3.11.30 \(\int (b d+2 c d x)^4 (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=249 \[ -\frac {3 d^4 \left (b^2-4 a c\right )^5 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{7/2}}-\frac {3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c} \]

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Rubi [A]  time = 0.17, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {685, 692, 621, 206} \begin {gather*} -\frac {3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}-\frac {3 d^4 \left (b^2-4 a c\right )^5 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{7/2}}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x]

[Out]

(-3*(b^2 - 4*a*c)^4*d^4*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*c^3) - ((b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^3*Sqr
t[a + b*x + c*x^2])/(4096*c^3) + ((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^5*Sqrt[a + b*x + c*x^2])/(1024*c^3) - ((b^2
- 4*a*c)*d^4*(b + 2*c*x)^5*(a + b*x + c*x^2)^(3/2))/(128*c^2) + (d^4*(b + 2*c*x)^5*(a + b*x + c*x^2)^(5/2))/(2
0*c) - (3*(b^2 - 4*a*c)^5*d^4*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16384*c^(7/2))

Rule 206

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

Rule 621

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

Rule 685

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 + 2*p + 1)), x] - Dist[(d*p*(b^2 - 4*a*c))/(b*e*(m + 2*p + 1)), Int[(d + e*x)^m*(a +
b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] &&  !(IGtQ[(m - 1)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p]))
&& RationalQ[m] && IntegerQ[2*p]

Rule 692

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

Rubi steps

\begin {align*} \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (b^2-4 a c\right ) \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx}{8 c}\\ &=-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}+\frac {\left (3 \left (b^2-4 a c\right )^2\right ) \int (b d+2 c d x)^4 \sqrt {a+b x+c x^2} \, dx}{256 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (b^2-4 a c\right )^3 \int \frac {(b d+2 c d x)^4}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^3}\\ &=-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (3 \left (b^2-4 a c\right )^4 d^2\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx}{8192 c^3}\\ &=-\frac {3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (3 \left (b^2-4 a c\right )^5 d^4\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16384 c^3}\\ &=-\frac {3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (3 \left (b^2-4 a c\right )^5 d^4\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8192 c^3}\\ &=-\frac {3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {3 \left (b^2-4 a c\right )^5 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 3.54, size = 265, normalized size = 1.06 \begin {gather*} \frac {1}{5} d^4 \left ((b+2 c x)^3 (a+x (b+c x))^{7/2}-\frac {3}{2} c \left (a-\frac {b^2}{4 c}\right ) (b+2 c x) \sqrt {a+x (b+c x)} \left (\frac {\left (b^2-4 a c\right ) \left (16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )+8 b^2 c \left (11 c x^2-20 a\right )+32 b c^2 x \left (13 a+8 c x^2\right )+15 b^4-40 b^3 c x\right )}{3072 c^3}-\frac {5 \sqrt {c} \sqrt {4 a-\frac {b^2}{c}} (a+x (b+c x))^3 \sinh ^{-1}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {4 a-\frac {b^2}{c}}}\right )}{2048 (b+2 c x) \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{7/2}}+(a+x (b+c x))^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x]

[Out]

(d^4*((b + 2*c*x)^3*(a + x*(b + c*x))^(7/2) - (3*(a - b^2/(4*c))*c*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*((a + x*(
b + c*x))^3 + ((b^2 - 4*a*c)*(15*b^4 - 40*b^3*c*x + 32*b*c^2*x*(13*a + 8*c*x^2) + 8*b^2*c*(-20*a + 11*c*x^2) +
16*c^2*(33*a^2 + 26*a*c*x^2 + 8*c^2*x^4)))/(3072*c^3) - (5*Sqrt[4*a - b^2/c]*Sqrt[c]*(a + x*(b + c*x))^3*ArcS
inh[(b + 2*c*x)/(Sqrt[4*a - b^2/c]*Sqrt[c])])/(2048*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(7/2)))
)/2))/5

