3.11.30 \(\int \frac {(A+B x) (b x+c x^2)^{3/2}}{d+e x} \, dx\)

Optimal. Leaf size=392 \[ \frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{64 c^2 e^4}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (8 A c e \left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )-B \left (3 b^4 e^4+8 b^3 c d e^3+48 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{64 c^{5/2} e^5}-\frac {d^{3/2} (B d-A e) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e^5}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2} \]

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Rubi [A]  time = 0.59, antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {814, 843, 620, 206, 724} \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (8 b^2 c d e^2+3 b^3 e^3-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{64 c^2 e^4}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (8 A c e \left (6 b^2 c d e^2+b^3 e^3-24 b c^2 d^2 e+16 c^3 d^3\right )-B \left (48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{64 c^{5/2} e^5}-\frac {d^{3/2} (B d-A e) (c d-b e)^{3/2} \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^5}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((8*A*c*e*(8*c^2*d^2 - 10*b*c*d*e + b^2*e^2) - B*(64*c^3*d^3 - 80*b*c^2*d^2*e + 8*b^2*c*d*e^2 + 3*b^3*e^3) - 2
*c*e*(8*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c^2*e^4) - ((8
*B*c*d - 3*b*B*e - 8*A*c*e - 6*B*c*e*x)*(b*x + c*x^2)^(3/2))/(24*c*e^2) - ((8*A*c*e*(16*c^3*d^3 - 24*b*c^2*d^2
*e + 6*b^2*c*d*e^2 + b^3*e^3) - B*(128*c^4*d^4 - 192*b*c^3*d^3*e + 48*b^2*c^2*d^2*e^2 + 8*b^3*c*d*e^3 + 3*b^4*
e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(5/2)*e^5) - (d^(3/2)*(B*d - A*e)*(c*d - b*e)^(3/2)*ArcTan
h[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e^5

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 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 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]

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]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx &=-\frac {(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}-\frac {\int \frac {\left (-\frac {1}{2} b d (8 B c d-3 b B e-8 A c e)+\frac {1}{2} \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{d+e x} \, dx}{8 c e^2}\\ &=\frac {\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )-2 c e \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c^2 e^4}-\frac {(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}+\frac {\int \frac {-\frac {1}{4} b d \left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )\right )-\frac {1}{4} \left (4 b c d e (2 c d-b e) (8 B c d-3 b B e-8 A c e)+\left (8 c^2 d^2-4 b c d e-b^2 e^2\right ) \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right )\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{32 c^2 e^4}\\ &=\frac {\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )-2 c e \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c^2 e^4}-\frac {(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}-\frac {\left (d^2 (B d-A e) (c d-b e)^2\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{e^5}-\frac {\left (8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )-B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{128 c^2 e^5}\\ &=\frac {\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )-2 c e \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c^2 e^4}-\frac {(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}+\frac {\left (2 d^2 (B d-A e) (c d-b e)^2\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^5}-\frac {\left (8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )-B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 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 )}{64 c^2 e^5}\\ &=\frac {\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )-2 c e \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c^2 e^4}-\frac {(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}-\frac {\left (8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )-B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{5/2} e^5}-\frac {d^{3/2} (B d-A e) (c d-b e)^{3/2} \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^5}\\ \end {align*}

