\(\int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx\) [111]

Optimal result

Integrand size = 26, antiderivative size = 147 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=4 \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x}+\frac {4}{5} d (b d+2 c d x)^{5/2}-2 \left (b^2-4 a c\right )^{5/4} d^{7/2} \arctan \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{5/4} d^{7/2} \text {arctanh}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \] Output:

4*(-4*a*c+b^2)*d^3*(2*c*d*x+b*d)^(1/2)+4/5*d*(2*c*d*x+b*d)^(5/2)-2*(-4*a*c 
+b^2)^(5/4)*d^(7/2)*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)^(5/4)*d^(7/2)*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 = 0.41 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.46 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\frac {\left (\frac {1}{5}-\frac {i}{5}\right ) (d (b+2 c x))^{7/2} \left ((2+2 i) \sqrt {b+2 c x} \left (5 b^2-20 a c+(b+2 c x)^2\right )+5 \left (b^2-4 a c\right )^{5/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-5 \left (b^2-4 a c\right )^{5/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-5 \left (b^2-4 a c\right )^{5/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )\right )}{(b+2 c x)^{7/2}} \] Input:

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

Output:

((1/5 - I/5)*(d*(b + 2*c*x))^(7/2)*((2 + 2*I)*Sqrt[b + 2*c*x]*(5*b^2 - 20* 
a*c + (b + 2*c*x)^2) + 5*(b^2 - 4*a*c)^(5/4)*ArcTan[1 - ((1 + I)*Sqrt[b + 
2*c*x])/(b^2 - 4*a*c)^(1/4)] - 5*(b^2 - 4*a*c)^(5/4)*ArcTan[1 + ((1 + I)*S 
qrt[b + 2*c*x])/(b^2 - 4*a*c)^(1/4)] - 5*(b^2 - 4*a*c)^(5/4)*ArcTanh[((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)^(7/2)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 168, 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, 756, 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.

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

\(\Big \downarrow \) 1116

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

\(\Big \downarrow \) 1116

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

\(\Big \downarrow \) 1118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 756

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   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. \(296\) vs. \(2(121)=242\).

Time = 1.45 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.02

method	result	size
pseudoelliptic	\(-\frac {4 d \left (-\frac {\left (d \left (2 c x +b \right )\right )^{\frac {5}{2}} \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}}}{5}+\frac {d^{2} \left (4 a c -b^{2}\right ) \left (8 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \sqrt {d \left (2 c x +b \right )}-d^{2} \sqrt {2}\, \left (4 a c -b^{2}\right ) \left (\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}-1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}+1\right )\right )\right )}{8}\right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}}}\)	\(297\)
derivativedivides	\(4 d \left (-4 a c \,d^{2} \sqrt {2 c d x +b d}+b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {d^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\)	\(302\)
default	\(4 d \left (-4 a c \,d^{2} \sqrt {2 c d x +b d}+b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {d^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}{2 c d x +b d -\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a \,d^{2} c -b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a \,d^{2} c -b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\)	\(302\)

Input:

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

Output:

-4*d*(-1/5*(d*(2*c*x+b))^(5/2)*(d^2*(4*a*c-b^2))^(3/4)+1/8*d^2*(4*a*c-b^2) 
*(8*(d^2*(4*a*c-b^2))^(3/4)*(d*(2*c*x+b))^(1/2)-d^2*2^(1/2)*(4*a*c-b^2)*(l 
n(((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))/((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)))+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))))/(d^2*(4*a*c-b^2))^(3/4)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 629, normalized size of antiderivative = 4.28 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\frac {8}{5} \, {\left (2 \, c^{2} d^{3} x^{2} + 2 \, b c d^{3} x + {\left (3 \, b^{2} - 10 \, a c\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} + \left ({\left (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}\right )} d^{14}\right )^{\frac {1}{4}} \log \left (-\sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{3} + \left ({\left (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}\right )} d^{14}\right )^{\frac {1}{4}}\right ) + i \, \left ({\left (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}\right )} d^{14}\right )^{\frac {1}{4}} \log \left (-\sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{3} + i \, \left ({\left (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}\right )} d^{14}\right )^{\frac {1}{4}}\right ) - i \, \left ({\left (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}\right )} d^{14}\right )^{\frac {1}{4}} \log \left (-\sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{3} - i \, \left ({\left (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}\right )} d^{14}\right )^{\frac {1}{4}}\right ) - \left ({\left (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}\right )} d^{14}\right )^{\frac {1}{4}} \log \left (-\sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{3} - \left ({\left (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}\right )} d^{14}\right )^{\frac {1}{4}}\right ) \] Input:

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

Output:

8/5*(2*c^2*d^3*x^2 + 2*b*c*d^3*x + (3*b^2 - 10*a*c)*d^3)*sqrt(2*c*d*x + b* 
d) + ((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)*d^14)^(1/4)*log(-sqrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d 
^3 + ((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)*d^14)^(1/4)) + I*((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)*d^14)^(1/4)*log(-s 
qrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d^3 + I*((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)*d^14)^(1/4)) - I 
*((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)*d^14)^(1/4)*log(-sqrt(2*c*d*x + b*d)*(b^2 - 4*a*c)*d^3 - 
 I*((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)*d^14)^(1/4)) - ((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)*d^14)^(1/4)*log(-sqrt( 
2*c*d*x + b*d)*(b^2 - 4*a*c)*d^3 - ((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)*d^14)^(1/4))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*d*x+b*d)^(7/2)/(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 = 3.61 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=4 \, \sqrt {2 \, c d x + b d} b^{2} d^{3} - 16 \, \sqrt {2 \, c d x + b d} a c d^{3} + \frac {4}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} d - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) \] Input:

