3.4.3 \(\int (d+e x)^2 (b x+c x^2)^{5/2} \, dx\) [303]

Optimal. Leaf size=266 \[ \frac {5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}} \]

[Out]

-5/6144*b^2*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^4+1/384*(9*b^2*e^2-32*b*c*d*e+32*c
^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c^3+9/112*e*(-b*e+2*c*d)*(c*x^2+b*x)^(7/2)/c^2+1/8*e*(e*x+d)*(c*x^2+b*x)^(
7/2)/c-5/16384*b^6*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(11/2)+5/16384*b^4
*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^5

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Rubi [A]
time = 0.14, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {756, 654, 626, 634, 212} \begin {gather*} -\frac {5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{6144 c^4}+\frac {(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{384 c^3}-\frac {5 b^6 \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}}+\frac {5 b^4 (b+2 c x) \sqrt {b x+c x^2} \left (9 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{16384 c^5}+\frac {9 e \left (b x+c x^2\right )^{7/2} (2 c d-b e)}{112 c^2}+\frac {e \left (b x+c x^2\right )^{7/2} (d+e x)}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b^4*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(16384*c^5) - (5*b^2*(32*c^2*d^2 -
32*b*c*d*e + 9*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(6144*c^4) + ((32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*
(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(384*c^3) + (9*e*(2*c*d - b*e)*(b*x + c*x^2)^(7/2))/(112*c^2) + (e*(d + e*x)*
(b*x + c*x^2)^(7/2))/(8*c) - (5*b^6*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2
]])/(16384*c^(11/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

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

Rule 634

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

Rule 654

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

Rule 756

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

Rubi steps

\begin {align*} \int (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx &=\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}+\frac {\int \left (\frac {1}{2} d (16 c d-7 b e)+\frac {9}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2} \, dx}{8 c}\\ &=\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (c d (16 c d-7 b e)-\frac {9}{2} b e (2 c d-b e)\right ) \int \left (b x+c x^2\right )^{5/2} \, dx}{16 c^2}\\ &=\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{768 c^3}\\ &=-\frac {5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}+\frac {\left (5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \int \sqrt {b x+c x^2} \, dx}{4096 c^4}\\ &=\frac {5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{32768 c^5}\\ &=\frac {5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {\left (5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{16384 c^5}\\ &=\frac {5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{5/2}}{384 c^3}+\frac {9 e (2 c d-b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.56, size = 286, normalized size = 1.08 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (945 b^7 e^2-210 b^6 c e (16 d+3 e x)+128 b^3 c^4 x^2 \left (14 d^2+12 d e x+3 e^2 x^2\right )+56 b^5 c^2 \left (60 d^2+40 d e x+9 e^2 x^2\right )+2048 c^7 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )-16 b^4 c^3 x \left (140 d^2+112 d e x+27 e^2 x^2\right )+1024 b c^6 x^4 \left (140 d^2+232 d e x+99 e^2 x^2\right )+256 b^2 c^5 x^3 \left (378 d^2+592 d e x+243 e^2 x^2\right )\right )+\frac {105 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{344064 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(945*b^7*e^2 - 210*b^6*c*e*(16*d + 3*e*x) + 128*b^3*c^4*x^2*(14*d^2 + 12*d*e*x + 3
*e^2*x^2) + 56*b^5*c^2*(60*d^2 + 40*d*e*x + 9*e^2*x^2) + 2048*c^7*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) - 16*b^
4*c^3*x*(140*d^2 + 112*d*e*x + 27*e^2*x^2) + 1024*b*c^6*x^4*(140*d^2 + 232*d*e*x + 99*e^2*x^2) + 256*b^2*c^5*x
^3*(378*d^2 + 592*d*e*x + 243*e^2*x^2)) + (105*b^6*(32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*Log[-(Sqrt[c]*Sqrt[x]
) + Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(344064*c^(11/2))

