3.1717 \(\int \frac {(c+d x)^{3/2}}{(a+b x)^2 (e+f x)} \, dx\)

Optimal. Leaf size=170 \[ -\frac {\sqrt {b c-a d} (-a d f-2 b c f+3 b d e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b e-a f)^2}-\frac {\sqrt {c+d x} (b c-a d)}{b (a+b x) (b e-a f)}+\frac {2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} (b e-a f)^2} \]

[Out]

-(-a*d*f-2*b*c*f+3*b*d*e)*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))*(-a*d+b*c)^(1/2)/b^(3/2)/(-a*f+b*e)^
2+2*(-c*f+d*e)^(3/2)*arctan(f^(1/2)*(d*x+c)^(1/2)/(-c*f+d*e)^(1/2))/(-a*f+b*e)^2/f^(1/2)-(-a*d+b*c)*(d*x+c)^(1
/2)/b/(-a*f+b*e)/(b*x+a)

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Rubi [A]  time = 0.27, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {98, 156, 63, 208, 205} \[ -\frac {\sqrt {b c-a d} (-a d f-2 b c f+3 b d e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b e-a f)^2}-\frac {\sqrt {c+d x} (b c-a d)}{b (a+b x) (b e-a f)}+\frac {2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} (b e-a f)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*c - a*d)*Sqrt[c + d*x])/(b*(b*e - a*f)*(a + b*x))) + (2*(d*e - c*f)^(3/2)*ArcTan[(Sqrt[f]*Sqrt[c + d*x])
/Sqrt[d*e - c*f]])/(Sqrt[f]*(b*e - a*f)^2) - (Sqrt[b*c - a*d]*(3*b*d*e - 2*b*c*f - a*d*f)*ArcTanh[(Sqrt[b]*Sqr
t[c + d*x])/Sqrt[b*c - a*d]])/(b^(3/2)*(b*e - a*f)^2)

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^2 (e+f x)} \, dx &=-\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}-\frac {\int \frac {\frac {1}{2} \left (a d^2 e-2 b c \left (\frac {3 d e}{2}-c f\right )\right )-\frac {1}{2} d (2 b d e-b c f-a d f) x}{(a+b x) \sqrt {c+d x} (e+f x)} \, dx}{b (b e-a f)}\\ &=-\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}+\frac {(d e-c f)^2 \int \frac {1}{\sqrt {c+d x} (e+f x)} \, dx}{(b e-a f)^2}+\frac {((b c-a d) (3 b d e-2 b c f-a d f)) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b (b e-a f)^2}\\ &=-\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}+\frac {\left (2 (d e-c f)^2\right ) \operatorname {Subst}\left (\int \frac {1}{e-\frac {c f}{d}+\frac {f x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d (b e-a f)^2}+\frac {((b c-a d) (3 b d e-2 b c f-a d f)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b d (b e-a f)^2}\\ &=-\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}+\frac {2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} (b e-a f)^2}-\frac {\sqrt {b c-a d} (3 b d e-2 b c f-a d f) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b e-a f)^2}\\ \end {align*}

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Mathematica [A]  time = 0.70, size = 265, normalized size = 1.56 \[ \frac {\frac {(-a d f-2 b c f+3 b d e) \left (\sqrt {b} \sqrt {c+d x} (-3 a d+4 b c+b d x)-3 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )\right )}{b^{3/2} (b e-a f)}+\frac {2 (b c-a d) \left (\sqrt {f} \sqrt {c+d x} \sqrt {d e-c f} (4 c f-3 d e+d f x)+3 (d e-c f)^2 \tan ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )\right )}{\sqrt {f} (b e-a f) \sqrt {d e-c f}}-\frac {3 b (c+d x)^{5/2}}{a+b x}}{3 (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*b*(c + d*x)^(5/2))/(a + b*x) + (2*(b*c - a*d)*(Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[c + d*x]*(-3*d*e + 4*c*f + d*
