\(\int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx\) [110]

Optimal result

Integrand size = 26, antiderivative size = 149 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \] Output:

4/3*(-4*a*c+b^2)*d^3*(2*c*d*x+b*d)^(3/2)+4/7*d*(2*c*d*x+b*d)^(7/2)+2*(-4*a 
*c+b^2)^(7/4)*d^(9/2)*arctan((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2 
))-2*(-4*a*c+b^2)^(7/4)*d^(9/2)*arctanh((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^( 
1/4)/d^(1/2))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.46 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {\left (\frac {1}{21}+\frac {i}{21}\right ) (d (b+2 c x))^{9/2} \left ((2-2 i) (b+2 c x)^{3/2} \left (7 b^2-28 a c+3 (b+2 c x)^2\right )-21 \left (b^2-4 a c\right )^{7/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+21 \left (b^2-4 a c\right )^{7/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-21 \left (b^2-4 a c\right )^{7/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )\right )}{(b+2 c x)^{9/2}} \] Input:

Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2),x]
 

Output:

((1/21 + I/21)*(d*(b + 2*c*x))^(9/2)*((2 - 2*I)*(b + 2*c*x)^(3/2)*(7*b^2 - 
 28*a*c + 3*(b + 2*c*x)^2) - 21*(b^2 - 4*a*c)^(7/4)*ArcTan[1 - ((1 + I)*Sq 
rt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)] + 21*(b^2 - 4*a*c)^(7/4)*ArcTan[1 + (( 
1 + I)*Sqrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)] - 21*(b^2 - 4*a*c)^(7/4)*ArcT 
anh[((1 + I)*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])/(Sqrt[b^2 - 4*a*c] + I*( 
b + 2*c*x))]))/(b + 2*c*x)^(9/2)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1116, 1116, 1118, 27, 25, 266, 827, 216, 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.

\(\displaystyle \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx\)

\(\Big \downarrow \) 1116

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

\(\Big \downarrow \) 1116

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

\(\Big \downarrow \) 1118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 216

\(\displaystyle d^2 \left (b^2-4 a c\right ) \left (\frac {4}{3} d (b d+2 c d x)^{3/2}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {1}{2} \int \frac {1}{-b d-2 c x d+\sqrt {b^2-4 a c} d}d\sqrt {b d+2 c x d}-\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}\right )\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\)

\(\Big \downarrow \) 219

\(\displaystyle d^2 \left (b^2-4 a c\right ) \left (\frac {4}{3} d (b d+2 c d x)^{3/2}-4 d^3 \left (b^2-4 a c\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}-\frac {\arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )}{2 \sqrt {d} \sqrt [4]{b^2-4 a c}}\right )\right )+\frac {4}{7} d (b d+2 c d x)^{7/2}\)

Input:

Int[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2),x]
 

Output:

(4*d*(b*d + 2*c*d*x)^(7/2))/7 + (b^2 - 4*a*c)*d^2*((4*d*(b*d + 2*c*d*x)^(3 
/2))/3 - 4*(b^2 - 4*a*c)*d^3*(-1/2*ArcTan[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a* 
c)^(1/4)*Sqrt[d])]/((b^2 - 4*a*c)^(1/4)*Sqrt[d]) + ArcTanh[Sqrt[b*d + 2*c* 
d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]/(2*(b^2 - 4*a*c)^(1/4)*Sqrt[d])))
 

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 216

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

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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 1116

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p 
 + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1)))   Int[(d 
 + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p 
+ 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
 

rule 1118

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[1/e   Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, 
d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(295\) vs. \(2(121)=242\).