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Maple [A]
time = 0.44, size = 435, normalized size = 1.64

method	result	size
risch	\(\frac {\left (43008 c^{7} e^{2} x^{7}+101376 b \,c^{6} e^{2} x^{6}+98304 c^{7} d e \,x^{6}+62208 b^{2} c^{5} e^{2} x^{5}+237568 b \,c^{6} d e \,x^{5}+57344 c^{7} d^{2} x^{5}+384 b^{3} c^{4} e^{2} x^{4}+151552 b^{2} c^{5} d e \,x^{4}+143360 b \,c^{6} d^{2} x^{4}-432 b^{4} c^{3} e^{2} x^{3}+1536 b^{3} c^{4} d e \,x^{3}+96768 b^{2} c^{5} d^{2} x^{3}+504 b^{5} c^{2} e^{2} x^{2}-1792 b^{4} c^{3} d e \,x^{2}+1792 b^{3} c^{4} d^{2} x^{2}-630 b^{6} c \,e^{2} x +2240 b^{5} c^{2} d e x -2240 b^{4} c^{3} d^{2} x +945 b^{7} e^{2}-3360 b^{6} c d e +3360 b^{5} c^{2} d^{2}\right ) x \left (c x +b \right )}{344064 c^{5} \sqrt {x \left (c x +b \right )}}-\frac {45 b^{8} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) e^{2}}{32768 c^{\frac {11}{2}}}+\frac {5 b^{7} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d e}{1024 c^{\frac {9}{2}}}-\frac {5 b^{6} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d^{2}}{1024 c^{\frac {7}{2}}}\)	\(377\)
default	\(e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )+d^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )\)	\(435\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^2*(1/8*x*(c*x^2+b*x)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5
/24*b^2/c*(1/8*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2)))))))+2*d*e*(1/7*(c*x^2+b*x)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x)
^(5/2)/c-5/24*b^2/c*(1/8*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c
^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))))))+d^2*(1/12*(2*c*x+b)*(c*x^2+b*x)^(5/2)/c-5/24*b^2/c*(1/8*(
2*c*x+b)*(c*x^2+b*x)^(3/2)/c-3/16*b^2/c*(1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x)^(1/2)))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (242) = 484\).
time = 0.29, size = 548, normalized size = 2.06 \begin {gather*} \frac {1}{6} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d^{2} x + \frac {5 \, \sqrt {c x^{2} + b x} b^{4} d^{2} x}{256 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d^{2} x}{96 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b^{5} d x e}{256 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d x e}{96 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d x e}{6 \, c} - \frac {5 \, b^{6} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {7}{2}}} + \frac {5 \, b^{7} d e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} b^{5} d^{2}}{512 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d^{2}}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d^{2}}{12 \, c} + \frac {45 \, \sqrt {c x^{2} + b x} b^{6} x e^{2}}{8192 \, c^{4}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} x e^{2}}{1024 \, c^{3}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} x e^{2}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} x e^{2}}{8 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b^{6} d e}{512 \, c^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} d e}{192 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} d e}{12 \, c^{2}} + \frac {2 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} d e}{7 \, c} - \frac {45 \, b^{8} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} + \frac {45 \, \sqrt {c x^{2} + b x} b^{7} e^{2}}{16384 \, c^{5}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5} e^{2}}{2048 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3} e^{2}}{128 \, c^{3}} - \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b e^{2}}{112 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*x^2 + b*x)^(5/2)*d^2*x + 5/256*sqrt(c*x^2 + b*x)*b^4*d^2*x/c^2 - 5/96*(c*x^2 + b*x)^(3/2)*b^2*d^2*x/c -
5/256*sqrt(c*x^2 + b*x)*b^5*d*x*e/c^3 + 5/96*(c*x^2 + b*x)^(3/2)*b^3*d*x*e/c^2 - 1/6*(c*x^2 + b*x)^(5/2)*b*d*
x*e/c - 5/1024*b^6*d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) + 5/1024*b^7*d*e*log(2*c*x + b + 2
*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) + 5/512*sqrt(c*x^2 + b*x)*b^5*d^2/c^3 - 5/192*(c*x^2 + b*x)^(3/2)*b^3*d^2/
c^2 + 1/12*(c*x^2 + b*x)^(5/2)*b*d^2/c + 45/8192*sqrt(c*x^2 + b*x)*b^6*x*e^2/c^4 - 15/1024*(c*x^2 + b*x)^(3/2)
*b^4*x*e^2/c^3 + 3/64*(c*x^2 + b*x)^(5/2)*b^2*x*e^2/c^2 + 1/8*(c*x^2 + b*x)^(7/2)*x*e^2/c - 5/512*sqrt(c*x^2 +
b*x)*b^6*d*e/c^4 + 5/192*(c*x^2 + b*x)^(3/2)*b^4*d*e/c^3 - 1/12*(c*x^2 + b*x)^(5/2)*b^2*d*e/c^2 + 2/7*(c*x^2
+ b*x)^(7/2)*d*e/c - 45/32768*b^8*e^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(11/2) + 45/16384*sqrt(c*
x^2 + b*x)*b^7*e^2/c^5 - 15/2048*(c*x^2 + b*x)^(3/2)*b^5*e^2/c^4 + 3/128*(c*x^2 + b*x)^(5/2)*b^3*e^2/c^3 - 9/1
12*(c*x^2 + b*x)^(7/2)*b*e^2/c^2

