### 3.307 $$\int \frac{(b x+c x^2)^{5/2}}{(d+e x)^2} \, dx$$

Optimal. Leaf size=314 $-\frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-b^3 e^3-112 b c^2 d^2 e+64 c^3 d^3\right )}{64 c e^5}+\frac{5 \left (144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2} e^6}-\frac{5 d^{3/2} (c d-b e)^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^6}-\frac{5 \left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)}$

[Out]

(-5*(64*c^3*d^3 - 112*b*c^2*d^2*e + 48*b^2*c*d*e^2 - b^3*e^3 - 2*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*x)*Sq
rt[b*x + c*x^2])/(64*c*e^5) - (5*(8*c*d - 7*b*e - 6*c*e*x)*(b*x + c*x^2)^(3/2))/(24*e^3) - (b*x + c*x^2)^(5/2)
/(e*(d + e*x)) + (5*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4)*ArcTanh[(
Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(3/2)*e^6) - (5*d^(3/2)*(c*d - b*e)^(3/2)*(2*c*d - b*e)*ArcTanh[(b*d + (2
*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^6)

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Rubi [A]  time = 0.393477, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {732, 814, 843, 620, 206, 724} $-\frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+48 b^2 c d e^2-b^3 e^3-112 b c^2 d^2 e+64 c^3 d^3\right )}{64 c e^5}+\frac{5 \left (144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2} e^6}-\frac{5 d^{3/2} (c d-b e)^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 e^6}-\frac{5 \left (b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-5*(64*c^3*d^3 - 112*b*c^2*d^2*e + 48*b^2*c*d*e^2 - b^3*e^3 - 2*c*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2)*x)*Sq
rt[b*x + c*x^2])/(64*c*e^5) - (5*(8*c*d - 7*b*e - 6*c*e*x)*(b*x + c*x^2)^(3/2))/(24*e^3) - (b*x + c*x^2)^(5/2)
/(e*(d + e*x)) + (5*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4)*ArcTanh[(
Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(3/2)*e^6) - (5*d^(3/2)*(c*d - b*e)^(3/2)*(2*c*d - b*e)*ArcTanh[(b*d + (2
*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^6)

Rule 732

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[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ
[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

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

Rule 620

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 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 724

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

Rubi steps

\begin{align*} \int \frac{\left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx &=-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac{5 \int \frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e}\\ &=-\frac{5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)}-\frac{5 \int \frac{\left (-b c d (8 c d-7 b e)-c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{d+e x} \, dx}{16 c e^3}\\ &=-\frac{5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3-2 c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{64 c e^5}-\frac{5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac{5 \int \frac{\frac{1}{2} b c d \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )+\frac{1}{2} c \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right ) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{64 c^2 e^5}\\ &=-\frac{5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3-2 c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{64 c e^5}-\frac{5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)}-\frac{\left (5 d^2 (c d-b e)^2 (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2 e^6}+\frac{\left (5 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c e^6}\\ &=-\frac{5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3-2 c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{64 c e^5}-\frac{5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac{\left (5 d^2 (c d-b e)^2 (2 c d-b e)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{e^6}+\frac{\left (5 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{64 c e^6}\\ &=-\frac{5 \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3-2 c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{64 c e^5}-\frac{5 (8 c d-7 b e-6 c e x) \left (b x+c x^2\right )^{3/2}}{24 e^3}-\frac{\left (b x+c x^2\right )^{5/2}}{e (d+e x)}+\frac{5 \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2} e^6}-\frac{5 d^{3/2} (c d-b e)^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{2 e^6}\\ \end{align*}