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Mathematica [A]  time = 1.41, size = 386, normalized size = 0.98 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (e \sqrt {x} \left (8 A c e \left (3 b^2 e^2+2 b c e (7 e x-15 d)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+B \left (-9 b^3 e^3+6 b^2 c e^2 (e x-4 d)+8 b c^2 e \left (30 d^2-14 d e x+9 e^2 x^2\right )-16 c^3 \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )\right )\right )-\frac {384 c^2 d^{3/2} (B d-A e) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {b+c x}}\right )+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (B \left (3 b^4 e^4+8 b^3 c d e^3+48 b^2 c^2 d^2 e^2-192 b c^3 d^3 e+128 c^4 d^4\right )-8 A c e \left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )\right )}{\sqrt {b} \sqrt {\frac {c x}{b}+1}}\right )}{192 c^{5/2} e^5 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*((3*(-8*A*c*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3) + B*(128*c^4*d^4 - 19
2*b*c^3*d^3*e + 48*b^2*c^2*d^2*e^2 + 8*b^3*c*d*e^3 + 3*b^4*e^4))*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*
Sqrt[1 + (c*x)/b]) + Sqrt[c]*(e*Sqrt[x]*(8*A*c*e*(3*b^2*e^2 + 2*b*c*e*(-15*d + 7*e*x) + 4*c^2*(6*d^2 - 3*d*e*x
+ 2*e^2*x^2)) + B*(-9*b^3*e^3 + 6*b^2*c*e^2*(-4*d + e*x) + 8*b*c^2*e*(30*d^2 - 14*d*e*x + 9*e^2*x^2) - 16*c^3
*(12*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 - 3*e^3*x^3))) - (384*c^2*d^(3/2)*(B*d - A*e)*(c*d - b*e)^(3/2)*ArcTanh[(Sq
rt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x])))/(192*c^(5/2)*e^5*Sqrt[x])

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IntegrateAlgebraic [A]  time = 8.70, size = 487, normalized size = 1.24 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (24 A b^2 c e^3-240 A b c^2 d e^2+112 A b c^2 e^3 x+192 A c^3 d^2 e-96 A c^3 d e^2 x+64 A c^3 e^3 x^2-9 b^3 B e^3-24 b^2 B c d e^2+6 b^2 B c e^3 x+240 b B c^2 d^2 e-112 b B c^2 d e^2 x+72 b B c^2 e^3 x^2-192 B c^3 d^3+96 B c^3 d^2 e x-64 B c^3 d e^2 x^2+48 B c^3 e^3 x^3\right )}{192 c^2 e^4}+\frac {\log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (8 A b^3 c e^4+48 A b^2 c^2 d e^3-192 A b c^3 d^2 e^2+128 A c^4 d^3 e-3 b^4 B e^4-8 b^3 B c d e^3-48 b^2 B c^2 d^2 e^2+192 b B c^3 d^3 e-128 B c^4 d^4\right )}{128 c^{5/2} e^5}-\frac {2 \left (A b d^{3/2} e^2 \sqrt {c d-b e}-A c d^{5/2} e \sqrt {c d-b e}+B c d^{7/2} \sqrt {c d-b e}-b B d^{5/2} e \sqrt {c d-b e}\right ) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(-192*B*c^3*d^3 + 240*b*B*c^2*d^2*e + 192*A*c^3*d^2*e - 24*b^2*B*c*d*e^2 - 240*A*b*c^2*d*e^
2 - 9*b^3*B*e^3 + 24*A*b^2*c*e^3 + 96*B*c^3*d^2*e*x - 112*b*B*c^2*d*e^2*x - 96*A*c^3*d*e^2*x + 6*b^2*B*c*e^3*x
+ 112*A*b*c^2*e^3*x - 64*B*c^3*d*e^2*x^2 + 72*b*B*c^2*e^3*x^2 + 64*A*c^3*e^3*x^2 + 48*B*c^3*e^3*x^3))/(192*c^
2*e^4) - (2*(B*c*d^(7/2)*Sqrt[c*d - b*e] - b*B*d^(5/2)*e*Sqrt[c*d - b*e] - A*c*d^(5/2)*e*Sqrt[c*d - b*e] + A*b
*d^(3/2)*e^2*Sqrt[c*d - b*e])*ArcTanh[(Sqrt[c]*d + Sqrt[c]*e*x - e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e]
)])/e^5 + ((-128*B*c^4*d^4 + 192*b*B*c^3*d^3*e + 128*A*c^4*d^3*e - 48*b^2*B*c^2*d^2*e^2 - 192*A*b*c^3*d^2*e^2
- 8*b^3*B*c*d*e^3 + 48*A*b^2*c^2*d*e^3 - 3*b^4*B*e^4 + 8*A*b^3*c*e^4)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x
^2]])/(128*c^(5/2)*e^5)