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Fricas [A]
time = 1.93, size = 623, normalized size = 2.34 \begin {gather*} \left [\frac {105 \, {\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (57344 \, c^{8} d^{2} x^{5} + 143360 \, b c^{7} d^{2} x^{4} + 96768 \, b^{2} c^{6} d^{2} x^{3} + 1792 \, b^{3} c^{5} d^{2} x^{2} - 2240 \, b^{4} c^{4} d^{2} x + 3360 \, b^{5} c^{3} d^{2} + 3 \, {\left (14336 \, c^{8} x^{7} + 33792 \, b c^{7} x^{6} + 20736 \, b^{2} c^{6} x^{5} + 128 \, b^{3} c^{5} x^{4} - 144 \, b^{4} c^{4} x^{3} + 168 \, b^{5} c^{3} x^{2} - 210 \, b^{6} c^{2} x + 315 \, b^{7} c\right )} e^{2} + 32 \, {\left (3072 \, c^{8} d x^{6} + 7424 \, b c^{7} d x^{5} + 4736 \, b^{2} c^{6} d x^{4} + 48 \, b^{3} c^{5} d x^{3} - 56 \, b^{4} c^{4} d x^{2} + 70 \, b^{5} c^{3} d x - 105 \, b^{6} c^{2} d\right )} e\right )} \sqrt {c x^{2} + b x}}{688128 \, c^{6}}, \frac {105 \, {\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (57344 \, c^{8} d^{2} x^{5} + 143360 \, b c^{7} d^{2} x^{4} + 96768 \, b^{2} c^{6} d^{2} x^{3} + 1792 \, b^{3} c^{5} d^{2} x^{2} - 2240 \, b^{4} c^{4} d^{2} x + 3360 \, b^{5} c^{3} d^{2} + 3 \, {\left (14336 \, c^{8} x^{7} + 33792 \, b c^{7} x^{6} + 20736 \, b^{2} c^{6} x^{5} + 128 \, b^{3} c^{5} x^{4} - 144 \, b^{4} c^{4} x^{3} + 168 \, b^{5} c^{3} x^{2} - 210 \, b^{6} c^{2} x + 315 \, b^{7} c\right )} e^{2} + 32 \, {\left (3072 \, c^{8} d x^{6} + 7424 \, b c^{7} d x^{5} + 4736 \, b^{2} c^{6} d x^{4} + 48 \, b^{3} c^{5} d x^{3} - 56 \, b^{4} c^{4} d x^{2} + 70 \, b^{5} c^{3} d x - 105 \, b^{6} c^{2} d\right )} e\right )} \sqrt {c x^{2} + b x}}{344064 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/688128*(105*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)
) + 2*(57344*c^8*d^2*x^5 + 143360*b*c^7*d^2*x^4 + 96768*b^2*c^6*d^2*x^3 + 1792*b^3*c^5*d^2*x^2 - 2240*b^4*c^4*
d^2*x + 3360*b^5*c^3*d^2 + 3*(14336*c^8*x^7 + 33792*b*c^7*x^6 + 20736*b^2*c^6*x^5 + 128*b^3*c^5*x^4 - 144*b^4*
c^4*x^3 + 168*b^5*c^3*x^2 - 210*b^6*c^2*x + 315*b^7*c)*e^2 + 32*(3072*c^8*d*x^6 + 7424*b*c^7*d*x^5 + 4736*b^2*
c^6*d*x^4 + 48*b^3*c^5*d*x^3 - 56*b^4*c^4*d*x^2 + 70*b^5*c^3*d*x - 105*b^6*c^2*d)*e)*sqrt(c*x^2 + b*x))/c^6, 1
/344064*(105*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (
57344*c^8*d^2*x^5 + 143360*b*c^7*d^2*x^4 + 96768*b^2*c^6*d^2*x^3 + 1792*b^3*c^5*d^2*x^2 - 2240*b^4*c^4*d^2*x +
3360*b^5*c^3*d^2 + 3*(14336*c^8*x^7 + 33792*b*c^7*x^6 + 20736*b^2*c^6*x^5 + 128*b^3*c^5*x^4 - 144*b^4*c^4*x^3
+ 168*b^5*c^3*x^2 - 210*b^6*c^2*x + 315*b^7*c)*e^2 + 32*(3072*c^8*d*x^6 + 7424*b*c^7*d*x^5 + 4736*b^2*c^6*d*x
^4 + 48*b^3*c^5*d*x^3 - 56*b^4*c^4*d*x^2 + 70*b^5*c^3*d*x - 105*b^6*c^2*d)*e)*sqrt(c*x^2 + b*x))/c^6]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x*(b + c*x))**(5/2)*(d + e*x)**2, x)

