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

Optimal. Leaf size=356 \[ \frac{\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{48 c e^3}+\frac{\sqrt{b x+c x^2} \left (-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4-288 b c^3 d^3 e+128 c^4 d^4\right )}{128 c^2 e^5}-\frac{(2 c d-b e) \left (112 b^2 c^2 d^2 e^2+16 b^3 c d e^3+3 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{5/2} e^6}+\frac{d^{5/2} (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^6}+\frac{\left (b x+c x^2\right )^{5/2}}{5 e} \]

[Out]

((128*c^4*d^4 - 288*b*c^3*d^3*e + 176*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e^3 - 3*b^4*e^4 - 2*c*e*(2*c*d - b*e)*(16*c
^2*d^2 - 16*b*c*d*e - 3*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(128*c^2*e^5) + ((16*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2 -
6*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(48*c*e^3) + (b*x + c*x^2)^(5/2)/(5*e) - ((2*c*d - b*e)*(128*c^4*
d^4 - 256*b*c^3*d^3*e + 112*b^2*c^2*d^2*e^2 + 16*b^3*c*d*e^3 + 3*b^4*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2
]])/(128*c^(5/2)*e^6) + (d^(5/2)*(c*d - b*e)^(5/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*
Sqrt[b*x + c*x^2])])/e^6

________________________________________________________________________________________

Rubi [A]  time = 0.434445, antiderivative size = 356, 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 = {734, 814, 843, 620, 206, 724} \[ \frac{\left (b x+c x^2\right )^{3/2} \left (3 b^2 e^2-6 c e x (2 c d-b e)-22 b c d e+16 c^2 d^2\right )}{48 c e^3}+\frac{\sqrt{b x+c x^2} \left (-2 c e x (2 c d-b e) \left (-3 b^2 e^2-16 b c d e+16 c^2 d^2\right )+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4-288 b c^3 d^3 e+128 c^4 d^4\right )}{128 c^2 e^5}-\frac{(2 c d-b e) \left (112 b^2 c^2 d^2 e^2+16 b^3 c d e^3+3 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{5/2} e^6}+\frac{d^{5/2} (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^6}+\frac{\left (b x+c x^2\right )^{5/2}}{5 e} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(5/2)/(d + e*x),x]

[Out]

((128*c^4*d^4 - 288*b*c^3*d^3*e + 176*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e^3 - 3*b^4*e^4 - 2*c*e*(2*c*d - b*e)*(16*c
^2*d^2 - 16*b*c*d*e - 3*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(128*c^2*e^5) + ((16*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2 -
6*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(48*c*e^3) + (b*x + c*x^2)^(5/2)/(5*e) - ((2*c*d - b*e)*(128*c^4*
d^4 - 256*b*c^3*d^3*e + 112*b^2*c^2*d^2*e^2 + 16*b^3*c*d*e^3 + 3*b^4*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2
]])/(128*c^(5/2)*e^6) + (d^(5/2)*(c*d - b*e)^(5/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*
Sqrt[b*x + c*x^2])])/e^6

Rule 734

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[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, 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] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[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} \, dx &=\frac{\left (b x+c x^2\right )^{5/2}}{5 e}-\frac{\int \frac{(b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e}\\ &=\frac{\left (16 c^2 d^2-22 b c d e+3 b^2 e^2-6 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^3}+\frac{\left (b x+c x^2\right )^{5/2}}{5 e}+\frac{\int \frac{\left (-\frac{1}{2} b d \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-\frac{1}{2} (2 c d-b e) \left (16 c^2 d^2-16 b c d e-3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{d+e x} \, dx}{16 c e^3}\\ &=\frac{\left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4-2 c e (2 c d-b e) \left (16 c^2 d^2-16 b c d e-3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{128 c^2 e^5}+\frac{\left (16 c^2 d^2-22 b c d e+3 b^2 e^2-6 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^3}+\frac{\left (b x+c x^2\right )^{5/2}}{5 e}-\frac{\int \frac{\frac{1}{4} b d \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )+\frac{1}{4} (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+112 b^2 c^2 d^2 e^2+16 b^3 c d e^3+3 b^4 e^4\right ) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{64 c^2 e^5}\\ &=\frac{\left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4-2 c e (2 c d-b e) \left (16 c^2 d^2-16 b c d e-3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{128 c^2 e^5}+\frac{\left (16 c^2 d^2-22 b c d e+3 b^2 e^2-6 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^3}+\frac{\left (b x+c x^2\right )^{5/2}}{5 e}+\frac{\left (d^3 (c d-b e)^3\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{e^6}-\frac{\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+112 b^2 c^2 d^2 e^2+16 b^3 c d e^3+3 b^4 e^4\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{256 c^2 e^6}\\ &=\frac{\left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4-2 c e (2 c d-b e) \left (16 c^2 d^2-16 b c d e-3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{128 c^2 e^5}+\frac{\left (16 c^2 d^2-22 b c d e+3 b^2 e^2-6 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^3}+\frac{\left (b x+c x^2\right )^{5/2}}{5 e}-\frac{\left (2 d^3 (c d-b e)^3\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 ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+112 b^2 c^2 d^2 e^2+16 b^3 c d e^3+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 )}{128 c^2 e^6}\\ &=\frac{\left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4-2 c e (2 c d-b e) \left (16 c^2 d^2-16 b c d e-3 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{128 c^2 e^5}+\frac{\left (16 c^2 d^2-22 b c d e+3 b^2 e^2-6 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^3}+\frac{\left (b x+c x^2\right )^{5/2}}{5 e}-\frac{(2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+112 b^2 c^2 d^2 e^2+16 b^3 c d e^3+3 b^4 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{5/2} e^6}+\frac{d^{5/2} (c d-b e)^{5/2} \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 )}{e^6}\\ \end{align*}