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IntegrateAlgebraic [B]  time = 1.93, size = 538, normalized size = 2.16 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-3840 a^4 b c^4 d^4-7680 a^4 c^5 d^4 x+4480 a^3 b^3 c^3 d^4+11520 a^3 b^2 c^4 d^4 x+7680 a^3 b c^5 d^4 x^2+5120 a^3 c^6 d^4 x^3+2048 a^2 b^5 c^2 d^4+33920 a^2 b^4 c^3 d^4 x+152960 a^2 b^3 c^4 d^4 x^2+313600 a^2 b^2 c^5 d^4 x^3+317440 a^2 b c^6 d^4 x^4+126976 a^2 c^7 d^4 x^5-280 a b^7 c d^4+176 a b^6 c^2 d^4 x+34976 a b^5 c^3 d^4 x^2+218560 a b^4 c^4 d^4 x^3+593920 a b^3 c^5 d^4 x^4+839680 a b^2 c^6 d^4 x^5+602112 a b c^7 d^4 x^6+172032 a c^8 d^4 x^7+15 b^9 d^4-10 b^8 c d^4 x+8 b^7 c^2 d^4 x^2+11696 b^6 c^3 d^4 x^3+89728 b^5 c^4 d^4 x^4+298240 b^4 c^5 d^4 x^5+537600 b^3 c^6 d^4 x^6+546816 b^2 c^7 d^4 x^7+294912 b c^8 d^4 x^8+65536 c^9 d^4 x^9\right )}{40960 c^3}+\frac {3 \left (-1024 a^5 c^5 d^4+1280 a^4 b^2 c^4 d^4-640 a^3 b^4 c^3 d^4+160 a^2 b^6 c^2 d^4-20 a b^8 c d^4+b^{10} d^4\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{16384 c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[a + b*x + c*x^2]*(15*b^9*d^4 - 280*a*b^7*c*d^4 + 2048*a^2*b^5*c^2*d^4 + 4480*a^3*b^3*c^3*d^4 - 3840*a^4*
b*c^4*d^4 - 10*b^8*c*d^4*x + 176*a*b^6*c^2*d^4*x + 33920*a^2*b^4*c^3*d^4*x + 11520*a^3*b^2*c^4*d^4*x - 7680*a^
4*c^5*d^4*x + 8*b^7*c^2*d^4*x^2 + 34976*a*b^5*c^3*d^4*x^2 + 152960*a^2*b^3*c^4*d^4*x^2 + 7680*a^3*b*c^5*d^4*x^
2 + 11696*b^6*c^3*d^4*x^3 + 218560*a*b^4*c^4*d^4*x^3 + 313600*a^2*b^2*c^5*d^4*x^3 + 5120*a^3*c^6*d^4*x^3 + 897
28*b^5*c^4*d^4*x^4 + 593920*a*b^3*c^5*d^4*x^4 + 317440*a^2*b*c^6*d^4*x^4 + 298240*b^4*c^5*d^4*x^5 + 839680*a*b
^2*c^6*d^4*x^5 + 126976*a^2*c^7*d^4*x^5 + 537600*b^3*c^6*d^4*x^6 + 602112*a*b*c^7*d^4*x^6 + 546816*b^2*c^7*d^4
*x^7 + 172032*a*c^8*d^4*x^7 + 294912*b*c^8*d^4*x^8 + 65536*c^9*d^4*x^9))/(40960*c^3) + (3*(b^10*d^4 - 20*a*b^8
*c*d^4 + 160*a^2*b^6*c^2*d^4 - 640*a^3*b^4*c^3*d^4 + 1280*a^4*b^2*c^4*d^4 - 1024*a^5*c^5*d^4)*Log[b + 2*c*x -
2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(16384*c^(7/2))