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Giac [A]
time = 1.82, size = 350, normalized size = 1.32 \begin {gather*} \frac {1}{344064} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, c^{2} x e^{2} + \frac {32 \, c^{9} d e + 33 \, b c^{8} e^{2}}{c^{7}}\right )} x + \frac {224 \, c^{9} d^{2} + 928 \, b c^{8} d e + 243 \, b^{2} c^{7} e^{2}}{c^{7}}\right )} x + \frac {1120 \, b c^{8} d^{2} + 1184 \, b^{2} c^{7} d e + 3 \, b^{3} c^{6} e^{2}}{c^{7}}\right )} x + \frac {3 \, {\left (2016 \, b^{2} c^{7} d^{2} + 32 \, b^{3} c^{6} d e - 9 \, b^{4} c^{5} e^{2}\right )}}{c^{7}}\right )} x + \frac {7 \, {\left (32 \, b^{3} c^{6} d^{2} - 32 \, b^{4} c^{5} d e + 9 \, b^{5} c^{4} e^{2}\right )}}{c^{7}}\right )} x - \frac {35 \, {\left (32 \, b^{4} c^{5} d^{2} - 32 \, b^{5} c^{4} d e + 9 \, b^{6} c^{3} e^{2}\right )}}{c^{7}}\right )} x + \frac {105 \, {\left (32 \, b^{5} c^{4} d^{2} - 32 \, b^{6} c^{3} d e + 9 \, b^{7} c^{2} e^{2}\right )}}{c^{7}}\right )} + \frac {5 \, {\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/344064*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(12*(14*c^2*x*e^2 + (32*c^9*d*e + 33*b*c^8*e^2)/c^7)*x + (224*c^9*d^
2 + 928*b*c^8*d*e + 243*b^2*c^7*e^2)/c^7)*x + (1120*b*c^8*d^2 + 1184*b^2*c^7*d*e + 3*b^3*c^6*e^2)/c^7)*x + 3*(
2016*b^2*c^7*d^2 + 32*b^3*c^6*d*e - 9*b^4*c^5*e^2)/c^7)*x + 7*(32*b^3*c^6*d^2 - 32*b^4*c^5*d*e + 9*b^5*c^4*e^2
)/c^7)*x - 35*(32*b^4*c^5*d^2 - 32*b^5*c^4*d*e + 9*b^6*c^3*e^2)/c^7)*x + 105*(32*b^5*c^4*d^2 - 32*b^6*c^3*d*e
+ 9*b^7*c^2*e^2)/c^7) + 5/32768*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*sqrt(c) - b))/c^(11/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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