\(\int (c+d x)^n \sqrt {a+b x^2} (A+B x+C x^2+D x^3) \, dx\) [166]

Optimal result

Integrand size = 34, antiderivative size = 537 \[ \int (c+d x)^n \sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {(C d (5+n)-c D (8+n)) (c+d x)^{1+n} \left (a+b x^2\right )^{3/2}}{b d^2 (4+n) (5+n)}+\frac {D (c+d x)^{2+n} \left (a+b x^2\right )^{3/2}}{b d^2 (5+n)}-\frac {\left (a d^2 (1+n) (C d (5+n)-c D (8+n))+b \left (12 c^3 D-3 c^2 C d (5+n)+B c d^2 \left (20+9 n+n^2\right )-A d^3 \left (20+9 n+n^2\right )\right )\right ) (c+d x)^{1+n} \sqrt {a+b x^2} \operatorname {AppellF1}\left (1+n,-\frac {1}{2},-\frac {1}{2},2+n,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{b d^4 (1+n) (4+n) (5+n) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}}}-\frac {\left (a d^2 D \left (8+6 n+n^2\right )-b \left (12 c^2 D-3 c C d (5+n)+B d^2 \left (20+9 n+n^2\right )\right )\right ) (c+d x)^{2+n} \sqrt {a+b x^2} \operatorname {AppellF1}\left (2+n,-\frac {1}{2},-\frac {1}{2},3+n,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{b d^4 (2+n) (4+n) (5+n) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}}} \] Output:

(C*d*(5+n)-c*D*(8+n))*(d*x+c)^(1+n)*(b*x^2+a)^(3/2)/b/d^2/(4+n)/(5+n)+D*(d 
*x+c)^(2+n)*(b*x^2+a)^(3/2)/b/d^2/(5+n)-(a*d^2*(1+n)*(C*d*(5+n)-c*D*(8+n)) 
+b*(12*D*c^3-3*c^2*C*d*(5+n)+B*c*d^2*(n^2+9*n+20)-A*d^3*(n^2+9*n+20)))*(d* 
x+c)^(1+n)*(b*x^2+a)^(1/2)*AppellF1(1+n,-1/2,-1/2,2+n,(d*x+c)/(c-(-a)^(1/2 
)*d/b^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/b/d^4/(1+n)/(4+n)/(5+n)/(1- 
(d*x+c)/(c-(-a)^(1/2)*d/b^(1/2)))^(1/2)/(1-(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2) 
))^(1/2)-(a*d^2*D*(n^2+6*n+8)-b*(12*D*c^2-3*c*C*d*(5+n)+B*d^2*(n^2+9*n+20) 
))*(d*x+c)^(2+n)*(b*x^2+a)^(1/2)*AppellF1(2+n,-1/2,-1/2,3+n,(d*x+c)/(c-(-a 
)^(1/2)*d/b^(1/2)),(d*x+c)/(c+(-a)^(1/2)*d/b^(1/2)))/b/d^4/(2+n)/(4+n)/(5+ 
n)/(1-(d*x+c)/(c-(-a)^(1/2)*d/b^(1/2)))^(1/2)/(1-(d*x+c)/(c+(-a)^(1/2)*d/b 
^(1/2)))^(1/2)
 

Mathematica [F]

\[ \int (c+d x)^n \sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int (c+d x)^n \sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx \] Input:

Integrate[(c + d*x)^n*Sqrt[a + b*x^2]*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

Integrate[(c + d*x)^n*Sqrt[a + b*x^2]*(A + B*x + C*x^2 + D*x^3), x]
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2185, 25, 2185, 25, 27, 719, 514, 150}

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.