Mathematica [A]  time = 1.82675, size = 334, normalized size = 1.06 $\frac{\sqrt{x (b+c x)} \left (\frac{\sqrt{c} e \sqrt{x} \left (2 b^2 c e^2 \left (-360 d^2-205 d e x+59 e^2 x^2\right )+15 b^3 e^3 (d+e x)+8 b c^2 e \left (110 d^2 e x+210 d^3-35 d e^2 x^2+17 e^3 x^3\right )-16 c^3 \left (-10 d^2 e^2 x^2+30 d^3 e x+60 d^4+5 d e^3 x^3-3 e^4 x^4\right )\right )}{d+e x}-\frac{15 \left (-144 b^2 c^2 d^2 e^2+16 b^3 c d e^3+b^4 e^4+256 b c^3 d^3 e-128 c^4 d^4\right ) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}-\frac{960 c^{3/2} d^{3/2} (2 c d-b e) (b e-c d)^{3/2} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b+c x}}\right )}{192 c^{3/2} e^6 \sqrt{x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*((Sqrt[c]*e*Sqrt[x]*(15*b^3*e^3*(d + e*x) + 2*b^2*c*e^2*(-360*d^2 - 205*d*e*x + 59*e^2*x^2)
+ 8*b*c^2*e*(210*d^3 + 110*d^2*e*x - 35*d*e^2*x^2 + 17*e^3*x^3) - 16*c^3*(60*d^4 + 30*d^3*e*x - 10*d^2*e^2*x^
2 + 5*d*e^3*x^3 - 3*e^4*x^4)))/(d + e*x) - (15*(-128*c^4*d^4 + 256*b*c^3*d^3*e - 144*b^2*c^2*d^2*e^2 + 16*b^3*
c*d*e^3 + b^4*e^4)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) - (960*c^(3/2)*d^(3/2)*(2*c
*d - b*e)*(-(c*d) + b*e)^(3/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x]))/(
192*c^(3/2)*e^6*Sqrt[x])

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Maple [B]  time = 0.272, size = 2534, normalized size = 8.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/24/e/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b^2+1/e/(b*e-c*d)*(c*(d/e+x)^2+(b*
e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*c-1/d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(5/2)*b+1/d/(b*e-c*d)/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(7/2)-5/128/e/(b*e-c*d)/c^(3
/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*b^5-85
/32/e^2*d/(b*e-c*d)*b^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c+5/e^3*d^2/(b*e-c*d)*(c*(
d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*b-45/2/e^5*d^4/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*
ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(
b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2*c^2+25/2/e^4*d^3/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(
b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/
e+x))*b^3*c+35/2/e^6*d^5/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d
*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^3*b-c/d/(b*e-c*d)*
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(5/2)*x-245/64/e^2*d/(b*e-c*d)*b^3*(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)+5/e^5*d^4/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/
2)*c^3+5/3/e^3*d^2/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2-5/e^6*d^5/(b*e-c*d)
*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(7/2)+5
/32/e/(b*e-c*d)*b^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x+5/64/e/(b*e-c*d)/c*b^4*(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)-25/8/e^2*d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(
b*e-c*d)/e^2)^(3/2)*b*c-5/2/e^4*d^3/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^3+
25/2/e^3*d^2/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^2*c-55/4/e^4*d^3/(b*e-c*d)*
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*c^2-75/128/e^2*d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(
d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^4+25/4/e^3*d^2/(b*e-c*d
)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*
b^3-5/2/e^3*d^2/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d
)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^4-5/e^7*d^6/(b*e-c*d)/(-d*(
b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^4-5/4/e^2*d/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(3/2)*x*c^2+15/e^5*d^4/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*
c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(5/2)*b+5/4/e/(b*e-c*d)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(3/2)*x*c*b-125/8/e^4*d^3/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e
*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*b^2