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

Output:

4*sqrt(2*c*d*x + b*d)*b^2*d^3 - 16*sqrt(2*c*d*x + b*d)*a*c*d^3 + 4/5*(2*c* 
d*x + b*d)^(5/2)*d - (sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*d^3 - 4*sqr 
t(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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)) - (sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*d^3 - 4*sqrt(2)*(-b^2*d 
^2 + 4*a*c*d^2)^(1/4)*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)^(1/4)*b^2*d^3 - 4*sqrt(2)*(-b^2*d^2 + 4* 
a*c*d^2)^(1/4)*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*(sqrt(2)*(- 
b^2*d^2 + 4*a*c*d^2)^(1/4)*b^2*d^3 - 4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4 
)*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 = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.14 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\frac {4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}{5}-4\,d^3\,\sqrt {b\,d+2\,c\,d\,x}\,\left (4\,a\,c-b^2\right )-2\,d^{7/2}\,\mathrm {atan}\left (\frac {b^2\,\sqrt {b\,d+2\,c\,d\,x}-4\,a\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{5/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}+d^{7/2}\,\mathrm {atan}\left (\frac {b^2\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}-a\,c\,\sqrt {b\,d+2\,c\,d\,x}\,4{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{5/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}\,2{}\mathrm {i} \] Input:

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

Output:

(4*d*(b*d + 2*c*d*x)^(5/2))/5 - 4*d^3*(b*d + 2*c*d*x)^(1/2)*(4*a*c - b^2) 
- 2*d^(7/2)*atan((b^2*(b*d + 2*c*d*x)^(1/2) - 4*a*c*(b*d + 2*c*d*x)^(1/2)) 
/(d^(1/2)*(b^2 - 4*a*c)^(5/4)))*(b^2 - 4*a*c)^(5/4) + d^(7/2)*atan((b^2*(b 
*d + 2*c*d*x)^(1/2)*1i - a*c*(b*d + 2*c*d*x)^(1/2)*4i)/(d^(1/2)*(b^2 - 4*a 
*c)^(5/4)))*(b^2 - 4*a*c)^(5/4)*2i
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 540, normalized size of antiderivative = 3.67 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\frac {\sqrt {d}\, d^{3} \left (-40 \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}-2 \sqrt {2 c x +b}}{\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}}\right ) a c +10 \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}-2 \sqrt {2 c x +b}}{\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}}\right ) b^{2}+40 \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+2 \sqrt {2 c x +b}}{\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}}\right ) a c -10 \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+2 \sqrt {2 c x +b}}{\left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}}\right ) b^{2}-20 \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {2 c x +b}\, \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {4 a c -b^{2}}+b +2 c x \right ) a c +5 \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {2 c x +b}\, \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {4 a c -b^{2}}+b +2 c x \right ) b^{2}+20 \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {2 c x +b}\, \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {4 a c -b^{2}}+b +2 c x \right ) a c -5 \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {2 c x +b}\, \left (4 a c -b^{2}\right )^{\frac {1}{4}} \sqrt {2}+\sqrt {4 a c -b^{2}}+b +2 c x \right ) b^{2}-160 \sqrt {2 c x +b}\, a c +48 \sqrt {2 c x +b}\, b^{2}+32 \sqrt {2 c x +b}\, b c x +32 \sqrt {2 c x +b}\, c^{2} x^{2}\right )}{10} \] Input:

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

Output:

(sqrt(d)*d**3*( - 40*(4*a*c - b**2)**(1/4)*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 + 1 
0*(4*a*c - b**2)**(1/4)*sqrt(2)*atan(((4*a*c - b**2)**(1/4)*sqrt(2) - 2*sq 
rt(b + 2*c*x))/((4*a*c - b**2)**(1/4)*sqrt(2)))*b**2 + 40*(4*a*c - b**2)** 
(1/4)*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 - 10*(4*a*c - b**2)**(1/4)*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)))*b**2 - 20*(4*a*c - b**2)**(1/4)*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 + 5*(4 
*a*c - b**2)**(1/4)*sqrt(2)*log( - sqrt(b + 2*c*x)*(4*a*c - b**2)**(1/4)*s 
qrt(2) + sqrt(4*a*c - b**2) + b + 2*c*x)*b**2 + 20*(4*a*c - b**2)**(1/4)*s 
qrt(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 - 5*(4*a*c - b**2)**(1/4)*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 - 16 
0*sqrt(b + 2*c*x)*a*c + 48*sqrt(b + 2*c*x)*b**2 + 32*sqrt(b + 2*c*x)*b*c*x 
 + 32*sqrt(b + 2*c*x)*c**2*x**2))/10