\(\displaystyle \int \sqrt {a+b x^2} (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int -(c+d x)^n \sqrt {b x^2+a} \left (-b (C d (n+5)-c D (n+8)) x^2 d^2+(a c D (n+2)-A b d (n+5)) d^2+\left (3 b D c^2+a d^2 D (n+2)-b B d^2 (n+5)\right ) x d\right )dx}{b d^3 (n+5)}+\frac {D \left (a+b x^2\right )^{3/2} (c+d x)^{n+2}}{b d^2 (n+5)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {D \left (a+b x^2\right )^{3/2} (c+d x)^{n+2}}{b d^2 (n+5)}-\frac {\int (c+d x)^n \sqrt {b x^2+a} \left (-b (C d (n+5)-c D (n+8)) x^2 d^2+(a c D (n+2)-A b d (n+5)) d^2+\left (3 b D c^2+a d^2 D (n+2)-b B d^2 (n+5)\right ) x d\right )dx}{b d^3 (n+5)}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {D \left (a+b x^2\right )^{3/2} (c+d x)^{n+2}}{b d^2 (n+5)}-\frac {\frac {\int -b d^3 (c+d x)^n \left (d \left (3 a c D n-a C d \left (n^2+6 n+5\right )+A b d \left (n^2+9 n+20\right )\right )-\left (a d^2 D \left (n^2+6 n+8\right )-b \left (12 D c^2-3 C d (n+5) c+B d^2 \left (n^2+9 n+20\right )\right )\right ) x\right ) \sqrt {b x^2+a}dx}{b d^2 (n+4)}-\frac {d \left (a+b x^2\right )^{3/2} (c+d x)^{n+1} (C d (n+5)-c D (n+8))}{n+4}}{b d^3 (n+5)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {D \left (a+b x^2\right )^{3/2} (c+d x)^{n+2}}{b d^2 (n+5)}-\frac {-\frac {\int b d^3 (c+d x)^n \left (d \left (3 a c D n-a C d \left (n^2+6 n+5\right )+A b d \left (n^2+9 n+20\right )\right )-\left (a d^2 D \left (n^2+6 n+8\right )-b \left (12 D c^2-3 C d (n+5) c+B d^2 \left (n^2+9 n+20\right )\right )\right ) x\right ) \sqrt {b x^2+a}dx}{b d^2 (n+4)}-\frac {d \left (a+b x^2\right )^{3/2} (c+d x)^{n+1} (C d (n+5)-c D (n+8))}{n+4}}{b d^3 (n+5)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {D \left (a+b x^2\right )^{3/2} (c+d x)^{n+2}}{b d^2 (n+5)}-\frac {-\frac {d \int (c+d x)^n \left (d \left (3 a c D n-a C d \left (n^2+6 n+5\right )+A b d \left (n^2+9 n+20\right )\right )-\left (a d^2 D \left (n^2+6 n+8\right )-b \left (12 D c^2-3 C d (n+5) c+B d^2 \left (n^2+9 n+20\right )\right )\right ) x\right ) \sqrt {b x^2+a}dx}{n+4}-\frac {d \left (a+b x^2\right )^{3/2} (c+d x)^{n+1} (C d (n+5)-c D (n+8))}{n+4}}{b d^3 (n+5)}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {D \left (a+b x^2\right )^{3/2} (c+d x)^{n+2}}{b d^2 (n+5)}-\frac {-\frac {d \left (-\frac {\left (a d^2 (n+1) (C d (n+5)-c D (n+8))+b \left (-A d^3 \left (n^2+9 n+20\right )+B c d^2 \left (n^2+9 n+20\right )+12 c^3 D-3 c^2 C d (n+5)\right )\right ) \int (c+d x)^n \sqrt {b x^2+a}dx}{d}-\frac {\left (a d^2 D \left (n^2+6 n+8\right )-b \left (B d^2 \left (n^2+9 n+20\right )+12 c^2 D-3 c C d (n+5)\right )\right ) \int (c+d x)^{n+1} \sqrt {b x^2+a}dx}{d}\right )}{n+4}-\frac {d \left (a+b x^2\right )^{3/2} (c+d x)^{n+1} (C d (n+5)-c D (n+8))}{n+4}}{b d^3 (n+5)}\)

\(\Big \downarrow \) 514

\(\displaystyle \frac {D \left (a+b x^2\right )^{3/2} (c+d x)^{n+2}}{b d^2 (n+5)}-\frac {-\frac {d \left (-\frac {\sqrt {a+b x^2} \left (a d^2 (n+1) (C d (n+5)-c D (n+8))+b \left (-A d^3 \left (n^2+9 n+20\right )+B c d^2 \left (n^2+9 n+20\right )+12 c^3 D-3 c^2 C d (n+5)\right )\right ) \int (c+d x)^n \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}}d(c+d x)}{d^2 \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}}}-\frac {\sqrt {a+b x^2} \left (a d^2 D \left (n^2+6 n+8\right )-b \left (B d^2 \left (n^2+9 n+20\right )+12 c^2 D-3 c C d (n+5)\right )\right ) \int (c+d x)^{n+1} \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}}d(c+d x)}{d^2 \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}}}\right )}{n+4}-\frac {d \left (a+b x^2\right )^{3/2} (c+d x)^{n+1} (C d (n+5)-c D (n+8))}{n+4}}{b d^3 (n+5)}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {D \left (a+b x^2\right )^{3/2} (c+d x)^{n+2}}{b d^2 (n+5)}-\frac {-\frac {d \left (-\frac {\sqrt {a+b x^2} (c+d x)^{n+1} \operatorname {AppellF1}\left (n+1,-\frac {1}{2},-\frac {1}{2},n+2,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right ) \left (a d^2 (n+1) (C d (n+5)-c D (n+8))+b \left (-A d^3 \left (n^2+9 n+20\right )+B c d^2 \left (n^2+9 n+20\right )+12 c^3 D-3 c^2 C d (n+5)\right )\right )}{d^2 (n+1) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}}}-\frac {\sqrt {a+b x^2} (c+d x)^{n+2} \operatorname {AppellF1}\left (n+2,-\frac {1}{2},-\frac {1}{2},n+3,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right ) \left (a d^2 D \left (n^2+6 n+8\right )-b \left (B d^2 \left (n^2+9 n+20\right )+12 c^2 D-3 c C d (n+5)\right )\right )}{d^2 (n+2) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {-a} d}{\sqrt {b}}+c}}}\right )}{n+4}-\frac {d \left (a+b x^2\right )^{3/2} (c+d x)^{n+1} (C d (n+5)-c D (n+8))}{n+4}}{b d^3 (n+5)}\)