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fricas [B]  time = 0.50, size = 923, normalized size = 3.71 \begin {gather*} \left [-\frac {15 \, {\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 )} \sqrt {c} d^{4} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (65536 \, c^{10} d^{4} x^{9} + 294912 \, b c^{9} d^{4} x^{8} + 6144 \, {\left (89 \, b^{2} c^{8} + 28 \, a c^{9}\right )} d^{4} x^{7} + 21504 \, {\left (25 \, b^{3} c^{7} + 28 \, a b c^{8}\right )} d^{4} x^{6} + 256 \, {\left (1165 \, b^{4} c^{6} + 3280 \, a b^{2} c^{7} + 496 \, a^{2} c^{8}\right )} d^{4} x^{5} + 128 \, {\left (701 \, b^{5} c^{5} + 4640 \, a b^{3} c^{6} + 2480 \, a^{2} b c^{7}\right )} d^{4} x^{4} + 16 \, {\left (731 \, b^{6} c^{4} + 13660 \, a b^{4} c^{5} + 19600 \, a^{2} b^{2} c^{6} + 320 \, a^{3} c^{7}\right )} d^{4} x^{3} + 8 \, {\left (b^{7} c^{3} + 4372 \, a b^{5} c^{4} + 19120 \, a^{2} b^{3} c^{5} + 960 \, a^{3} b c^{6}\right )} d^{4} x^{2} - 2 \, {\left (5 \, b^{8} c^{2} - 88 \, a b^{6} c^{3} - 16960 \, a^{2} b^{4} c^{4} - 5760 \, a^{3} b^{2} c^{5} + 3840 \, a^{4} c^{6}\right )} d^{4} x + {\left (15 \, b^{9} c - 280 \, a b^{7} c^{2} + 2048 \, a^{2} b^{5} c^{3} + 4480 \, a^{3} b^{3} c^{4} - 3840 \, a^{4} b c^{5}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{163840 \, c^{4}}, \frac {15 \, {\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 )} \sqrt {-c} d^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (65536 \, c^{10} d^{4} x^{9} + 294912 \, b c^{9} d^{4} x^{8} + 6144 \, {\left (89 \, b^{2} c^{8} + 28 \, a c^{9}\right )} d^{4} x^{7} + 21504 \, {\left (25 \, b^{3} c^{7} + 28 \, a b c^{8}\right )} d^{4} x^{6} + 256 \, {\left (1165 \, b^{4} c^{6} + 3280 \, a b^{2} c^{7} + 496 \, a^{2} c^{8}\right )} d^{4} x^{5} + 128 \, {\left (701 \, b^{5} c^{5} + 4640 \, a b^{3} c^{6} + 2480 \, a^{2} b c^{7}\right )} d^{4} x^{4} + 16 \, {\left (731 \, b^{6} c^{4} + 13660 \, a b^{4} c^{5} + 19600 \, a^{2} b^{2} c^{6} + 320 \, a^{3} c^{7}\right )} d^{4} x^{3} + 8 \, {\left (b^{7} c^{3} + 4372 \, a b^{5} c^{4} + 19120 \, a^{2} b^{3} c^{5} + 960 \, a^{3} b c^{6}\right )} d^{4} x^{2} - 2 \, {\left (5 \, b^{8} c^{2} - 88 \, a b^{6} c^{3} - 16960 \, a^{2} b^{4} c^{4} - 5760 \, a^{3} b^{2} c^{5} + 3840 \, a^{4} c^{6}\right )} d^{4} x + {\left (15 \, b^{9} c - 280 \, a b^{7} c^{2} + 2048 \, a^{2} b^{5} c^{3} + 4480 \, a^{3} b^{3} c^{4} - 3840 \, a^{4} b c^{5}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{81920 \, c^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/163840*(15*(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(
c)*d^4*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(65536*c^10*d
^4*x^9 + 294912*b*c^9*d^4*x^8 + 6144*(89*b^2*c^8 + 28*a*c^9)*d^4*x^7 + 21504*(25*b^3*c^7 + 28*a*b*c^8)*d^4*x^6
+ 256*(1165*b^4*c^6 + 3280*a*b^2*c^7 + 496*a^2*c^8)*d^4*x^5 + 128*(701*b^5*c^5 + 4640*a*b^3*c^6 + 2480*a^2*b*
c^7)*d^4*x^4 + 16*(731*b^6*c^4 + 13660*a*b^4*c^5 + 19600*a^2*b^2*c^6 + 320*a^3*c^7)*d^4*x^3 + 8*(b^7*c^3 + 437
2*a*b^5*c^4 + 19120*a^2*b^3*c^5 + 960*a^3*b*c^6)*d^4*x^2 - 2*(5*b^8*c^2 - 88*a*b^6*c^3 - 16960*a^2*b^4*c^4 - 5
760*a^3*b^2*c^5 + 3840*a^4*c^6)*d^4*x + (15*b^9*c - 280*a*b^7*c^2 + 2048*a^2*b^5*c^3 + 4480*a^3*b^3*c^4 - 3840
*a^4*b*c^5)*d^4)*sqrt(c*x^2 + b*x + a))/c^4, 1/81920*(15*(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(-c)*d^4*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x
^2 + b*c*x + a*c)) + 2*(65536*c^10*d^4*x^9 + 294912*b*c^9*d^4*x^8 + 6144*(89*b^2*c^8 + 28*a*c^9)*d^4*x^7 + 215
04*(25*b^3*c^7 + 28*a*b*c^8)*d^4*x^6 + 256*(1165*b^4*c^6 + 3280*a*b^2*c^7 + 496*a^2*c^8)*d^4*x^5 + 128*(701*b^
5*c^5 + 4640*a*b^3*c^6 + 2480*a^2*b*c^7)*d^4*x^4 + 16*(731*b^6*c^4 + 13660*a*b^4*c^5 + 19600*a^2*b^2*c^6 + 320
*a^3*c^7)*d^4*x^3 + 8*(b^7*c^3 + 4372*a*b^5*c^4 + 19120*a^2*b^3*c^5 + 960*a^3*b*c^6)*d^4*x^2 - 2*(5*b^8*c^2 -
88*a*b^6*c^3 - 16960*a^2*b^4*c^4 - 5760*a^3*b^2*c^5 + 3840*a^4*c^6)*d^4*x + (15*b^9*c - 280*a*b^7*c^2 + 2048*a
^2*b^5*c^3 + 4480*a^3*b^3*c^4 - 3840*a^4*b*c^5)*d^4)*sqrt(c*x^2 + b*x + a))/c^4]