f*x) + 3*(d*e - c*f)^2*ArcTan[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]]))/(Sqrt[f]*(b*e - a*f)*Sqrt[d*e - c*f])
+ ((3*b*d*e - 2*b*c*f - a*d*f)*(Sqrt[b]*Sqrt[c + d*x]*(4*b*c - 3*a*d + b*d*x) - 3*(b*c - a*d)^(3/2)*ArcTanh[(
Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]]))/(b^(3/2)*(b*e - a*f)))/(3*(b*c - a*d)*(b*e - a*f))

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fricas [A]  time = 3.11, size = 1134, normalized size = 6.67 \[ \left [-\frac {{\left (3 \, a b d e - {\left (2 \, a b c + a^{2} d\right )} f + {\left (3 \, b^{2} d e - {\left (2 \, b^{2} c + a b d\right )} f\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (a b d e - a b c f + {\left (b^{2} d e - b^{2} c f\right )} x\right )} \sqrt {-\frac {d e - c f}{f}} \log \left (\frac {d f x - d e + 2 \, c f - 2 \, \sqrt {d x + c} f \sqrt {-\frac {d e - c f}{f}}}{f x + e}\right ) + 2 \, {\left ({\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f\right )} \sqrt {d x + c}}{2 \, {\left (a b^{3} e^{2} - 2 \, a^{2} b^{2} e f + a^{3} b f^{2} + {\left (b^{4} e^{2} - 2 \, a b^{3} e f + a^{2} b^{2} f^{2}\right )} x\right )}}, -\frac {{\left (3 \, a b d e - {\left (2 \, a b c + a^{2} d\right )} f + {\left (3 \, b^{2} d e - {\left (2 \, b^{2} c + a b d\right )} f\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (a b d e - a b c f + {\left (b^{2} d e - b^{2} c f\right )} x\right )} \sqrt {-\frac {d e - c f}{f}} \log \left (\frac {d f x - d e + 2 \, c f - 2 \, \sqrt {d x + c} f \sqrt {-\frac {d e - c f}{f}}}{f x + e}\right ) + {\left ({\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f\right )} \sqrt {d x + c}}{a b^{3} e^{2} - 2 \, a^{2} b^{2} e f + a^{3} b f^{2} + {\left (b^{4} e^{2} - 2 \, a b^{3} e f + a^{2} b^{2} f^{2}\right )} x}, -\frac {4 \, {\left (a b d e - a b c f + {\left (b^{2} d e - b^{2} c f\right )} x\right )} \sqrt {\frac {d e - c f}{f}} \arctan \left (-\frac {\sqrt {d x + c} f \sqrt {\frac {d e - c f}{f}}}{d e - c f}\right ) + {\left (3 \, a b d e - {\left (2 \, a b c + a^{2} d\right )} f + {\left (3 \, b^{2} d e - {\left (2 \, b^{2} c + a b d\right )} f\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left ({\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f\right )} \sqrt {d x + c}}{2 \, {\left (a b^{3} e^{2} - 2 \, a^{2} b^{2} e f + a^{3} b f^{2} + {\left (b^{4} e^{2} - 2 \, a b^{3} e f + a^{2} b^{2} f^{2}\right )} x\right )}}, -\frac {{\left (3 \, a b d e - {\left (2 \, a b c + a^{2} d\right )} f + {\left (3 \, b^{2} d e - {\left (2 \, b^{2} c + a b d\right )} f\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + 2 \, {\left (a b d e - a b c f + {\left (b^{2} d e - b^{2} c f\right )} x\right )} \sqrt {\frac {d e - c