235*(sqrt(b)*x - sqrt(b*x^2 + a))^9*a*b^3*c^2 - 2 
10*(sqrt(b)*x - sqrt(b*x^2 + a))^9*a^2*b^2*d^2 - 480*(sqrt(b)*x - sqrt(b*x 
^2 + a))^8*a^2*b^(5/2)*c*d + 390*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a^2*b^3*c 
^2 + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^7*a^3*b^2*d^2 + 960*(sqrt(b)*x - sqr 
t(b*x^2 + a))^6*a^3*b^(5/2)*c*d + 390*(sqrt(b)*x - sqrt(b*x^2 + a))^5*a^3* 
b^3*c^2 + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^5*a^4*b^2*d^2 - 960*(sqrt(b)*x 
- sqrt(b*x^2 + a))^4*a^4*b^(5/2)*c*d + 235*(sqrt(b)*x - sqrt(b*x^2 + a))^3 
*a^4*b^3*c^2 - 210*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a^5*b^2*d^2 + 96*(sqrt( 
b)*x - sqrt(b*x^2 + a))^2*a^5*b^(5/2)*c*d + 15*(sqrt(b)*x - sqrt(b*x^2 + a 
))*a^5*b^3*c^2 + 150*(sqrt(b)*x - sqrt(b*x^2 + a))*a^6*b^2*d^2 - 96*a^6*b^ 
(5/2)*c*d)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^6*a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2}{x^7} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(c + d*x)^2)/x^7,x)
 

Output:

int(((a + b*x^2)^(3/2)*(c + d*x)^2)/x^7, x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.03 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2}}{x^7} \, dx=\frac {-40 \sqrt {b \,x^{2}+a}\, a^{3} c^{2}-96 \sqrt {b \,x^{2}+a}\, a^{3} c d x -60 \sqrt {b \,x^{2}+a}\, a^{3} d^{2} x^{2}-70 \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{2} x^{2}-192 \sqrt {b \,x^{2}+a}\, a^{2} b c d \,x^{3}-150 \sqrt {b \,x^{2}+a}\, a^{2} b \,d^{2} x^{4}-15 \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} x^{4}-96 \sqrt {b \,x^{2}+a}\, a \,b^{2} c d \,x^{5}+90 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d^{2} x^{6}-15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{2} x^{6}-90 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d^{2} x^{6}+15 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{2} x^{6}-64 \sqrt {b}\, a \,b^{2} c d \,x^{6}}{240 a^{2} x^{6}} \] Input:

int((d*x+c)^2*(b*x^2+a)^(3/2)/x^7,x)
 

Output:

( - 40*sqrt(a + b*x**2)*a**3*c**2 - 96*sqrt(a + b*x**2)*a**3*c*d*x - 60*sq 
rt(a + b*x**2)*a**3*d**2*x**2 - 70*sqrt(a + b*x**2)*a**2*b*c**2*x**2 - 192 
*sqrt(a + b*x**2)*a**2*b*c*d*x**3 - 150*sqrt(a + b*x**2)*a**2*b*d**2*x**4 
- 15*sqrt(a + b*x**2)*a*b**2*c**2*x**4 - 96*sqrt(a + b*x**2)*a*b**2*c*d*x* 
*5 + 90*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b* 
*2*d**2*x**6 - 15*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqr 
t(a))*b**3*c**2*x**6 - 90*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b 
)*x)/sqrt(a))*a*b**2*d**2*x**6 + 15*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a 
) + sqrt(b)*x)/sqrt(a))*b**3*c**2*x**6 - 64*sqrt(b)*a*b**2*c*d*x**6)/(240* 
a**2*x**6)