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giac [B]  time = 0.33, size = 547, normalized size = 2.20 \begin {gather*} \frac {1}{40960} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (16 \, {\left (2 \, c^{6} d^{4} x + 9 \, b c^{5} d^{4}\right )} x + \frac {3 \, {\left (89 \, b^{2} c^{13} d^{4} + 28 \, a c^{14} d^{4}\right )}}{c^{9}}\right )} x + \frac {21 \, {\left (25 \, b^{3} c^{12} d^{4} + 28 \, a b c^{13} d^{4}\right )}}{c^{9}}\right )} x + \frac {1165 \, b^{4} c^{11} d^{4} + 3280 \, a b^{2} c^{12} d^{4} + 496 \, a^{2} c^{13} d^{4}}{c^{9}}\right )} x + \frac {701 \, b^{5} c^{10} d^{4} + 4640 \, a b^{3} c^{11} d^{4} + 2480 \, a^{2} b c^{12} d^{4}}{c^{9}}\right )} x + \frac {731 \, b^{6} c^{9} d^{4} + 13660 \, a b^{4} c^{10} d^{4} + 19600 \, a^{2} b^{2} c^{11} d^{4} + 320 \, a^{3} c^{12} d^{4}}{c^{9}}\right )} x + \frac {b^{7} c^{8} d^{4} + 4372 \, a b^{5} c^{9} d^{4} + 19120 \, a^{2} b^{3} c^{10} d^{4} + 960 \, a^{3} b c^{11} d^{4}}{c^{9}}\right )} x - \frac {5 \, b^{8} c^{7} d^{4} - 88 \, a b^{6} c^{8} d^{4} - 16960 \, a^{2} b^{4} c^{9} d^{4} - 5760 \, a^{3} b^{2} c^{10} d^{4} + 3840 \, a^{4} c^{11} d^{4}}{c^{9}}\right )} x + \frac {15 \, b^{9} c^{6} d^{4} - 280 \, a b^{7} c^{7} d^{4} + 2048 \, a^{2} b^{5} c^{8} d^{4} + 4480 \, a^{3} b^{3} c^{9} d^{4} - 3840 \, a^{4} b c^{10} d^{4}}{c^{9}}\right )} + \frac {3 \, {\left (b^{10} d^{4} - 20 \, a b^{8} c d^{4} + 160 \, a^{2} b^{6} c^{2} d^{4} - 640 \, a^{3} b^{4} c^{3} d^{4} + 1280 \, a^{4} b^{2} c^{4} d^{4} - 1024 \, a^{5} c^{5} d^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16384 \, c^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

1/40960*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(2*(16*(2*c^6*d^4*x + 9*b*c^5*d^4)*x + 3*(89*b^2*c^13*d^4 + 28
*a*c^14*d^4)/c^9)*x + 21*(25*b^3*c^12*d^4 + 28*a*b*c^13*d^4)/c^9)*x + (1165*b^4*c^11*d^4 + 3280*a*b^2*c^12*d^4
+ 496*a^2*c^13*d^4)/c^9)*x + (701*b^5*c^10*d^4 + 4640*a*b^3*c^11*d^4 + 2480*a^2*b*c^12*d^4)/c^9)*x + (731*b^6
*c^9*d^4 + 13660*a*b^4*c^10*d^4 + 19600*a^2*b^2*c^11*d^4 + 320*a^3*c^12*d^4)/c^9)*x + (b^7*c^8*d^4 + 4372*a*b^
5*c^9*d^4 + 19120*a^2*b^3*c^10*d^4 + 960*a^3*b*c^11*d^4)/c^9)*x - (5*b^8*c^7*d^4 - 88*a*b^6*c^8*d^4 - 16960*a^
2*b^4*c^9*d^4 - 5760*a^3*b^2*c^10*d^4 + 3840*a^4*c^11*d^4)/c^9)*x + (15*b^9*c^6*d^4 - 280*a*b^7*c^7*d^4 + 2048
*a^2*b^5*c^8*d^4 + 4480*a^3*b^3*c^9*d^4 - 3840*a^4*b*c^10*d^4)/c^9) + 3/16384*(b^10*d^4 - 20*a*b^8*c*d^4 + 160
*a^2*b^6*c^2*d^4 - 640*a^3*b^4*c^3*d^4 + 1280*a^4*b^2*c^4*d^4 - 1024*a^5*c^5*d^4)*log(abs(-2*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)