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fricas [A]  time = 32.56, size = 1695, normalized size = 4.32

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c -
6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e^4)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 384*(B
*c^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 + A*c^4)*d^2*e)*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*B*c^4*e^4*x^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c^4)*
d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c^2)*e^4 - 8*(8*B*c^4*d*e^3 - (9*B*b*c^3
+ 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 - 8*(7*B*b*c^3 + 6*A*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)*x)
*sqrt(c*x^2 + b*x))/(c^3*e^5), -1/384*(768*(B*c^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 + A*c^4)*d^2*e)*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)) + 3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2
*A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e
^4)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(48*B*c^4*e^4*x^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c
^3 + 4*A*c^4)*d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c^2)*e^4 - 8*(8*B*c^4*d*e^3
- (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 - 8*(7*B*b*c^3 + 6*A*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*
b*c^3)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^3*e^5), -1/192*(3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B
*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e^4)*sqrt(-c)*arctan(s
qrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + 192*(B*c^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 + A*c^4)*d^2*e)*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*B*c^4*e^4*x^3 - 192*
B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c^4)*d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c^
2)*e^4 - 8*(8*B*c^4*d*e^3 - (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 - 8*(7*B*b*c^3 + 6*A*c^4)*d*e
^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^3*e^5), -1/192*(384*(B*c^4*d^3 + A*b*c^3*d*e^2 -
(B*b*c^3 + A*c^4)*d^2*e)*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))
+ 3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b
^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e^4)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (48*B*c^4*e^4*x
^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c^4)*d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8
*A*b^2*c^2)*e^4 - 8*(8*B*c^4*d*e^3 - (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 - 8*(7*B*b*c^3 + 6*A
*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^3*e^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r: Bad Argument Type

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maple [B]  time = 0.05, size = 2334, normalized size = 5.95

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8/e^2*d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))
/c^(1/2)*b^2*A+1/2/e^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c*d^2*B-1/e^5*d^4/(-(b*e-c*
d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)
*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*c^2*A-1/2/e^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e
)^(1/2)*x*c*d*A-1/4/e^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b*B*d+1/16/e^2/c^(3/2)*ln(
((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*b^3*B*d-1/e^3
*d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)
^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2*A+3/8/e^3*d^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)
/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/c^(1/2)*b^2*B-1/8/e^2/c*((x+d/e)^2*c-(b*e-
c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2*B*d+3/2/e^3*d^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^
2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b*A+2/e^4*d^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)
^(1/2))/(x+d/e))*b*c*A-2/e^5*d^4/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*
e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*c*B+1/3/e*((x+d/e)^2
*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*A-1/3/e^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^
(3/2)*B*d+1/4*B/e*(c*x^2+b*x)^(3/2)*x-3/32*B/e*b^2/c*(c*x^2+b*x)^(1/2)*x-3/2/e^4*d^3*ln(((x+d/e)*c+1/2*(b*e-2*
c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b*B+3/128*B/e*b^4/c^(5/2)*l
n((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/8*B/e/c*(c*x^2+b*x)^(3/2)*b+1/e^4*d^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((
-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(
x+d/e)/e)^(1/2))/(x+d/e))*b^2*B+1/e^6*d^5/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/
e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*c^2*B-3/64*B/
e*b^3/c^2*(c*x^2+b*x)^(1/2)+1/e^3*d^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c*A-1/e^4*d^3*
((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c*B-1/e^4*d^3*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/
2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(3/2)*A+1/4/e*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*
e-2*c*d)*(x+d/e)/e)^(1/2)*x*b*A+1/e^5*d^4*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^
2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(3/2)*B-1/16/e/c^(3/2)*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c
-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*b^3*A+1/8/e/c*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/
e)^(1/2)*b^2*A-5/4/e^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*d*A+5/4/e^3*((x+d/e)^2*c-(b
*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*d^2*B

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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((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d),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(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{d + e x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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