f}{f}} \arctan \left (-\frac {\sqrt {d x + c} f \sqrt {\frac {d e - c f}{f}}}{d e - c f}\right ) + {\left ({\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f\right )} \sqrt {d x + c}}{a b^{3} e^{2} - 2 \, a^{2} b^{2} e f + a^{3} b f^{2} + {\left (b^{4} e^{2} - 2 \, a b^{3} e f + a^{2} b^{2} f^{2}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*((3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*f)*x)*sqrt((b*c - a*d)/b)*log((b*d*x
+ 2*b*c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*(a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*f)
*x)*sqrt(-(d*e - c*f)/f)*log((d*f*x - d*e + 2*c*f - 2*sqrt(d*x + c)*f*sqrt(-(d*e - c*f)/f))/(f*x + e)) + 2*((b
^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^
3*e*f + a^2*b^2*f^2)*x), -((3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*f)*x)*sqrt(-(b*c
- a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + (a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*f)
*x)*sqrt(-(d*e - c*f)/f)*log((d*f*x - d*e + 2*c*f - 2*sqrt(d*x + c)*f*sqrt(-(d*e - c*f)/f))/(f*x + e)) + ((b^2
*c - a*b*d)*e - (a*b*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^3*
e*f + a^2*b^2*f^2)*x), -1/2*(4*(a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*f)*x)*sqrt((d*e - c*f)/f)*arctan(-sqrt(d*
x + c)*f*sqrt((d*e - c*f)/f)/(d*e - c*f)) + (3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*
f)*x)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d + 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)) + 2*((b
^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^
3*e*f + a^2*b^2*f^2)*x), -((3*a*b*d*e - (2*a*b*c + a^2*d)*f + (3*b^2*d*e - (2*b^2*c + a*b*d)*f)*x)*sqrt(-(b*c
- a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)) + 2*(a*b*d*e - a*b*c*f + (b^2*d*e - b^2*c*
f)*x)*sqrt((d*e - c*f)/f)*arctan(-sqrt(d*x + c)*f*sqrt((d*e - c*f)/f)/(d*e - c*f)) + ((b^2*c - a*b*d)*e - (a*b
*c - a^2*d)*f)*sqrt(d*x + c))/(a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2 + (b^4*e^2 - 2*a*b^3*e*f + a^2*b^2*f^2)*x
)]

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giac [A]  time = 1.40, size = 250, normalized size = 1.47 \[ -\frac {{\left (2 \, b^{2} c^{2} f - a b c d f - a^{2} d^{2} f - 3 \, b^{2} c d e + 3 \, a b d^{2} e\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{2} b f^{2} - 2 \, a b^{2} f e + b^{3} e^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (c^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} f}{\sqrt {-c f^{2} + d f e}}\right )}{{\left (a^{2} f^{2} - 2 \, a b f e + b^{2} e^{2}\right )} \sqrt {-c f^{2} + d f e}} + \frac {\sqrt {d x + c} b c d - \sqrt {d x + c} a d^{2}}{{\left (a b f - b^{2} e\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b^2*c^2*f - a*b*c*d*f - a^2*d^2*f - 3*b^2*c*d*e + 3*a*b*d^2*e)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d)
)/((a^2*b*f^2 - 2*a*b^2*f*e + b^3*e^2)*sqrt(-b^2*c + a*b*d)) + 2*(c^2*f^2 - 2*c*d*f*e + d^2*e^2)*arctan(sqrt(d
*x + c)*f/sqrt(-c*f^2 + d*f*e))/((a^2*f^2 - 2*a*b*f*e + b^2*e^2)*sqrt(-c*f^2 + d*f*e)) + (sqrt(d*x + c)*b*c*d
- sqrt(d*x + c)*a*d^2)/((a*b*f - b^2*e)*((d*x + c)*b - b*c + a*d))

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maple [B]  time = 0.02, size = 549, normalized size = 3.23 \[ \frac {a^{2} d^{2} f \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a f -b e \right )^{2} \sqrt {\left (a d -b c \right ) b}\, b}+\frac {a c d f \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a f -b e \right )^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {3 a \,d^{2} e \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a f -b e \right )^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {2 b \,c^{2} f \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a f -b e \right )^{2} \sqrt {\left (a d -b c \right ) b}}+\frac {3 b c d e \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a f -b e \right )^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {2 c^{2} f^{2} \arctanh \left (\frac {\sqrt {d x +c}\, f}{\sqrt {\left (c f -d e \right ) f}}\right )}{\left (a f -b e \right )^{2} \sqrt {\left (c f -d e \right ) f}}+\frac {4 c d e f \arctanh \left (\frac {\sqrt {d x +c}\, f}{\sqrt {\left (c f -d e \right ) f}}\right )}{\left (a f -b e \right )^{2} \sqrt {\left (c f -d e \right ) f}}-\frac {2 d^{2} e^{2} \arctanh \left (\frac {\sqrt {d x +c}\, f}{\sqrt {\left (c f -d e \right ) f}}\right )}{\left (a f -b e \right )^{2} \sqrt {\left (c f -d e \right ) f}}-\frac {\sqrt {d x +c}\, a^{2} d^{2} f}{\left (a f -b e \right )^{2} \left (b d x +a d \right ) b}+\frac {\sqrt {d x +c}\, a c d f}{\left (a f -b e \right )^{2} \left (b d x +a d \right )}+\frac {\sqrt {d x +c}\, a \,d^{2} e}{\left (a f -b e \right )^{2} \left (b d x +a d \right )}-\frac {\sqrt {d x +c}\, b c d e}{\left (a f -b e \right )^{2} \left (b d x +a d \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*f-b*e)^2/((c*f-d*e)*f)^(1/2)*arctanh((d*x+c)^(1/2)*f/((c*f-d*e)*f)^(1/2))*c^2*f^2+4*d/(a*f-b*e)^2/((c*f-
d*e)*f)^(1/2)*arctanh((d*x+c)^(1/2)*f/((c*f-d*e)*f)^(1/2))*c*e*f-2*d^2/(a*f-b*e)^2/((c*f-d*e)*f)^(1/2)*arctanh
((d*x+c)^(1/2)*f/((c*f-d*e)*f)^(1/2))*e^2-d^2/(a*f-b*e)^2/b*(d*x+c)^(1/2)/(b*d*x+a*d)*a^2*f+d/(a*f-b*e)^2*(d*x
+c)^(1/2)/(b*d*x+a*d)*a*f*c+d^2/(a*f-b*e)^2*(d*x+c)^(1/2)/(b*d*x+a*d)*e*a-d/(a*f-b*e)^2*(d*x+c)^(1/2)/(b*d*x+a
*d)*e*b*c+d^2/(a*f-b*e)^2/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*a^2*f+d/(a*f-b*e)^
2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*a*f*c-2/(a*f-b*e)^2/((a*d-b*c)*b)^(1/2)*arct
an((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*c^2*f*b-3*d^2/(a*f-b*e)^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a
*d-b*c)*b)^(1/2)*b)*e*a+3*d/(a*f-b*e)^2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*e*b*c

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(3/2)/(b*x+a)^2/(f*x+e),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(c*f-d*e>0)', see `assume?` for
more details)Is c*f-d*e positive or negative?

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mupad [B]  time = 3.82, size = 7774, normalized size = 45.73 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (atan(((((2*(c + d*x)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c^2*d^4*f^5 + 9*a
^2*b^2*d^6*e^2*f^3 + 33*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*c^3*d^3*f^5 - 16
*b^4*c*d^5*e^3*f^2 - 28*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/(b^3*e^2 + a^2*b
*f^2 - 2*a*b^2*e*f) + ((-b^3*(a*d - b*c))^(1/2)*((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^5*e^4*f^3 + 12*a^3*b^
4*d^5*e^3*f^4 - 8*a^4*b^3*d^5*e^2*f^5 + 2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a*b^6*d^5*e^5*f^2 + 2*
a^5*b^2*d^5*e*f^6 - 2*b^7*c*d^4*e^5*f^2 + 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 - 8*a*b^6*c^2*d^3*e^3*
f^4 - 4*a^2*b^5*c*d^4*e^3*f^4 - 4*a^3*b^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^2*b^5*c^2*d^3*e^2*f^5
))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a*b^3*e^2*f) + ((-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a*d*f
+ 2*b*c*f - 3*b*d*e)*(4*a^5*b^3*d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^3*e^3*f^4 + 8*a^3*b^5*d^3*e^2*f^5 -
8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*c*d^2*e^4*f^3 + 32*a*b^7*c*d^2*e^3*f
^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48*a^2*b^6*c*d^2*e^2*f^5))/((b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)*(b^3*e^2 + a^2
*b*f^2 - 2*a*b^2*e*f)))*(a*d*f + 2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)))*(-b^3*(a*d - b
*c))^(1/2)*(a*d*f + 2*b*c*f - 3*b*d*e)*1i)/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)) + (((2*(c + d*x)^(1/2)*(a
^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2*f^3 + 33*b^4*c^2*
d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*c^3*d^3*f^5 - 16*b^4*c*d^5*e^3*f^2 - 28*b^4*c^3*
d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f) - ((-b^3*(a*
d - b*c))^(1/2)*((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^5*e^4*f^3 + 12*a^3*b^4*d^5*e^3*f^4 - 8*a^4*b^3*d^5*e^
2*f^5 + 2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a*b^6*d^5*e^5*f^2 + 2*a^5*b^2*d^5*e*f^6 - 2*b^7*c*d^4*
e^5*f^2 + 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 - 8*a*b^6*c^2*d^3*e^3*f^4 - 4*a^2*b^5*c*d^4*e^3*f^4 -
4*a^3*b^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^2*b^5*c^2*d^3*e^2*f^5))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*
b^2*e*f^2 - 3*a*b^3*e^2*f) - ((-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a*d*f + 2*b*c*f - 3*b*d*e)*(4*a^5*b^3*
d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^3*e^3*f^4 + 8*a^3*b^5*d^3*e^2*f^5 - 8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d
^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*c*d^2*e^4*f^3 + 32*a*b^7*c*d^2*e^3*f^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48
*a^2*b^6*c*d^2*e^2*f^5))/((b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)*(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f)))*(a*d*f +
2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)))*(-b^3*(a*d - b*c))^(1/2)*(a*d*f + 2*b*c*f - 3*
b*d*e)*1i)/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)))/((4*(a^3*c^2*d^6*f^5 - 2*b^3*c^5*d^3*f^5 + a^3*d^8*e^2*f
^3 + 19*b^3*c^2*d^6*e^3*f^2 - 22*b^3*c^3*d^5*e^2*f^3 + 6*a*b^2*d^8*e^4*f - 2*a^3*c*d^7*e*f^4 - 6*b^3*c*d^7*e^4
*f - a*b^2*c^4*d^4*f^5 + 2*a^2*b*c^3*d^5*f^5 - 5*a^2*b*d^8*e^3*f^2 + 11*b^3*c^4*d^4*e*f^4 - 14*a*b^2*c*d^7*e^3
*f^2 + 12*a^2*b*c*d^7*e^2*f^3 - 9*a^2*b*c^2*d^6*e*f^4 + 9*a*b^2*c^2*d^6*e^2*f^3))/(b^4*e^3 - a^3*b*f^3 + 3*a^2
*b^2*e*f^2 - 3*a*b^3*e^2*f) + (((2*(c + d*x)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*
b^2*c^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2*f^3 + 33*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a
*b^3*c^3*d^3*f^5 - 16*b^4*c*d^5*e^3*f^2 - 28*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f
^4))/(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f) + ((-b^3*(a*d - b*c))^(1/2)*((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^
5*e^4*f^3 + 12*a^3*b^4*d^5*e^3*f^4 - 8*a^4*b^3*d^5*e^2*f^5 + 2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a
*b^6*d^5*e^5*f^2 + 2*a^5*b^2*d^5*e*f^6 - 2*b^7*c*d^4*e^5*f^2 + 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 -
8*a*b^6*c^2*d^3*e^3*f^4 - 4*a^2*b^5*c*d^4*e^3*f^4 - 4*a^3*b^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^
2*b^5*c^2*d^3*e^2*f^5))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a*b^3*e^2*f) + ((-b^3*(a*d - b*c))^(1/2)*(c
+ d*x)^(1/2)*(a*d*f + 2*b*c*f - 3*b*d*e)*(4*a^5*b^3*d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^3*e^3*f^4 + 8*a
^3*b^5*d^3*e^2*f^5 - 8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*c*d^2*e^4*f^3 +
32*a*b^7*c*d^2*e^3*f^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48*a^2*b^6*c*d^2*e^2*f^5))/((b^5*e^2 + a^2*b^3*f^2 - 2*a*b^
4*e*f)*(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f)))*(a*d*f + 2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*
e*f)))*(-b^3*(a*d - b*c))^(1/2)*(a*d*f + 2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)) - (((2*
(c + d*x)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2
*f^3 + 33*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*c^3*d^3*f^5 - 16*b^4*c*d^5*e^3
*f^2 - 28*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/(b^3*e^2 + a^2*b*f^2 - 2*a*b^2
*e*f) - ((-b^3*(a*d - b*c))^(1/2)*((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^5*e^4*f^3 + 12*a^3*b^4*d^5*e^3*f^4
- 8*a^4*b^3*d^5*e^2*f^5 + 2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a*b^6*d^5*e^5*f^2 + 2*a^5*b^2*d^5*e*
f^6 - 2*b^7*c*d^4*e^5*f^2 + 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 - 8*a*b^6*c^2*d^3*e^3*f^4 - 4*a^2*b^
5*c*d^4*e^3*f^4 - 4*a^3*b^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^2*b^5*c^2*d^3*e^2*f^5))/(b^4*e^3 -
a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a*b^3*e^2*f) - ((-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a*d*f + 2*b*c*f - 3*
b*d*e)*(4*a^5*b^3*d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^3*e^3*f^4 + 8*a^3*b^5*d^3*e^2*f^5 - 8*a^4*b^4*c*d^
2*f^7 - 12*a*b^7*d^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*c*d^2*e^4*f^3 + 32*a*b^7*c*d^2*e^3*f^4 + 32*a^3*b^
5*c*d^2*e*f^6 - 48*a^2*b^6*c*d^2*e^2*f^5))/((b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)*(b^3*e^2 + a^2*b*f^2 - 2*a*b
^2*e*f)))*(a*d*f + 2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f)))*(-b^3*(a*d - b*c))^(1/2)*(a*
d*f + 2*b*c*f - 3*b*d*e))/(2*(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f))))*(-b^3*(a*d - b*c))^(1/2)*(a*d*f + 2*b*c*
f - 3*b*d*e)*1i)/(b^5*e^2 + a^2*b^3*f^2 - 2*a*b^4*e*f) - (atan(((((2*(c + d*x)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*
e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2*f^3 + 33*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c
*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*c^3*d^3*f^5 - 16*b^4*c*d^5*e^3*f^2 - 28*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*
d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f) + (((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a
^2*b^5*d^5*e^4*f^3 + 12*a^3*b^4*d^5*e^3*f^4 - 8*a^4*b^3*d^5*e^2*f^5 + 2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*
f^7 + 2*a*b^6*d^5*e^5*f^2 + 2*a^5*b^2*d^5*e*f^6 - 2*b^7*c*d^4*e^5*f^2 + 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^
4*e*f^6 - 8*a*b^6*c^2*d^3*e^3*f^4 - 4*a^2*b^5*c*d^4*e^3*f^4 - 4*a^3*b^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^
6 + 12*a^2*b^5*c^2*d^3*e^2*f^5))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a*b^3*e^2*f) + (2*(f*(c*f - d*e)^3
)^(1/2)*(c + d*x)^(1/2)*(4*a^5*b^3*d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^3*e^3*f^4 + 8*a^3*b^5*d^3*e^2*f^5
- 8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*c*d^2*e^4*f^3 + 32*a*b^7*c*d^2*e^
3*f^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48*a^2*b^6*c*d^2*e^2*f^5))/(f*(a*f - b*e)^2*(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*
f)))*(f*(c*f - d*e)^3)^(1/2))/(f*(a*f - b*e)^2))*(f*(c*f - d*e)^3)^(1/2)*1i)/(f*(a*f - b*e)^2) + (((2*(c + d*x
)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2*f^3 + 3
3*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*c^3*d^3*f^5 - 16*b^4*c*d^5*e^3*f^2 - 2
8*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f) -
(((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^5*e^4*f^3 + 12*a^3*b^4*d^5*e^3*f^4 - 8*a^4*b^3*d^5*e^2*f^5 + 2*b^7*c
^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a*b^6*d^5*e^5*f^2 + 2*a^5*b^2*d^5*e*f^6 - 2*b^7*c*d^4*e^5*f^2 + 6*a*b
^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 - 8*a*b^6*c^2*d^3*e^3*f^4 - 4*a^2*b^5*c*d^4*e^3*f^4 - 4*a^3*b^4*c*d^4
*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^2*b^5*c^2*d^3*e^2*f^5))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a
*b^3*e^2*f) - (2*(f*(c*f - d*e)^3)^(1/2)*(c + d*x)^(1/2)*(4*a^5*b^3*d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^
3*e^3*f^4 + 8*a^3*b^5*d^3*e^2*f^5 - 8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*
c*d^2*e^4*f^3 + 32*a*b^7*c*d^2*e^3*f^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48*a^2*b^6*c*d^2*e^2*f^5))/(f*(a*f - b*e)^2*
(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f)))*(f*(c*f - d*e)^3)^(1/2))/(f*(a*f - b*e)^2))*(f*(c*f - d*e)^3)^(1/2)*1i)/
(f*(a*f - b*e)^2))/((4*(a^3*c^2*d^6*f^5 - 2*b^3*c^5*d^3*f^5 + a^3*d^8*e^2*f^3 + 19*b^3*c^2*d^6*e^3*f^2 - 22*b^
3*c^3*d^5*e^2*f^3 + 6*a*b^2*d^8*e^4*f - 2*a^3*c*d^7*e*f^4 - 6*b^3*c*d^7*e^4*f - a*b^2*c^4*d^4*f^5 + 2*a^2*b*c^
3*d^5*f^5 - 5*a^2*b*d^8*e^3*f^2 + 11*b^3*c^4*d^4*e*f^4 - 14*a*b^2*c*d^7*e^3*f^2 + 12*a^2*b*c*d^7*e^2*f^3 - 9*a
^2*b*c^2*d^6*e*f^4 + 9*a*b^2*c^2*d^6*e^2*f^3))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a*b^3*e^2*f) + (((2*
(c + d*x)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2
*f^3 + 33*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*c^3*d^3*f^5 - 16*b^4*c*d^5*e^3
*f^2 - 28*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/(b^3*e^2 + a^2*b*f^2 - 2*a*b^2
*e*f) + (((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^5*e^4*f^3 + 12*a^3*b^4*d^5*e^3*f^4 - 8*a^4*b^3*d^5*e^2*f^5 +
2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a*b^6*d^5*e^5*f^2 + 2*a^5*b^2*d^5*e*f^6 - 2*b^7*c*d^4*e^5*f^2
+ 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 - 8*a*b^6*c^2*d^3*e^3*f^4 - 4*a^2*b^5*c*d^4*e^3*f^4 - 4*a^3*b
^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^2*b^5*c^2*d^3*e^2*f^5))/(b^4*e^3 - a^3*b*f^3 + 3*a^2*b^2*e*f
^2 - 3*a*b^3*e^2*f) + (2*(f*(c*f - d*e)^3)^(1/2)*(c + d*x)^(1/2)*(4*a^5*b^3*d^3*f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^
2*b^6*d^3*e^3*f^4 + 8*a^3*b^5*d^3*e^2*f^5 - 8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d^3*e^4*f^3 - 12*a^4*b^4*d^3*e*f^6
- 8*b^8*c*d^2*e^4*f^3 + 32*a*b^7*c*d^2*e^3*f^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48*a^2*b^6*c*d^2*e^2*f^5))/(f*(a*f -
b*e)^2*(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f)))*(f*(c*f - d*e)^3)^(1/2))/(f*(a*f - b*e)^2))*(f*(c*f - d*e)^3)^(1
/2))/(f*(a*f - b*e)^2) - (((2*(c + d*x)^(1/2)*(a^4*d^6*f^5 + 4*b^4*d^6*e^4*f + 8*b^4*c^4*d^2*f^5 - 3*a^2*b^2*c
^2*d^4*f^5 + 9*a^2*b^2*d^6*e^2*f^3 + 33*b^4*c^2*d^4*e^2*f^3 + 2*a^3*b*c*d^5*f^5 - 6*a^3*b*d^6*e*f^4 - 4*a*b^3*
c^3*d^3*f^5 - 16*b^4*c*d^5*e^3*f^2 - 28*b^4*c^3*d^3*e*f^4 - 18*a*b^3*c*d^5*e^2*f^3 + 18*a*b^3*c^2*d^4*e*f^4))/
(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f) - (((2*(2*a^4*b^3*c^2*d^3*f^7 - 8*a^2*b^5*d^5*e^4*f^3 + 12*a^3*b^4*d^5*e^3
*f^4 - 8*a^4*b^3*d^5*e^2*f^5 + 2*b^7*c^2*d^3*e^4*f^3 - 2*a^5*b^2*c*d^4*f^7 + 2*a*b^6*d^5*e^5*f^2 + 2*a^5*b^2*d
^5*e*f^6 - 2*b^7*c*d^4*e^5*f^2 + 6*a*b^6*c*d^4*e^4*f^3 + 6*a^4*b^3*c*d^4*e*f^6 - 8*a*b^6*c^2*d^3*e^3*f^4 - 4*a
^2*b^5*c*d^4*e^3*f^4 - 4*a^3*b^4*c*d^4*e^2*f^5 - 8*a^3*b^4*c^2*d^3*e*f^6 + 12*a^2*b^5*c^2*d^3*e^2*f^5))/(b^4*e
^3 - a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a*b^3*e^2*f) - (2*(f*(c*f - d*e)^3)^(1/2)*(c + d*x)^(1/2)*(4*a^5*b^3*d^3*
f^7 + 4*b^8*d^3*e^5*f^2 + 8*a^2*b^6*d^3*e^3*f^4 + 8*a^3*b^5*d^3*e^2*f^5 - 8*a^4*b^4*c*d^2*f^7 - 12*a*b^7*d^3*e
^4*f^3 - 12*a^4*b^4*d^3*e*f^6 - 8*b^8*c*d^2*e^4*f^3 + 32*a*b^7*c*d^2*e^3*f^4 + 32*a^3*b^5*c*d^2*e*f^6 - 48*a^2
*b^6*c*d^2*e^2*f^5))/(f*(a*f - b*e)^2*(b^3*e^2 + a^2*b*f^2 - 2*a*b^2*e*f)))*(f*(c*f - d*e)^3)^(1/2))/(f*(a*f -
b*e)^2))*(f*(c*f - d*e)^3)^(1/2))/(f*(a*f - b*e)^2)))*(f*(c*f - d*e)^3)^(1/2)*2i)/(f*(a*f - b*e)^2) - (d*(a*d
- b*c)*(c + d*x)^(1/2))/(b*(a*f - b*e)*(a*d - b*c + b*(c + d*x)))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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