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maple [B]  time = 0.06, size = 920, normalized size = 3.69 \begin {gather*} \frac {3 a^{5} c^{\frac {3}{2}} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16}-\frac {15 a^{4} b^{2} \sqrt {c}\, d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64}+\frac {15 a^{3} b^{4} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 \sqrt {c}}-\frac {15 a^{2} b^{6} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {3}{2}}}+\frac {15 a \,b^{8} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4096 c^{\frac {5}{2}}}-\frac {3 b^{10} d^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16384 c^{\frac {7}{2}}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, a^{4} c^{2} d^{4} x}{16}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{2} c \,d^{4} x}{16}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{4} d^{4} x}{128}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a \,b^{6} d^{4} x}{256 c}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{8} d^{4} x}{4096 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, a^{4} b c \,d^{4}}{32}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a^{3} b^{3} d^{4}}{32}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{3} c^{2} d^{4} x}{8}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{5} d^{4}}{256 c}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} b^{2} c \,d^{4} x}{32}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a \,b^{7} d^{4}}{512 c^{2}}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{4} d^{4} x}{128}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{9} d^{4}}{8192 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{6} d^{4} x}{512 c}+\frac {8 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} c^{3} d^{4} x^{3}}{5}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{3} b c \,d^{4}}{16}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} b^{3} d^{4}}{64}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{2} c^{2} d^{4} x}{10}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{5} d^{4}}{256 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,b^{2} c \,d^{4} x}{20}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{7} d^{4}}{1024 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{4} d^{4} x}{160}+\frac {12 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} b \,c^{2} d^{4} x^{2}}{5}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a^{2} b c \,d^{4}}{20}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,b^{3} d^{4}}{40}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} a \,c^{2} d^{4} x}{5}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{5} d^{4}}{320 c}+\frac {27 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} b^{2} c \,d^{4} x}{20}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} a b c \,d^{4}}{10}+\frac {11 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} b^{3} d^{4}}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^(5/2),x)

[Out]

-3/5*d^4*c^2*a*x*(c*x^2+b*x+a)^(7/2)+3/32*d^4*c*a^4*(c*x^2+b*x+a)^(1/2)*b+1/10*d^4*c^2*a^2*x*(c*x^2+b*x+a)^(5/
2)+1/20*d^4*c*a^2*(c*x^2+b*x+a)^(5/2)*b+1/8*d^4*c^2*a^3*(c*x^2+b*x+a)^(3/2)*x+1/16*d^4*c*a^3*(c*x^2+b*x+a)^(3/
2)*b+3/16*d^4*c^2*a^4*(c*x^2+b*x+a)^(1/2)*x-1/512*d^4/c*b^6*(c*x^2+b*x+a)^(3/2)*x+3/256*d^4/c*b^5*(c*x^2+b*x+a
)^(3/2)*a+3/4096*d^4/c^2*b^8*(c*x^2+b*x+a)^(1/2)*x-3/16*d^4*c*b^2*a^3*(c*x^2+b*x+a)^(1/2)*x-3/256*d^4/c*b^6*(c
*x^2+b*x+a)^(1/2)*x*a-1/20*d^4*c*b^2*a*x*(c*x^2+b*x+a)^(5/2)-3/32*d^4*c*b^2*a^2*(c*x^2+b*x+a)^(3/2)*x-3/64*d^4
*b^3*a^2*(c*x^2+b*x+a)^(3/2)+1/160*d^4*b^4*x*(c*x^2+b*x+a)^(5/2)-3/32*d^4*b^3*a^3*(c*x^2+b*x+a)^(1/2)+3/16*d^4
*c^(3/2)*a^5*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/8192*d^4/c^3*b^9*(c*x^2+b*x+a)^(1/2)-3/16384*d^4*b^
10/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+8/5*d^4*c^3*x^3*(c*x^2+b*x+a)^(7/2)+15/4096*d^4*b^8/c^(
5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/320*d^4/c*b^5*(c*x^2+b*x+a)^(5/2)-1/1024*d^4/c^2*b^7*(c*x
^2+b*x+a)^(3/2)-1/40*d^4*b^3*a*(c*x^2+b*x+a)^(5/2)+9/256*d^4/c*b^5*(c*x^2+b*x+a)^(1/2)*a^2-3/512*d^4/c^2*b^7*(
c*x^2+b*x+a)^(1/2)*a+11/40*d^4*b^3*(c*x^2+b*x+a)^(7/2)-3/10*d^4*c*b*a*(c*x^2+b*x+a)^(7/2)+12/5*d^4*c^2*b*x^2*(
c*x^2+b*x+a)^(7/2)+27/20*d^4*c*b^2*x*(c*x^2+b*x+a)^(7/2)+9/128*d^4*b^4*(c*x^2+b*x+a)^(1/2)*x*a^2+3/128*d^4*b^4
*(c*x^2+b*x+a)^(3/2)*x*a-15/64*d^4*c^(1/2)*b^2*a^4*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+15/128*d^4*b^4/
c^(1/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-15/512*d^4*b^6/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*a^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^(5/2),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 details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (b\,d+2\,c\,d\,x\right )}^4\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x)