Input:

Int[(c + d*x)^n*Sqrt[a + b*x^2]*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

(D*(c + d*x)^(2 + n)*(a + b*x^2)^(3/2))/(b*d^2*(5 + n)) - (-((d*(C*d*(5 + 
n) - c*D*(8 + n))*(c + d*x)^(1 + n)*(a + b*x^2)^(3/2))/(4 + n)) - (d*(-((( 
a*d^2*(1 + n)*(C*d*(5 + n) - c*D*(8 + n)) + b*(12*c^3*D - 3*c^2*C*d*(5 + n 
) + B*c*d^2*(20 + 9*n + n^2) - A*d^3*(20 + 9*n + n^2)))*(c + d*x)^(1 + n)* 
Sqrt[a + b*x^2]*AppellF1[1 + n, -1/2, -1/2, 2 + n, (c + d*x)/(c - (Sqrt[-a 
]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(d^2*(1 + n)*Sqrt[1 
- (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b])]*Sqrt[1 - (c + d*x)/(c + (Sqrt[-a]* 
d)/Sqrt[b])])) - ((a*d^2*D*(8 + 6*n + n^2) - b*(12*c^2*D - 3*c*C*d*(5 + n) 
 + B*d^2*(20 + 9*n + n^2)))*(c + d*x)^(2 + n)*Sqrt[a + b*x^2]*AppellF1[2 + 
 n, -1/2, -1/2, 3 + n, (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c 
+ (Sqrt[-a]*d)/Sqrt[b])])/(d^2*(2 + n)*Sqrt[1 - (c + d*x)/(c - (Sqrt[-a]*d 
)/Sqrt[b])]*Sqrt[1 - (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])))/(4 + n))/(b* 
d^3*(5 + n))
 

Defintions of rubi rules used

rule 25

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

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 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 514

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( 
c + d*x)/(c + d*q))^p)   Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - 
x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && 
NeQ[b*c^2 + a*d^2, 0]
 

rule 719

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

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [F]

\[\int \left (d x +c \right )^{n} \sqrt {b \,x^{2}+a}\, \left (D x^{3}+C \,x^{2}+B x +A \right )d x\]

Input:

int((d*x+c)^n*(b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x)
 

Output:

int((d*x+c)^n*(b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x)
 

Fricas [F]

\[ \int (c+d x)^n \sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {b x^{2} + a} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="fric 
as")
 

Output:

integral((D*x^3 + C*x^2 + B*x + A)*sqrt(b*x^2 + a)*(d*x + c)^n, x)
 

Sympy [F]

\[ \int (c+d x)^n \sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int \sqrt {a + b x^{2}} \left (c + d x\right )^{n} \left (A + B x + C x^{2} + D x^{3}\right )\, dx \] Input:

integrate((d*x+c)**n*(b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A),x)
 

Output:

Integral(sqrt(a + b*x**2)*(c + d*x)**n*(A + B*x + C*x**2 + D*x**3), x)
 

Maxima [F]

\[ \int (c+d x)^n \sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {b x^{2} + a} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="maxi 
ma")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(b*x^2 + a)*(d*x + c)^n, x)
 

Giac [F]

\[ \int (c+d x)^n \sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int { {\left (D x^{3} + C x^{2} + B x + A\right )} \sqrt {b x^{2} + a} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="giac 
")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)*sqrt(b*x^2 + a)*(d*x + c)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^n \sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int \sqrt {b\,x^2+a}\,{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \] Input:

int((a + b*x^2)^(1/2)*(c + d*x)^n*(A + B*x + C*x^2 + x^3*D),x)
 

Output:

int((a + b*x^2)^(1/2)*(c + d*x)^n*(A + B*x + C*x^2 + x^3*D), x)
 

Reduce [F]

\[ \int (c+d x)^n \sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int \left (d x +c \right )^{n} \sqrt {b \,x^{2}+a}\, \left (D x^{3}+C \,x^{2}+B x +A \right )d x \] Input:

int((d*x+c)^n*(b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x)
                                                                                    
                                                                                    
 

Output:

int((d*x+c)^n*(b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A),x)