Time = 1.52 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.99

method	result	size
pseudoelliptic	\(\frac {4 d \left (-\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} d^{2} \left (4 a c -b^{2}\right ) \left (d \left (2 c x +b \right )\right )^{\frac {3}{2}}}{3}+\frac {\left (d \left (2 c x +b \right )\right )^{\frac {7}{2}} \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{7}+\frac {\sqrt {2}\, d^{4} \left (4 a c -b^{2}\right )^{2} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}-1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+1\right )+\ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right )\right )}{8}\right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\)	\(296\)
derivativedivides	\(4 d \left (-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {d^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\)	\(303\)
default	\(4 d \left (-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {d^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\)	\(303\)

Input:

int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

4*d/(d^2*(4*a*c-b^2))^(1/4)*(-1/3*(d^2*(4*a*c-b^2))^(1/4)*d^2*(4*a*c-b^2)* 
(d*(2*c*x+b))^(3/2)+1/7*(d*(2*c*x+b))^(7/2)*(d^2*(4*a*c-b^2))^(1/4)+1/8*2^ 
(1/2)*d^4*(4*a*c-b^2)^2*(2*arctan(2^(1/2)/(d^2*(4*a*c-b^2))^(1/4)*(d*(2*c* 
x+b))^(1/2)-1)+2*arctan(2^(1/2)/(d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x+b))^(1/2 
)+1)+ln(((d^2*(4*a*c-b^2))^(1/2)-(d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x+b))^(1/ 
2)*2^(1/2)+d*(2*c*x+b))/((d^2*(4*a*c-b^2))^(1/4)*(d*(2*c*x+b))^(1/2)*2^(1/ 
2)+(d^2*(4*a*c-b^2))^(1/2)+d*(2*c*x+b)))))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 1005, normalized size of antiderivative = 6.74 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

8/21*(12*c^3*d^4*x^3 + 18*b*c^2*d^4*x^2 + 4*(4*b^2*c - 7*a*c^2)*d^4*x + (5 
*b^3 - 14*a*b*c)*d^4)*sqrt(2*c*d*x + b*d) + ((b^14 - 28*a*b^12*c + 336*a^2 
*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 2867 
2*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(1/4)*log(-(b^10 - 20*a*b^8*c + 160*a 
^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(2*c*d 
*x + b*d)*d^13 + ((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^ 
3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c 
^7)*d^18)^(3/4)) - I*((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^ 
8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a 
^7*c^7)*d^18)^(1/4)*log(-(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^ 
4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(2*c*d*x + b*d)*d^13 + I*((b^ 
14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 
- 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(3/4)) + I* 
((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6* 
c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(1/4)*l 
og(-(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2* 
c^4 - 1024*a^5*c^5)*sqrt(2*c*d*x + b*d)*d^13 - I*((b^14 - 28*a*b^12*c + 33 
6*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 
 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(3/4)) - ((b^14 - 28*a*b^12*c + 
336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\text {Timed out} \] Input:

integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (121) = 242\).

Time = 0.17 (sec) , antiderivative size = 531, normalized size of antiderivative = 3.56 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {4}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{3} - \frac {16}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c d^{3} + \frac {4}{7} \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} d - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) + \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) \] Input:

integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

4/3*(2*c*d*x + b*d)^(3/2)*b^2*d^3 - 16/3*(2*c*d*x + b*d)^(3/2)*a*c*d^3 + 4 
/7*(2*c*d*x + b*d)^(7/2)*d - (sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*b^2*d^3 
 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*a*c*d^3)*arctan(1/2*sqrt(2)*(sqr 
t(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a 
*c*d^2)^(1/4)) - (sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*b^2*d^3 - 4*sqrt(2) 
*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*a*c*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b^2* 
d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/ 
4)) + 1/2*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*b^2*d^3 - 4*sqrt(2)*(-b^2* 
d^2 + 4*a*c*d^2)^(3/4)*a*c*d^3)*log(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4* 
a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2)) - 1/2*(sq 
rt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*b^2*d^3 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d 
^2)^(3/4)*a*c*d^3)*log(2*c*d*x + b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4 
)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2))
 

Mupad [B] (verification not implemented)

Time = 5.43 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.13 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{7}-\frac {4\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (4\,a\,c-b^2\right )}{3}+2\,d^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}+d^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}\,1{}\mathrm {i}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}\,2{}\mathrm {i} \] Input:

int((b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2),x)
 

Output:

(4*d*(b*d + 2*c*d*x)^(7/2))/7 - (4*d^3*(b*d + 2*c*d*x)^(3/2)*(4*a*c - b^2) 
)/3 + 2*d^(9/2)*atan(((b*d + 2*c*d*x)^(1/2)*(b^2 - 4*a*c)^(7/4))/(d^(1/2)* 
(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(b^2 - 4*a*c)^(7/4) + d^(9/2)*atan(((b*d 
+ 2*c*d*x)^(1/2)*(b^2 - 4*a*c)^(7/4)*1i)/(d^(1/2)*(b^4 + 16*a^2*c^2 - 8*a* 
b^2*c)))*(b^2 - 4*a*c)^(7/4)*2i
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.85 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {\sqrt {d}\, d^{4} \left (-168 \left (4 a c -b^{2}\right )^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}-2 \sqrt {2 c x +b}}{\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}}\right ) a c +42 \left (4 a c -b^{2}\right )^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}-2 \sqrt {2 c x +b}}{\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}}\right ) b^{2}+168 \left (4 a c -b^{2}\right )^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+2 \sqrt {2 c x +b}}{\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}}\right ) a c -42 \left (4 a c -b^{2}\right )^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+2 \sqrt {2 c x +b}}{\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}}\right ) b^{2}+84 \left (4 a c -b^{2}\right )^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {2 c x +b}\, \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {4 a c -b^{2}}+b +2 c x \right ) a c -21 \left (4 a c -b^{2}\right )^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {2 c x +b}\, \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {4 a c -b^{2}}+b +2 c x \right ) b^{2}-84 \left (4 a c -b^{2}\right )^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {2 c x +b}\, \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {4 a c -b^{2}}+b +2 c x \right ) a c +21 \left (4 a c -b^{2}\right )^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {2 c x +b}\, \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {4 a c -b^{2}}+b +2 c x \right ) b^{2}-224 \sqrt {2 c x +b}\, a b c -448 \sqrt {2 c x +b}\, a \,c^{2} x +80 \sqrt {2 c x +b}\, b^{3}+256 \sqrt {2 c x +b}\, b^{2} c x +288 \sqrt {2 c x +b}\, b \,c^{2} x^{2}+192 \sqrt {2 c x +b}\, c^{3} x^{3}\right )}{42} \] Input:

int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x)
 

Output:

(sqrt(d)*d**4*( - 168*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)** 
(1/4)*sqrt(2) - 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*a*c + 
42*(4*a*c - b**2)**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*s 
qrt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*b**2 + 168*(4*a*c - b**2) 
**(3/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) + 2*sqrt(b + 2*c*x))/( 
(4*a*c - b**2)**(1/4)*sqrt(2)))*a*c - 42*(4*a*c - b**2)**(3/4)*sqrt(2)*ata 
n(((4*a*c - b**2)**(1/4)*sqrt(2) + 2*sqrt(b + 2*c*x))/((4*a*c - b**2)**(1/ 
4)*sqrt(2)))*b**2 + 84*(4*a*c - b**2)**(3/4)*sqrt(2)*log( - sqrt(b + 2*c*x 
)*(4*a*c - b**2)**(1/4)*sqrt(2) + sqrt(4*a*c - b**2) + b + 2*c*x)*a*c - 21 
*(4*a*c - b**2)**(3/4)*sqrt(2)*log( - sqrt(b + 2*c*x)*(4*a*c - b**2)**(1/4 
)*sqrt(2) + sqrt(4*a*c - b**2) + b + 2*c*x)*b**2 - 84*(4*a*c - b**2)**(3/4 
)*sqrt(2)*log(sqrt(b + 2*c*x)*(4*a*c - b**2)**(1/4)*sqrt(2) + sqrt(4*a*c - 
 b**2) + b + 2*c*x)*a*c + 21*(4*a*c - b**2)**(3/4)*sqrt(2)*log(sqrt(b + 2* 
c*x)*(4*a*c - b**2)**(1/4)*sqrt(2) + sqrt(4*a*c - b**2) + b + 2*c*x)*b**2 
- 224*sqrt(b + 2*c*x)*a*b*c - 448*sqrt(b + 2*c*x)*a*c**2*x + 80*sqrt(b + 2 
*c*x)*b**3 + 256*sqrt(b + 2*c*x)*b**2*c*x + 288*sqrt(b + 2*c*x)*b*c**2*x** 
2 + 192*sqrt(b + 2*c*x)*c**3*x**3))/42