3.3.95 \(\int (d+e x)^3 (b x+c x^2)^{3/2} \, dx\) [295]

Optimal. Leaf size=271 \[ -\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}} \]

[Out]

1/128*(-b*e+2*c*d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(3/2)/c^4+1/7*e*(e*x+d)^2*(c*x^2+b*x)
^(5/2)/c+1/280*e*(128*c^2*d^2-98*b*c*d*e+21*b^2*e^2+30*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x)^(5/2)/c^3+3/1024*b^4*(-
b*e+2*c*d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(11/2)-3/1024*b^2*(-b*e+2*c*
d)*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c^5

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Rubi [A]
time = 0.23, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {756, 793, 626, 634, 212} \begin {gather*} -\frac {3 b^2 (b+2 c x) \sqrt {b x+c x^2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{1024 c^5}+\frac {(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{128 c^4}+\frac {e \left (b x+c x^2\right )^{5/2} \left (21 b^2 e^2+30 c e x (2 c d-b e)-98 b c d e+128 c^2 d^2\right )}{280 c^3}+\frac {3 b^4 (2 c d-b e) \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}}+\frac {e \left (b x+c x^2\right )^{5/2} (d+e x)^2}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(b*x + c*x^2)^(3/2),x]

[Out]

(-3*b^2*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(1024*c^5) + ((2*c*d
- b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(128*c^4) + (e*(d + e*x)^2*(b*x +
c*x^2)^(5/2))/(7*c) + (e*(128*c^2*d^2 - 98*b*c*d*e + 21*b^2*e^2 + 30*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(5/2))
/(280*c^3) + (3*b^4*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/
(1024*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 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]

Rule 793

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rubi steps

\begin {align*} \int (d+e x)^3 \left (b x+c x^2\right )^{3/2} \, dx &=\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {\int (d+e x) \left (\frac {1}{2} d (14 c d-5 b e)+\frac {9}{2} e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{16 c^3}\\ &=\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}-\frac {\left (3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \sqrt {b x+c x^2} \, dx}{256 c^4}\\ &=-\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2048 c^5}\\ &=-\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {\left (3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 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 )}{1024 c^5}\\ &=-\frac {3 b^2 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{1024 c^5}+\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{128 c^4}+\frac {e (d+e x)^2 \left (b x+c x^2\right )^{5/2}}{7 c}+\frac {e \left (128 c^2 d^2-98 b c d e+21 b^2 e^2+30 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{5/2}}{280 c^3}+\frac {3 b^4 (2 c d-b e) \left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 313, normalized size = 1.15 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (315 b^6 e^3-210 b^5 c e^2 (7 d+e x)+28 b^4 c^2 e \left (90 d^2+35 d e x+6 e^2 x^2\right )+32 b^2 c^4 x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )-16 b^3 c^3 \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )+256 c^6 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+128 b c^5 x^2 \left (105 d^3+231 d^2 e x+182 d e^2 x^2+50 e^3 x^3\right )\right )+\frac {105 b^4 \left (-16 c^3 d^3+24 b c^2 d^2 e-14 b^2 c d e^2+3 b^3 e^3\right ) \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{35840 c^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(b*x + c*x^2)^(3/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(315*b^6*e^3 - 210*b^5*c*e^2*(7*d + e*x) + 28*b^4*c^2*e*(90*d^2 + 35*d*e*x + 6*e^2
*x^2) + 32*b^2*c^4*x*(35*d^3 + 42*d^2*e*x + 21*d*e^2*x^2 + 4*e^3*x^3) - 16*b^3*c^3*(105*d^3 + 105*d^2*e*x + 49
*d*e^2*x^2 + 9*e^3*x^3) + 256*c^6*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 128*b*c^5*x^2*(105*d
^3 + 231*d^2*e*x + 182*d*e^2*x^2 + 50*e^3*x^3)) + (105*b^4*(-16*c^3*d^3 + 24*b*c^2*d^2*e - 14*b^2*c*d*e^2 + 3*
b^3*e^3)*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(35840*c^(11/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(508\) vs. \(2(247)=494\).
time = 0.47, size = 509, normalized size = 1.88

method	result	size
risch	\(\frac {\left (5120 c^{6} e^{3} x^{6}+6400 b \,c^{5} e^{3} x^{5}+17920 c^{6} d \,e^{2} x^{5}+128 b^{2} c^{4} e^{3} x^{4}+23296 b \,c^{5} d \,e^{2} x^{4}+21504 c^{6} d^{2} e \,x^{4}-144 b^{3} c^{3} e^{3} x^{3}+672 b^{2} c^{4} d \,e^{2} x^{3}+29568 b \,c^{5} d^{2} e \,x^{3}+8960 c^{6} d^{3} x^{3}+168 b^{4} c^{2} e^{3} x^{2}-784 b^{3} c^{3} d \,e^{2} x^{2}+1344 b^{2} c^{4} d^{2} e \,x^{2}+13440 b \,c^{5} d^{3} x^{2}-210 b^{5} c \,e^{3} x +980 b^{4} c^{2} d \,e^{2} x -1680 b^{3} c^{3} d^{2} e x +1120 b^{2} c^{4} d^{3} x +315 b^{6} e^{3}-1470 b^{5} c d \,e^{2}+2520 b^{4} c^{2} d^{2} e -1680 b^{3} c^{3} d^{3}\right ) x \left (c x +b \right )}{35840 c^{5} \sqrt {x \left (c x +b \right )}}-\frac {9 b^{7} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) e^{3}}{2048 c^{\frac {11}{2}}}+\frac {21 b^{6} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d \,e^{2}}{1024 c^{\frac {9}{2}}}-\frac {9 b^{5} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d^{2} e}{256 c^{\frac {7}{2}}}+\frac {3 b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) d^{3}}{128 c^{\frac {5}{2}}}\)	\(437\)
default	\(e^{3} \left (\frac {x^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{7 c}-\frac {9 b \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )}{14 c}\right )+3 d \,e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )+3 d^{2} e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )+d^{3} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )\)	\(509\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

e^3*(1/7*x^2*(c*x^2+b*x)^(5/2)/c-9/14*b/c*(1/6*x*(c*x^2+b*x)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x)^(5/2)/c-1/2*b/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)))))))+3*d*e^2*(1/6*x*(c*x^2+b*x)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x)^(5/2)/c-1/2*b/
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))))))+3*d^2*e*(1/5*(c*x^2+b*x)^(5/2)/c-1/2*b/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^
3*(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. 615 vs. \(2 (255) = 510\).
time = 0.34, size = 615, normalized size = 2.27 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d^{3} x - \frac {3 \, \sqrt {c x^{2} + b x} b^{2} d^{3} x}{32 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} b^{3} d^{2} x e}{64 \, c^{2}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d^{2} x e}{8 \, c} + \frac {3 \, b^{4} d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {5}{2}}} - \frac {9 \, b^{5} d^{2} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b^{3} d^{3}}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} b d^{3}}{8 \, c} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} x^{2} e^{3}}{7 \, c} - \frac {21 \, \sqrt {c x^{2} + b x} b^{4} d x e^{2}}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d x e^{2}}{32 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} d x e^{2}}{2 \, c} + \frac {9 \, \sqrt {c x^{2} + b x} b^{4} d^{2} e}{128 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d^{2} e}{16 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d^{2} e}{5 \, c} + \frac {21 \, b^{6} d e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} + \frac {9 \, \sqrt {c x^{2} + b x} b^{5} x e^{3}}{512 \, c^{4}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} x e^{3}}{64 \, c^{3}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b x e^{3}}{28 \, c^{2}} - \frac {21 \, \sqrt {c x^{2} + b x} b^{5} d e^{2}}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d e^{2}}{64 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d e^{2}}{20 \, c^{2}} - \frac {9 \, b^{7} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2048 \, c^{\frac {11}{2}}} + \frac {9 \, \sqrt {c x^{2} + b x} b^{6} e^{3}}{1024 \, c^{5}} - \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} e^{3}}{128 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} e^{3}}{40 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

1/4*(c*x^2 + b*x)^(3/2)*d^3*x - 3/32*sqrt(c*x^2 + b*x)*b^2*d^3*x/c + 9/64*sqrt(c*x^2 + b*x)*b^3*d^2*x*e/c^2 -
3/8*(c*x^2 + b*x)^(3/2)*b*d^2*x*e/c + 3/128*b^4*d^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 9/2
56*b^5*d^2*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) - 3/64*sqrt(c*x^2 + b*x)*b^3*d^3/c^2 + 1/8*(
c*x^2 + b*x)^(3/2)*b*d^3/c + 1/7*(c*x^2 + b*x)^(5/2)*x^2*e^3/c - 21/256*sqrt(c*x^2 + b*x)*b^4*d*x*e^2/c^3 + 7/
32*(c*x^2 + b*x)^(3/2)*b^2*d*x*e^2/c^2 + 1/2*(c*x^2 + b*x)^(5/2)*d*x*e^2/c + 9/128*sqrt(c*x^2 + b*x)*b^4*d^2*e
/c^3 - 3/16*(c*x^2 + b*x)^(3/2)*b^2*d^2*e/c^2 + 3/5*(c*x^2 + b*x)^(5/2)*d^2*e/c + 21/1024*b^6*d*e^2*log(2*c*x
+ b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(9/2) + 9/512*sqrt(c*x^2 + b*x)*b^5*x*e^3/c^4 - 3/64*(c*x^2 + b*x)^(3/2)*
b^3*x*e^3/c^3 - 3/28*(c*x^2 + b*x)^(5/2)*b*x*e^3/c^2 - 21/512*sqrt(c*x^2 + b*x)*b^5*d*e^2/c^4 + 7/64*(c*x^2 +
b*x)^(3/2)*b^3*d*e^2/c^3 - 7/20*(c*x^2 + b*x)^(5/2)*b*d*e^2/c^2 - 9/2048*b^7*e^3*log(2*c*x + b + 2*sqrt(c*x^2
+ b*x)*sqrt(c))/c^(11/2) + 9/1024*sqrt(c*x^2 + b*x)*b^6*e^3/c^5 - 3/128*(c*x^2 + b*x)^(3/2)*b^4*e^3/c^4 + 3/40
*(c*x^2 + b*x)^(5/2)*b^2*e^3/c^3

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Fricas [A]
time = 1.68, size = 676, normalized size = 2.49 \begin {gather*} \left [-\frac {105 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (8960 \, c^{7} d^{3} x^{3} + 13440 \, b c^{6} d^{3} x^{2} + 1120 \, b^{2} c^{5} d^{3} x - 1680 \, b^{3} c^{4} d^{3} + {\left (5120 \, c^{7} x^{6} + 6400 \, b c^{6} x^{5} + 128 \, b^{2} c^{5} x^{4} - 144 \, b^{3} c^{4} x^{3} + 168 \, b^{4} c^{3} x^{2} - 210 \, b^{5} c^{2} x + 315 \, b^{6} c\right )} e^{3} + 14 \, {\left (1280 \, c^{7} d x^{5} + 1664 \, b c^{6} d x^{4} + 48 \, b^{2} c^{5} d x^{3} - 56 \, b^{3} c^{4} d x^{2} + 70 \, b^{4} c^{3} d x - 105 \, b^{5} c^{2} d\right )} e^{2} + 168 \, {\left (128 \, c^{7} d^{2} x^{4} + 176 \, b c^{6} d^{2} x^{3} + 8 \, b^{2} c^{5} d^{2} x^{2} - 10 \, b^{3} c^{4} d^{2} x + 15 \, b^{4} c^{3} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x}}{71680 \, c^{6}}, -\frac {105 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (8960 \, c^{7} d^{3} x^{3} + 13440 \, b c^{6} d^{3} x^{2} + 1120 \, b^{2} c^{5} d^{3} x - 1680 \, b^{3} c^{4} d^{3} + {\left (5120 \, c^{7} x^{6} + 6400 \, b c^{6} x^{5} + 128 \, b^{2} c^{5} x^{4} - 144 \, b^{3} c^{4} x^{3} + 168 \, b^{4} c^{3} x^{2} - 210 \, b^{5} c^{2} x + 315 \, b^{6} c\right )} e^{3} + 14 \, {\left (1280 \, c^{7} d x^{5} + 1664 \, b c^{6} d x^{4} + 48 \, b^{2} c^{5} d x^{3} - 56 \, b^{3} c^{4} d x^{2} + 70 \, b^{4} c^{3} d x - 105 \, b^{5} c^{2} d\right )} e^{2} + 168 \, {\left (128 \, c^{7} d^{2} x^{4} + 176 \, b c^{6} d^{2} x^{3} + 8 \, b^{2} c^{5} d^{2} x^{2} - 10 \, b^{3} c^{4} d^{2} x + 15 \, b^{4} c^{3} d^{2}\right )} e\right )} \sqrt {c x^{2} + b x}}{35840 \, c^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