Mathematica [B]  time = 3.78271, size = 727, normalized size = 2.04 \[ \frac{(x (b+c x))^{5/2} \left (3 e^5 (b+c x)^3 \sqrt{b e-c d} \left (b c x \sqrt{\frac{c x}{b}+1} \left (248 b^2 c^2 x^2+10 b^3 c x-15 b^4+336 b c^3 x^3+128 c^4 x^4\right )+15 b^{11/2} \sqrt{c} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )-10 c d \left (e^4 (b+c x)^3 \sqrt{b e-c d} \left (b c x \sqrt{\frac{c x}{b}+1} \left (118 b^2 c x+15 b^3+136 b c^2 x^2+48 c^3 x^3\right )-15 b^{9/2} \sqrt{c} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )-8 b c^{3/2} d \sqrt{x} \left (b^2 e^3 \sqrt{b e-c d} \left (b \sqrt{c} \sqrt{x} \left (33 b^2+26 b c x+8 c^2 x^2\right ) \left (\frac{c x}{b}+1\right )^{7/2}+15 \sqrt{b} (b+c x)^3 \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )-6 \sqrt{c} \sqrt{d} (b+c x) \left (b \sqrt{c} \sqrt{d} e^2 \sqrt{b e-c d} \left (b \sqrt{c} \sqrt{x} (5 b+2 c x) \left (\frac{c x}{b}+1\right )^{5/2}+3 \sqrt{b} (b+c x)^2 \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )-4 (c d-b e) \left (\sqrt{c} \sqrt{d} e (b+c x) \sqrt{b e-c d} \left (b \sqrt{c} \sqrt{x} \left (\frac{c x}{b}+1\right )^{3/2}+\sqrt{b} (b+c x) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )-2 b \left (\frac{c x}{b}+1\right )^{3/2} (c d-b e) \left (\sqrt{b+c x} (b e-c d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )+\sqrt{b} \sqrt{c} \sqrt{d} \sqrt{\frac{c x}{b}+1} \sqrt{b e-c d} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )\right )\right )\right )\right )\right )\right )}{1920 b c^3 e^6 x^3 (b+c x)^5 \sqrt{\frac{c x}{b}+1} \sqrt{b e-c d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((x*(b + c*x))^(5/2)*(3*e^5*Sqrt[-(c*d) + b*e]*(b + c*x)^3*(b*c*x*Sqrt[1 + (c*x)/b]*(-15*b^4 + 10*b^3*c*x + 24
8*b^2*c^2*x^2 + 336*b*c^3*x^3 + 128*c^4*x^4) + 15*b^(11/2)*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])
- 10*c*d*(e^4*Sqrt[-(c*d) + b*e]*(b + c*x)^3*(b*c*x*Sqrt[1 + (c*x)/b]*(15*b^3 + 118*b^2*c*x + 136*b*c^2*x^2 +
48*c^3*x^3) - 15*b^(9/2)*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]) - 8*b*c^(3/2)*d*Sqrt[x]*(b^2*e^3
*Sqrt[-(c*d) + b*e]*(b*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(7/2)*(33*b^2 + 26*b*c*x + 8*c^2*x^2) + 15*Sqrt[b]*(b + c
*x)^3*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]) - 6*Sqrt[c]*Sqrt[d]*(b + c*x)*(b*Sqrt[c]*Sqrt[d]*e^2*Sqrt[-(c*d) + b
*e]*(b*Sqrt[c]*Sqrt[x]*(5*b + 2*c*x)*(1 + (c*x)/b)^(5/2) + 3*Sqrt[b]*(b + c*x)^2*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqr
t[b]]) - 4*(c*d - b*e)*(Sqrt[c]*Sqrt[d]*e*Sqrt[-(c*d) + b*e]*(b + c*x)*(b*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2)
+ Sqrt[b]*(b + c*x)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]) - 2*b*(c*d - b*e)*(1 + (c*x)/b)^(3/2)*(Sqrt[b]*Sqrt[c]
*Sqrt[d]*Sqrt[-(c*d) + b*e]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]] + (-(c*d) + b*e)*Sqrt[b + c*x
]*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])))))))/(1920*b*c^3*e^6*Sqrt[-(c*d) + b*e]*x^3*(
b + c*x)^5*Sqrt[1 + (c*x)/b])

________________________________________________________________________________________

Maple [B]  time = 0.233, size = 1932, normalized size = 5.4 \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),x)

[Out]

