\(\int \frac {(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^7} \, dx\) [1228]

Optimal result

Integrand size = 26, antiderivative size = 155 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx=-\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{1024 c^{7/2} \sqrt {b^2-4 a c} d^7} \]

[Out]

-5/192*(c*x^2+b*x+a)^(3/2)/c^2/d^7/(2*c*x+b)^4-1/12*(c*x^2+b*x+a)^(5/2)/c/d^7/(2*c*x+b)^6+5/1024*arctan(2*c^(1
/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(7/2)/d^7/(-4*a*c+b^2)^(1/2)-5/512*(c*x^2+b*x+a)^(1/2)/c^3/d^7/(
2*c*x+b)^2

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {698, 702, 211} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx=\frac {5 \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{1024 c^{7/2} d^7 \sqrt {b^2-4 a c}}-\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6} \]

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7,x]

[Out]

(-5*Sqrt[a + b*x + c*x^2])/(512*c^3*d^7*(b + 2*c*x)^2) - (5*(a + b*x + c*x^2)^(3/2))/(192*c^2*d^7*(b + 2*c*x)^
4) - (a + b*x + c*x^2)^(5/2)/(12*c*d^7*(b + 2*c*x)^6) + (5*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 -
4*a*c]])/(1024*c^(7/2)*Sqrt[b^2 - 4*a*c]*d^7)

Rule 211

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

Rule 698

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

Rule 702

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^5} \, dx}{24 c d^2} \\ & = -\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^3} \, dx}{128 c^2 d^4} \\ & = -\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{1024 c^3 d^6} \\ & = -\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \text {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{256 c^2 d^6} \\ & = -\frac {5 \sqrt {a+b x+c x^2}}{512 c^3 d^7 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{192 c^2 d^7 (b+2 c x)^4}-\frac {\left (a+b x+c x^2\right )^{5/2}}{12 c d^7 (b+2 c x)^6}+\frac {5 \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{1024 c^{7/2} \sqrt {b^2-4 a c} d^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.27 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx=\frac {-2 c \left (128 a^3 c^2+8 a^2 c \left (5 b^2+68 b c x+68 c^2 x^2\right )+a \left (15 b^4+200 b^3 c x+1144 b^2 c^2 x^2+1888 b c^3 x^3+944 c^4 x^4\right )+x \left (15 b^5+175 b^4 c x+848 b^3 c^2 x^2+1744 b^2 c^3 x^3+1584 b c^4 x^4+528 c^5 x^5\right )\right )-15 (b+2 c x)^6 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \text {arctanh}\left (2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}\right )}{3072 c^4 d^7 (b+2 c x)^6 \sqrt {a+x (b+c x)}} \]

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7,x]

[Out]

(-2*c*(128*a^3*c^2 + 8*a^2*c*(5*b^2 + 68*b*c*x + 68*c^2*x^2) + a*(15*b^4 + 200*b^3*c*x + 1144*b^2*c^2*x^2 + 18
88*b*c^3*x^3 + 944*c^4*x^4) + x*(15*b^5 + 175*b^4*c*x + 848*b^3*c^2*x^2 + 1744*b^2*c^3*x^3 + 1584*b*c^4*x^4 +
528*c^5*x^5)) - 15*(b + 2*c*x)^6*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*ArcTanh[2*Sqrt[(c*(a + x*(b + c*x)
))/(-b^2 + 4*a*c)]])/(3072*c^4*d^7*(b + 2*c*x)^6*Sqrt[a + x*(b + c*x)])

Maple [A] (verified)

Time = 5.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(-\frac {\frac {15 \left (2 c x +b \right )^{6} \operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )}{256}+\left (\frac {33 c^{4} x^{4}}{8}+\frac {\left (33 b \,x^{3}+13 a \,x^{2}\right ) c^{3}}{4}+\left (\frac {43}{8} b^{2} x^{2}+\frac {13}{4} a b x +a^{2}\right ) c^{2}+\frac {5 b^{2} \left (4 b x +a \right ) c}{16}+\frac {15 b^{4}}{128}\right ) \sqrt {4 c^{2} a -b^{2} c}\, \sqrt {c \,x^{2}+b x +a}}{12 \sqrt {4 c^{2} a -b^{2} c}\, c^{3} \left (2 c x +b \right )^{6} d^{7}}\)	\(167\)
default	\(\frac {-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}+\frac {2 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}+\frac {3 c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {10 c^{2} \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{5}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 c}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}}{128 d^{7} c^{7}}\)	\(464\)