[-1/71680*(105*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7*e^3)*sqrt(c)*log(2*c*x + b - 2*sqrt
(c*x^2 + b*x)*sqrt(c)) - 2*(8960*c^7*d^3*x^3 + 13440*b*c^6*d^3*x^2 + 1120*b^2*c^5*d^3*x - 1680*b^3*c^4*d^3 + (
5120*c^7*x^6 + 6400*b*c^6*x^5 + 128*b^2*c^5*x^4 - 144*b^3*c^4*x^3 + 168*b^4*c^3*x^2 - 210*b^5*c^2*x + 315*b^6*
c)*e^3 + 14*(1280*c^7*d*x^5 + 1664*b*c^6*d*x^4 + 48*b^2*c^5*d*x^3 - 56*b^3*c^4*d*x^2 + 70*b^4*c^3*d*x - 105*b^
5*c^2*d)*e^2 + 168*(128*c^7*d^2*x^4 + 176*b*c^6*d^2*x^3 + 8*b^2*c^5*d^2*x^2 - 10*b^3*c^4*d^2*x + 15*b^4*c^3*d^
2)*e)*sqrt(c*x^2 + b*x))/c^6, -1/35840*(105*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7*e^3)*s
qrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (8960*c^7*d^3*x^3 + 13440*b*c^6*d^3*x^2 + 1120*b^2*c^5*d^3*
x - 1680*b^3*c^4*d^3 + (5120*c^7*x^6 + 6400*b*c^6*x^5 + 128*b^2*c^5*x^4 - 144*b^3*c^4*x^3 + 168*b^4*c^3*x^2 -
210*b^5*c^2*x + 315*b^6*c)*e^3 + 14*(1280*c^7*d*x^5 + 1664*b*c^6*d*x^4 + 48*b^2*c^5*d*x^3 - 56*b^3*c^4*d*x^2 +
70*b^4*c^3*d*x - 105*b^5*c^2*d)*e^2 + 168*(128*c^7*d^2*x^4 + 176*b*c^6*d^2*x^3 + 8*b^2*c^5*d^2*x^2 - 10*b^3*c
^4*d^2*x + 15*b^4*c^3*d^2)*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 {3}{2}} \left (d + e x\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+b*x)**(3/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)*(d + e*x)**3, x)

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Giac [A]
time = 2.84, size = 365, normalized size = 1.35 \begin {gather*} \frac {1}{35840} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, c x e^{3} + \frac {14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac {168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac {560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} - 9 \, b^{3} c^{4} e^{3}}{c^{6}}\right )} x + \frac {7 \, {\left (240 \, b c^{6} d^{3} + 24 \, b^{2} c^{5} d^{2} e - 14 \, b^{3} c^{4} d e^{2} + 3 \, b^{4} c^{3} e^{3}\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (16 \, b^{2} c^{5} d^{3} - 24 \, b^{3} c^{4} d^{2} e + 14 \, b^{4} c^{3} d e^{2} - 3 \, b^{5} c^{2} e^{3}\right )}}{c^{6}}\right )} x - \frac {105 \, {\left (16 \, b^{3} c^{4} d^{3} - 24 \, b^{4} c^{3} d^{2} e + 14 \, b^{5} c^{2} d e^{2} - 3 \, b^{6} c e^{3}\right )}}{c^{6}}\right )} - \frac {3 \, {\left (16 \, b^{4} c^{3} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 14 \, b^{6} c d e^{2} - 3 \, b^{7} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

1/35840*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(10*(4*c*x*e^3 + (14*c^7*d*e^2 + 5*b*c^6*e^3)/c^6)*x + (168*c^7*d^2*e +
182*b*c^6*d*e^2 + b^2*c^5*e^3)/c^6)*x + (560*c^7*d^3 + 1848*b*c^6*d^2*e + 42*b^2*c^5*d*e^2 - 9*b^3*c^4*e^3)/c^
6)*x + 7*(240*b*c^6*d^3 + 24*b^2*c^5*d^2*e - 14*b^3*c^4*d*e^2 + 3*b^4*c^3*e^3)/c^6)*x + 35*(16*b^2*c^5*d^3 - 2
4*b^3*c^4*d^2*e + 14*b^4*c^3*d*e^2 - 3*b^5*c^2*e^3)/c^6)*x - 105*(16*b^3*c^4*d^3 - 24*b^4*c^3*d^2*e + 14*b^5*c
^2*d*e^2 - 3*b^6*c*e^3)/c^6) - 3/2048*(16*b^4*c^3*d^3 - 24*b^5*c^2*d^2*e + 14*b^6*c*d*e^2 - 3*b^7*e^3)*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 )}^{3/2}\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^(3/2)*(d + e*x)^3,x)

[Out]

int((b*x + c*x^2)^(3/2)*(d + e*x)^3, x)

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