[Out]

int((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d^{4} \left (\int a^{2} b^{4} \sqrt {a + b x + c x^{2}}\, dx + \int b^{6} x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 16 c^{6} x^{8} \sqrt {a + b x + c x^{2}}\, dx + \int 2 a b^{5} x \sqrt {a + b x + c x^{2}}\, dx + \int 32 a c^{5} x^{6} \sqrt {a + b x + c x^{2}}\, dx + \int 16 a^{2} c^{4} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 64 b c^{5} x^{7} \sqrt {a + b x + c x^{2}}\, dx + \int 104 b^{2} c^{4} x^{6} \sqrt {a + b x + c x^{2}}\, dx + \int 88 b^{3} c^{3} x^{5} \sqrt {a + b x + c x^{2}}\, dx + \int 41 b^{4} c^{2} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 10 b^{5} c x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 96 a b c^{4} x^{5} \sqrt {a + b x + c x^{2}}\, dx + \int 112 a b^{2} c^{3} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 64 a b^{3} c^{2} x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 18 a b^{4} c x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 32 a^{2} b c^{3} x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 24 a^{2} b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 8 a^{2} b^{3} c x \sqrt {a + b x + c x^{2}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**(5/2),x)

[Out]

d**4*(Integral(a**2*b**4*sqrt(a + b*x + c*x**2), x) + Integral(b**6*x**2*sqrt(a + b*x + c*x**2), x) + Integral
(16*c**6*x**8*sqrt(a + b*x + c*x**2), x) + Integral(2*a*b**5*x*sqrt(a + b*x + c*x**2), x) + Integral(32*a*c**5
*x**6*sqrt(a + b*x + c*x**2), x) + Integral(16*a**2*c**4*x**4*sqrt(a + b*x + c*x**2), x) + Integral(64*b*c**5*
x**7*sqrt(a + b*x + c*x**2), x) + Integral(104*b**2*c**4*x**6*sqrt(a + b*x + c*x**2), x) + Integral(88*b**3*c*
*3*x**5*sqrt(a + b*x + c*x**2), x) + Integral(41*b**4*c**2*x**4*sqrt(a + b*x + c*x**2), x) + Integral(10*b**5*
c*x**3*sqrt(a + b*x + c*x**2), x) + Integral(96*a*b*c**4*x**5*sqrt(a + b*x + c*x**2), x) + Integral(112*a*b**2
*c**3*x**4*sqrt(a + b*x + c*x**2), x) + Integral(64*a*b**3*c**2*x**3*sqrt(a + b*x + c*x**2), x) + Integral(18*
a*b**4*c*x**2*sqrt(a + b*x + c*x**2), x) + Integral(32*a**2*b*c**3*x**3*sqrt(a + b*x + c*x**2), x) + Integral(
24*a**2*b**2*c**2*x**2*sqrt(a + b*x + c*x**2), x) + Integral(8*a**2*b**3*c*x*sqrt(a + b*x + c*x**2), x))

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