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^7,x,method=_RETURNVERBOSE)

[Out]

-1/12*(15/256*(2*c*x+b)^6*arctanh(2*c*(c*x^2+b*x+a)^(1/2)/(4*a*c^2-b^2*c)^(1/2))+(33/8*c^4*x^4+1/4*(33*b*x^3+1
3*a*x^2)*c^3+(43/8*b^2*x^2+13/4*a*b*x+a^2)*c^2+5/16*b^2*(4*b*x+a)*c+15/128*b^4)*(4*a*c^2-b^2*c)^(1/2)*(c*x^2+b
*x+a)^(1/2))/(4*a*c^2-b^2*c)^(1/2)/c^3/(2*c*x+b)^6/d^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (131) = 262\).

Time = 2.15 (sec) , antiderivative size = 914, normalized size of antiderivative = 5.90 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx=\left [-\frac {15 \, {\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \sqrt {-b^{2} c + 4 \, a c^{2}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {-b^{2} c + 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, {\left (15 \, b^{6} c - 20 \, a b^{4} c^{2} - 32 \, a^{2} b^{2} c^{3} - 512 \, a^{3} c^{4} + 528 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} + 1056 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} + 16 \, {\left (43 \, b^{4} c^{3} - 146 \, a b^{2} c^{4} - 104 \, a^{2} c^{5}\right )} x^{2} + 32 \, {\left (5 \, b^{5} c^{2} - 7 \, a b^{3} c^{3} - 52 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{6144 \, {\left (64 \, {\left (b^{2} c^{10} - 4 \, a c^{11}\right )} d^{7} x^{6} + 192 \, {\left (b^{3} c^{9} - 4 \, a b c^{10}\right )} d^{7} x^{5} + 240 \, {\left (b^{4} c^{8} - 4 \, a b^{2} c^{9}\right )} d^{7} x^{4} + 160 \, {\left (b^{5} c^{7} - 4 \, a b^{3} c^{8}\right )} d^{7} x^{3} + 60 \, {\left (b^{6} c^{6} - 4 \, a b^{4} c^{7}\right )} d^{7} x^{2} + 12 \, {\left (b^{7} c^{5} - 4 \, a b^{5} c^{6}\right )} d^{7} x + {\left (b^{8} c^{4} - 4 \, a b^{6} c^{5}\right )} d^{7}\right )}}, -\frac {15 \, {\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \sqrt {b^{2} c - 4 \, a c^{2}} \arctan \left (\frac {\sqrt {b^{2} c - 4 \, a c^{2}} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (15 \, b^{6} c - 20 \, a b^{4} c^{2} - 32 \, a^{2} b^{2} c^{3} - 512 \, a^{3} c^{4} + 528 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} x^{4} + 1056 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{3} + 16 \, {\left (43 \, b^{4} c^{3} - 146 \, a b^{2} c^{4} - 104 \, a^{2} c^{5}\right )} x^{2} + 32 \, {\left (5 \, b^{5} c^{2} - 7 \, a b^{3} c^{3} - 52 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3072 \, {\left (64 \, {\left (b^{2} c^{10} - 4 \, a c^{11}\right )} d^{7} x^{6} + 192 \, {\left (b^{3} c^{9} - 4 \, a b c^{10}\right )} d^{7} x^{5} + 240 \, {\left (b^{4} c^{8} - 4 \, a b^{2} c^{9}\right )} d^{7} x^{4} + 160 \, {\left (b^{5} c^{7} - 4 \, a b^{3} c^{8}\right )} d^{7} x^{3} + 60 \, {\left (b^{6} c^{6} - 4 \, a b^{4} c^{7}\right )} d^{7} x^{2} + 12 \, {\left (b^{7} c^{5} - 4 \, a b^{5} c^{6}\right )} d^{7} x + {\left (b^{8} c^{4} - 4 \, a b^{6} c^{5}\right )} d^{7}\right )}}\right ] \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^7,x, algorithm="fricas")

[Out]

[-1/6144*(15*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b
^6)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x
+ a))/(4*c^2*x^2 + 4*b*c*x + b^2)) + 4*(15*b^6*c - 20*a*b^4*c^2 - 32*a^2*b^2*c^3 - 512*a^3*c^4 + 528*(b^2*c^5
- 4*a*c^6)*x^4 + 1056*(b^3*c^4 - 4*a*b*c^5)*x^3 + 16*(43*b^4*c^3 - 146*a*b^2*c^4 - 104*a^2*c^5)*x^2 + 32*(5*b
^5*c^2 - 7*a*b^3*c^3 - 52*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x + a))/(64*(b^2*c^10 - 4*a*c^11)*d^7*x^6 + 192*(b^3*c^
9 - 4*a*b*c^10)*d^7*x^5 + 240*(b^4*c^8 - 4*a*b^2*c^9)*d^7*x^4 + 160*(b^5*c^7 - 4*a*b^3*c^8)*d^7*x^3 + 60*(b^6*
c^6 - 4*a*b^4*c^7)*d^7*x^2 + 12*(b^7*c^5 - 4*a*b^5*c^6)*d^7*x + (b^8*c^4 - 4*a*b^6*c^5)*d^7), -1/3072*(15*(64*
c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6)*sqrt(b^2*c -
4*a*c^2)*arctan(1/2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)/(c^2*x^2 + b*c*x + a*c)) + 2*(15*b^6*c - 20*a*
b^4*c^2 - 32*a^2*b^2*c^3 - 512*a^3*c^4 + 528*(b^2*c^5 - 4*a*c^6)*x^4 + 1056*(b^3*c^4 - 4*a*b*c^5)*x^3 + 16*(43
*b^4*c^3 - 146*a*b^2*c^4 - 104*a^2*c^5)*x^2 + 32*(5*b^5*c^2 - 7*a*b^3*c^3 - 52*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x
+ a))/(64*(b^2*c^10 - 4*a*c^11)*d^7*x^6 + 192*(b^3*c^9 - 4*a*b*c^10)*d^7*x^5 + 240*(b^4*c^8 - 4*a*b^2*c^9)*d^7
*x^4 + 160*(b^5*c^7 - 4*a*b^3*c^8)*d^7*x^3 + 60*(b^6*c^6 - 4*a*b^4*c^7)*d^7*x^2 + 12*(b^7*c^5 - 4*a*b^5*c^6)*d
^7*x + (b^8*c^4 - 4*a*b^6*c^5)*d^7)]

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx}{d^{7}} \]

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**7,x)

[Out]

(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*
c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x*
*2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x**5 + 4
48*b*c**6*x**6 + 128*c**7*x**7), x) + Integral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*
c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x
) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560
*b**3*c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x) + Integral(2*a*c*x**2*sqrt(a + b*x
+ c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x
**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x +
84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*
x**7), x))/d**7

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^7,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1249 vs. \(2 (131) = 262\).

Time = 0.46 (sec) , antiderivative size = 1249, normalized size of antiderivative = 8.06 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^7,x, algorithm="giac")

[Out]

5/512*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b^2*c - 4*a*c^2))/(sqrt(b^2*c - 4*a*c
^2)*c^3*d^7) + 1/1536*(1056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*c^5 + 5808*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^10*b*c^(9/2) + 14480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^2*c^4 + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^9*a*c^5 + 21600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^3*c^(7/2) + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^8*a*b*c^(9/2) + 21600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^4*c^3 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^7*a^2*c^5 + 15456*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^5*c^(5/2) - 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^6*a*b^3*c^(7/2) + 10080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b*c^(9/2) + 8208*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*b^6*c^2 - 5760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^4*c^3 + 12960*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^5*a^2*b^2*c^4 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*c^5 + 3240*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^4*b^7*c^(3/2) - 4320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^5*c^(5/2) + 7200*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^4*a^2*b^3*c^(7/2) + 7200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b*c^(9/2) + 910*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*b^8*c - 1600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^6*c^2 + 960*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^3*a^2*b^4*c^3 + 7040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^2*c^4 + 160*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^3*a^4*c^5 + 165*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^9*sqrt(c) - 240*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^2*a*b^7*c^(3/2) - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^5*c^(5/2) + 3360*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^3*c^(7/2) + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b*c^(9/2) + 1
5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^10 + 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^8*c - 480*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a^2*b^6*c^2 + 1440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^4*c^3 - 1200*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a^4*b^2*c^4 + 1056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*c^5 + 15*a*b^9*sqrt(c) - 12
0*a^2*b^7*c^(3/2) + 400*a^3*b^5*c^(5/2) - 640*a^4*b^3*c^(7/2) + 528*a^5*b*c^(9/2))/((2*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^6*c^3*d^7)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^7} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^7} \,d x \]

[In]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7,x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^7, x)