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 10.5872, size = 4049, normalized size = 12.89 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(15*(128*c^4*d^5 - 256*b*c^3*d^4*e + 144*b^2*c^2*d^3*e^2 - 16*b^3*c*d^2*e^3 - b^4*d*e^4 + (128*c^4*d^4
*e - 256*b*c^3*d^3*e^2 + 144*b^2*c^2*d^2*e^3 - 16*b^3*c*d*e^4 - b^4*e^5)*x)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x
^2 + b*x)*sqrt(c)) - 960*(2*c^4*d^4 - 3*b*c^3*d^3*e + b^2*c^2*d^2*e^2 + (2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + b^2*c
^2*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x +
d)) - 2*(48*c^4*e^5*x^4 - 960*c^4*d^4*e + 1680*b*c^3*d^3*e^2 - 720*b^2*c^2*d^2*e^3 + 15*b^3*c*d*e^4 - 8*(10*c^
4*d*e^4 - 17*b*c^3*e^5)*x^3 + 2*(80*c^4*d^2*e^3 - 140*b*c^3*d*e^4 + 59*b^2*c^2*e^5)*x^2 - 5*(96*c^4*d^3*e^2 -
176*b*c^3*d^2*e^3 + 82*b^2*c^2*d*e^4 - 3*b^3*c*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*e^7*x + c^2*d*e^6), -1/384*(192
0*(2*c^4*d^4 - 3*b*c^3*d^3*e + b^2*c^2*d^2*e^2 + (2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + b^2*c^2*d*e^3)*x)*sqrt(-c*d^
2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + 15*(128*c^4*d^5 - 256*b*c^3*d^4*e
+ 144*b^2*c^2*d^3*e^2 - 16*b^3*c*d^2*e^3 - b^4*d*e^4 + (128*c^4*d^4*e - 256*b*c^3*d^3*e^2 + 144*b^2*c^2*d^2*e
^3 - 16*b^3*c*d*e^4 - b^4*e^5)*x)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(48*c^4*e^5*x^4 - 9
60*c^4*d^4*e + 1680*b*c^3*d^3*e^2 - 720*b^2*c^2*d^2*e^3 + 15*b^3*c*d*e^4 - 8*(10*c^4*d*e^4 - 17*b*c^3*e^5)*x^3
+ 2*(80*c^4*d^2*e^3 - 140*b*c^3*d*e^4 + 59*b^2*c^2*e^5)*x^2 - 5*(96*c^4*d^3*e^2 - 176*b*c^3*d^2*e^3 + 82*b^2*
c^2*d*e^4 - 3*b^3*c*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*e^7*x + c^2*d*e^6), -1/192*(15*(128*c^4*d^5 - 256*b*c^3*d^
4*e + 144*b^2*c^2*d^3*e^2 - 16*b^3*c*d^2*e^3 - b^4*d*e^4 + (128*c^4*d^4*e - 256*b*c^3*d^3*e^2 + 144*b^2*c^2*d^
2*e^3 - 16*b^3*c*d*e^4 - b^4*e^5)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - 480*(2*c^4*d^4 - 3*b*
c^3*d^3*e + b^2*c^2*d^2*e^2 + (2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + b^2*c^2*d*e^3)*x)*sqrt(c*d^2 - b*d*e)*log((b*d
+ (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - (48*c^4*e^5*x^4 - 960*c^4*d^4*e + 16
80*b*c^3*d^3*e^2 - 720*b^2*c^2*d^2*e^3 + 15*b^3*c*d*e^4 - 8*(10*c^4*d*e^4 - 17*b*c^3*e^5)*x^3 + 2*(80*c^4*d^2*
e^3 - 140*b*c^3*d*e^4 + 59*b^2*c^2*e^5)*x^2 - 5*(96*c^4*d^3*e^2 - 176*b*c^3*d^2*e^3 + 82*b^2*c^2*d*e^4 - 3*b^3
*c*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*e^7*x + c^2*d*e^6), -1/192*(960*(2*c^4*d^4 - 3*b*c^3*d^3*e + b^2*c^2*d^2*e^
2 + (2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + b^2*c^2*d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(
c*x^2 + b*x)/((c*d - b*e)*x)) + 15*(128*c^4*d^5 - 256*b*c^3*d^4*e + 144*b^2*c^2*d^3*e^2 - 16*b^3*c*d^2*e^3 - b
^4*d*e^4 + (128*c^4*d^4*e - 256*b*c^3*d^3*e^2 + 144*b^2*c^2*d^2*e^3 - 16*b^3*c*d*e^4 - b^4*e^5)*x)*sqrt(-c)*ar
ctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (48*c^4*e^5*x^4 - 960*c^4*d^4*e + 1680*b*c^3*d^3*e^2 - 720*b^2*c^2*d^
2*e^3 + 15*b^3*c*d*e^4 - 8*(10*c^4*d*e^4 - 17*b*c^3*e^5)*x^3 + 2*(80*c^4*d^2*e^3 - 140*b*c^3*d*e^4 + 59*b^2*c^
2*e^5)*x^2 - 5*(96*c^4*d^3*e^2 - 176*b*c^3*d^2*e^3 + 82*b^2*c^2*d*e^4 - 3*b^3*c*e^5)*x)*sqrt(c*x^2 + b*x))/(c^
2*e^7*x + c^2*d*e^6)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out