-1/e^7*d^6/(-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+1/5/e*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
-d*(b*e-c*d)/e^2)^(5/2)-1/4/e^2*(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*d+5/128/e^2*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^4+
5/16/e^3*d^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
))/c^(1/2)*b^3-15/8/e^4*d^3*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^2+5/2/e^5*d^4*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+1/e^4*d^3/(-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
-5/32/e^2*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*d-5/64/e^2/c*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)*d-1/2/e^4*d^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*c^2-9/4/e^4*d^3*(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-3/64/e/c*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+3/4/e^3*d^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*b*c-3/e^5*d^4/(-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+3/e^6*d^5/(-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*c^2-3/128/e/c^2*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)-1/e^6*d^5*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)+1/3/e^3*d^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c+1/e^5*d^4*(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2+11/8/e^3*d^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^2+3/256/e/c^(5/2)*b^5*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))+1/8/e*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*b+1/16/e/c*(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b^2-11/24/e^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e
^2)^(3/2)*b*d

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 31.8437, size = 3449, normalized size = 9.69 \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),x, algorithm="fricas")

[Out]

[-1/3840*(15*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e^3 - 10*b^4*c*d*e^4 - 3*b^
5*e^5)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 3840*(c^5*d^4 - 2*b*c^4*d^3*e + b^2*c^3*d^2*e^2)
*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*(384
*c^5*e^5*x^4 + 1920*c^5*d^4*e - 4320*b*c^4*d^3*e^2 + 2640*b^2*c^3*d^2*e^3 - 150*b^3*c^2*d*e^4 - 45*b^4*c*e^5 -
48*(10*c^5*d*e^4 - 21*b*c^4*e^5)*x^3 + 8*(80*c^5*d^2*e^3 - 170*b*c^4*d*e^4 + 93*b^2*c^3*e^5)*x^2 - 10*(96*c^5
*d^3*e^2 - 208*b*c^4*d^2*e^3 + 118*b^2*c^3*d*e^4 - 3*b^3*c^2*e^5)*x)*sqrt(c*x^2 + b*x))/(c^3*e^6), 1/3840*(768
0*(c^5*d^4 - 2*b*c^4*d^3*e + b^2*c^3*d^2*e^2)*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*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e^3 - 10*b^4*c
*d*e^4 - 3*b^5*e^5)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(384*c^5*e^5*x^4 + 1920*c^5*d^4*e
- 4320*b*c^4*d^3*e^2 + 2640*b^2*c^3*d^2*e^3 - 150*b^3*c^2*d*e^4 - 45*b^4*c*e^5 - 48*(10*c^5*d*e^4 - 21*b*c^4*
e^5)*x^3 + 8*(80*c^5*d^2*e^3 - 170*b*c^4*d*e^4 + 93*b^2*c^3*e^5)*x^2 - 10*(96*c^5*d^3*e^2 - 208*b*c^4*d^2*e^3
+ 118*b^2*c^3*d*e^4 - 3*b^3*c^2*e^5)*x)*sqrt(c*x^2 + b*x))/(c^3*e^6), 1/1920*(15*(256*c^5*d^5 - 640*b*c^4*d^4*
e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*s
qrt(-c)/(c*x)) + 1920*(c^5*d^4 - 2*b*c^4*d^3*e + b^2*c^3*d^2*e^2)*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)) + (384*c^5*e^5*x^4 + 1920*c^5*d^4*e - 4320*b*c^4*d^3*
e^2 + 2640*b^2*c^3*d^2*e^3 - 150*b^3*c^2*d*e^4 - 45*b^4*c*e^5 - 48*(10*c^5*d*e^4 - 21*b*c^4*e^5)*x^3 + 8*(80*c
^5*d^2*e^3 - 170*b*c^4*d*e^4 + 93*b^2*c^3*e^5)*x^2 - 10*(96*c^5*d^3*e^2 - 208*b*c^4*d^2*e^3 + 118*b^2*c^3*d*e^
4 - 3*b^3*c^2*e^5)*x)*sqrt(c*x^2 + b*x))/(c^3*e^6), 1/1920*(3840*(c^5*d^4 - 2*b*c^4*d^3*e + b^2*c^3*d^2*e^2)*s
qrt(-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*(256*c^5*d^5 - 640*b*
c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5)*sqrt(-c)*arctan(sqrt(c*x^2
+ b*x)*sqrt(-c)/(c*x)) + (384*c^5*e^5*x^4 + 1920*c^5*d^4*e - 4320*b*c^4*d^3*e^2 + 2640*b^2*c^3*d^2*e^3 - 150*b
^3*c^2*d*e^4 - 45*b^4*c*e^5 - 48*(10*c^5*d*e^4 - 21*b*c^4*e^5)*x^3 + 8*(80*c^5*d^2*e^3 - 170*b*c^4*d*e^4 + 93*
b^2*c^3*e^5)*x^2 - 10*(96*c^5*d^3*e^2 - 208*b*c^4*d^2*e^3 + 118*b^2*c^3*d*e^4 - 3*b^3*c^2*e^5)*x)*sqrt(c*x^2 +
b*x))/(c^3*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),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError