3.1.14 \(\int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^3} \, dx\)

Optimal. Leaf size=208 \[ -\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}+\frac {\sqrt {c+d x} \left (a^2 (-d) f-3 a b d e+4 b^2 c e\right )}{4 a^2 b (a+b x) (b c-a d)}+\frac {\left (a^3 d^2 f+3 a^2 b d^2 e-12 a b^2 c d e+8 b^3 c^2 e\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 a^3 b^{3/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x} (b e-a f)}{2 a b (a+b x)^2} \]

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Rubi [A]  time = 0.27, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {149, 151, 156, 63, 208} \begin {gather*} \frac {\left (3 a^2 b d^2 e+a^3 d^2 f-12 a b^2 c d e+8 b^3 c^2 e\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 a^3 b^{3/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x} \left (a^2 (-d) f-3 a b d e+4 b^2 c e\right )}{4 a^2 b (a+b x) (b c-a d)}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}+\frac {\sqrt {c+d x} (b e-a f)}{2 a b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*e - a*f)*Sqrt[c + d*x])/(2*a*b*(a + b*x)^2) + ((4*b^2*c*e - 3*a*b*d*e - a^2*d*f)*Sqrt[c + d*x])/(4*a^2*b*(
b*c - a*d)*(a + b*x)) - (2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a^3 + ((8*b^3*c^2*e - 12*a*b^2*c*d*e + 3*
a^2*b*d^2*e + a^3*d^2*f)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(4*a^3*b^(3/2)*(b*c - a*d)^(3/2))

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 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

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 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 {\sqrt {c+d x} (e+f x)}{x (a+b x)^3} \, dx &=\frac {(b e-a f) \sqrt {c+d x}}{2 a b (a+b x)^2}-\frac {\int \frac {-2 b c e-\frac {1}{2} d (3 b e+a f) x}{x (a+b x)^2 \sqrt {c+d x}} \, dx}{2 a b}\\ &=\frac {(b e-a f) \sqrt {c+d x}}{2 a b (a+b x)^2}+\frac {\left (4 b^2 c e-3 a b d e-a^2 d f\right ) \sqrt {c+d x}}{4 a^2 b (b c-a d) (a+b x)}-\frac {\int \frac {-2 b c (b c-a d) e-\frac {1}{4} d \left (4 b^2 c e-a d (3 b e+a f)\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx}{2 a^2 b (b c-a d)}\\ &=\frac {(b e-a f) \sqrt {c+d x}}{2 a b (a+b x)^2}+\frac {\left (4 b^2 c e-3 a b d e-a^2 d f\right ) \sqrt {c+d x}}{4 a^2 b (b c-a d) (a+b x)}+\frac {(c e) \int \frac {1}{x \sqrt {c+d x}} \, dx}{a^3}-\frac {\left (8 b^3 c^2 e-12 a b^2 c d e+3 a^2 b d^2 e+a^3 d^2 f\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{8 a^3 b (b c-a d)}\\ &=\frac {(b e-a f) \sqrt {c+d x}}{2 a b (a+b x)^2}+\frac {\left (4 b^2 c e-3 a b d e-a^2 d f\right ) \sqrt {c+d x}}{4 a^2 b (b c-a d) (a+b x)}+\frac {(2 c e) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 d}-\frac {\left (8 b^3 c^2 e-12 a b^2 c d e+3 a^2 b d^2 e+a^3 d^2 f\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{4 a^3 b d (b c-a d)}\\ &=\frac {(b e-a f) \sqrt {c+d x}}{2 a b (a+b x)^2}+\frac {\left (4 b^2 c e-3 a b d e-a^2 d f\right ) \sqrt {c+d x}}{4 a^2 b (b c-a d) (a+b x)}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}+\frac {\left (8 b^3 c^2 e-12 a b^2 c d e+3 a^2 b d^2 e+a^3 d^2 f\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 a^3 b^{3/2} (b c-a d)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 260, normalized size = 1.25 \begin {gather*} \frac {-\frac {(c+d x)^{3/2} \left (a^2 d f-5 a b d e+4 b^2 c e\right )}{2 a (a+b x) (a d-b c)}+\frac {4 e (b c-a d) \left (\sqrt {c+d x}-\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )\right )}{a^2}+\frac {\left (a^3 d^2 f+3 a^2 b d^2 e-12 a b^2 c d e+8 b^3 c^2 e\right ) \left (\sqrt {b} \sqrt {c+d x}-\sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )\right )}{2 a^2 b^{3/2} (a d-b c)}+\frac {(c+d x)^{3/2} (b e-a f)}{(a+b x)^2}}{2 a (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((b*e - a*f)*(c + d*x)^(3/2))/(a + b*x)^2 - ((4*b^2*c*e - 5*a*b*d*e + a^2*d*f)*(c + d*x)^(3/2))/(2*a*(-(b*c)
+ a*d)*(a + b*x)) + (4*(b*c - a*d)*e*(Sqrt[c + d*x] - Sqrt[c]*ArcTanh[Sqrt[c + d*x]/Sqrt[c]]))/a^2 + ((8*b^3*c
^2*e - 12*a*b^2*c*d*e + 3*a^2*b*d^2*e + a^3*d^2*f)*(Sqrt[b]*Sqrt[c + d*x] - Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*S
qrt[c + d*x])/Sqrt[b*c - a*d]]))/(2*a^2*b^(3/2)*(-(b*c) + a*d)))/(2*a*(b*c - a*d))

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IntegrateAlgebraic [A]  time = 1.08, size = 268, normalized size = 1.29 \begin {gather*} -\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3}-\frac {d \sqrt {c+d x} \left (a^3 d^2 f-a^2 b d f (c+d x)-a^2 b c d f-5 a^2 b d^2 e-3 a b^2 d e (c+d x)+9 a b^2 c d e-4 b^3 c^2 e+4 b^3 c e (c+d x)\right )}{4 a^2 b (a d-b c) (a d+b (c+d x)-b c)^2}+\frac {\left (a^3 d^2 f+3 a^2 b d^2 e-12 a b^2 c d e+8 b^3 c^2 e\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{4 a^3 b^{3/2} (b c-a d) \sqrt {a d-b c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(d*Sqrt[c + d*x]*(-4*b^3*c^2*e + 9*a*b^2*c*d*e - 5*a^2*b*d^2*e - a^2*b*c*d*f + a^3*d^2*f + 4*b^3*c*e*(c +
d*x) - 3*a*b^2*d*e*(c + d*x) - a^2*b*d*f*(c + d*x)))/(a^2*b*(-(b*c) + a*d)*(-(b*c) + a*d + b*(c + d*x))^2) +
((8*b^3*c^2*e - 12*a*b^2*c*d*e + 3*a^2*b*d^2*e + a^3*d^2*f)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/
(b*c - a*d)])/(4*a^3*b^(3/2)*(b*c - a*d)*Sqrt[-(b*c) + a*d]) - (2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a^
3

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fricas [B]  time = 3.16, size = 2216, normalized size = 10.65

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/8*((a^5*d^2*f + (a^3*b^2*d^2*f + (8*b^5*c^2 - 12*a*b^4*c*d + 3*a^2*b^3*d^2)*e)*x^2 + (8*a^2*b^3*c^2 - 12*a
^3*b^2*c*d + 3*a^4*b*d^2)*e + 2*(a^4*b*d^2*f + (8*a*b^4*c^2 - 12*a^2*b^3*c*d + 3*a^3*b^2*d^2)*e)*x)*sqrt(b^2*c
- a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) - 8*((b^6*c^2 - 2*a*b^5*c
*d + a^2*b^4*d^2)*e*x^2 + 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*e*x + (a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4
*b^2*d^2)*e)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*((6*a^2*b^4*c^2 - 11*a^3*b^3*c*d + 5*a^4
*b^2*d^2)*e - (2*a^3*b^3*c^2 - 3*a^4*b^2*c*d + a^5*b*d^2)*f + ((4*a*b^5*c^2 - 7*a^2*b^4*c*d + 3*a^3*b^3*d^2)*e
- (a^3*b^3*c*d - a^4*b^2*d^2)*f)*x)*sqrt(d*x + c))/(a^5*b^4*c^2 - 2*a^6*b^3*c*d + a^7*b^2*d^2 + (a^3*b^6*c^2
- 2*a^4*b^5*c*d + a^5*b^4*d^2)*x^2 + 2*(a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*x), -1/4*((a^5*d^2*f + (a^3
*b^2*d^2*f + (8*b^5*c^2 - 12*a*b^4*c*d + 3*a^2*b^3*d^2)*e)*x^2 + (8*a^2*b^3*c^2 - 12*a^3*b^2*c*d + 3*a^4*b*d^2
)*e + 2*(a^4*b*d^2*f + (8*a*b^4*c^2 - 12*a^2*b^3*c*d + 3*a^3*b^2*d^2)*e)*x)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-
b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) - 4*((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*e*x^2 + 2*(a*b^5*c^2 -
2*a^2*b^4*c*d + a^3*b^3*d^2)*e*x + (a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*e)*sqrt(c)*log((d*x - 2*sqrt(d*
x + c)*sqrt(c) + 2*c)/x) - ((6*a^2*b^4*c^2 - 11*a^3*b^3*c*d + 5*a^4*b^2*d^2)*e - (2*a^3*b^3*c^2 - 3*a^4*b^2*c*
d + a^5*b*d^2)*f + ((4*a*b^5*c^2 - 7*a^2*b^4*c*d + 3*a^3*b^3*d^2)*e - (a^3*b^3*c*d - a^4*b^2*d^2)*f)*x)*sqrt(d
*x + c))/(a^5*b^4*c^2 - 2*a^6*b^3*c*d + a^7*b^2*d^2 + (a^3*b^6*c^2 - 2*a^4*b^5*c*d + a^5*b^4*d^2)*x^2 + 2*(a^4
*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*x), 1/8*(16*((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*e*x^2 + 2*(a*b^5*c^
2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*e*x + (a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*e)*sqrt(-c)*arctan(sqrt(d*x
+ c)*sqrt(-c)/c) - (a^5*d^2*f + (a^3*b^2*d^2*f + (8*b^5*c^2 - 12*a*b^4*c*d + 3*a^2*b^3*d^2)*e)*x^2 + (8*a^2*b
^3*c^2 - 12*a^3*b^2*c*d + 3*a^4*b*d^2)*e + 2*(a^4*b*d^2*f + (8*a*b^4*c^2 - 12*a^2*b^3*c*d + 3*a^3*b^2*d^2)*e)*
x)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) + 2*((6*a^2*
b^4*c^2 - 11*a^3*b^3*c*d + 5*a^4*b^2*d^2)*e - (2*a^3*b^3*c^2 - 3*a^4*b^2*c*d + a^5*b*d^2)*f + ((4*a*b^5*c^2 -
7*a^2*b^4*c*d + 3*a^3*b^3*d^2)*e - (a^3*b^3*c*d - a^4*b^2*d^2)*f)*x)*sqrt(d*x + c))/(a^5*b^4*c^2 - 2*a^6*b^3*c
*d + a^7*b^2*d^2 + (a^3*b^6*c^2 - 2*a^4*b^5*c*d + a^5*b^4*d^2)*x^2 + 2*(a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*
d^2)*x), -1/4*((a^5*d^2*f + (a^3*b^2*d^2*f + (8*b^5*c^2 - 12*a*b^4*c*d + 3*a^2*b^3*d^2)*e)*x^2 + (8*a^2*b^3*c^
2 - 12*a^3*b^2*c*d + 3*a^4*b*d^2)*e + 2*(a^4*b*d^2*f + (8*a*b^4*c^2 - 12*a^2*b^3*c*d + 3*a^3*b^2*d^2)*e)*x)*sq
rt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) - 8*((b^6*c^2 - 2*a*b^5*c*d + a^2*
b^4*d^2)*e*x^2 + 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*e*x + (a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)
*e)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) - ((6*a^2*b^4*c^2 - 11*a^3*b^3*c*d + 5*a^4*b^2*d^2)*e - (2*a^3*b
^3*c^2 - 3*a^4*b^2*c*d + a^5*b*d^2)*f + ((4*a*b^5*c^2 - 7*a^2*b^4*c*d + 3*a^3*b^3*d^2)*e - (a^3*b^3*c*d - a^4*
b^2*d^2)*f)*x)*sqrt(d*x + c))/(a^5*b^4*c^2 - 2*a^6*b^3*c*d + a^7*b^2*d^2 + (a^3*b^6*c^2 - 2*a^4*b^5*c*d + a^5*
b^4*d^2)*x^2 + 2*(a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*x)]

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giac [A]  time = 1.39, size = 300, normalized size = 1.44 \begin {gather*} -\frac {{\left (a^{3} d^{2} f + 8 \, b^{3} c^{2} e - 12 \, a b^{2} c d e + 3 \, a^{2} b d^{2} e\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (a^{3} b^{2} c - a^{4} b d\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, c \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right ) e}{a^{3} \sqrt {-c}} - \frac {{\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{2} f + \sqrt {d x + c} a^{2} b c d^{2} f - \sqrt {d x + c} a^{3} d^{3} f - 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c d e + 4 \, \sqrt {d x + c} b^{3} c^{2} d e + 3 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} d^{2} e - 9 \, \sqrt {d x + c} a b^{2} c d^{2} e + 5 \, \sqrt {d x + c} a^{2} b d^{3} e}{4 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a)^3,x, algorithm="giac")

[Out]

-1/4*(a^3*d^2*f + 8*b^3*c^2*e - 12*a*b^2*c*d*e + 3*a^2*b*d^2*e)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(
(a^3*b^2*c - a^4*b*d)*sqrt(-b^2*c + a*b*d)) + 2*c*arctan(sqrt(d*x + c)/sqrt(-c))*e/(a^3*sqrt(-c)) - 1/4*((d*x
+ c)^(3/2)*a^2*b*d^2*f + sqrt(d*x + c)*a^2*b*c*d^2*f - sqrt(d*x + c)*a^3*d^3*f - 4*(d*x + c)^(3/2)*b^3*c*d*e +
4*sqrt(d*x + c)*b^3*c^2*d*e + 3*(d*x + c)^(3/2)*a*b^2*d^2*e - 9*sqrt(d*x + c)*a*b^2*c*d^2*e + 5*sqrt(d*x + c)
*a^2*b*d^3*e)/((a^2*b^2*c - a^3*b*d)*((d*x + c)*b - b*c + a*d)^2)

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maple [B]  time = 0.02, size = 424, normalized size = 2.04 \begin {gather*} \frac {3 \left (d x +c \right )^{\frac {3}{2}} b \,d^{2} e}{4 \left (b d x +a d \right )^{2} \left (a d -b c \right ) a}+\frac {3 d^{2} e \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{4 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}\, a}-\frac {\left (d x +c \right )^{\frac {3}{2}} b^{2} c d e}{\left (b d x +a d \right )^{2} \left (a d -b c \right ) a^{2}}-\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 d -b c \right ) \sqrt {\left (a d -b c \right ) b}\, a^{2}}+\frac {2 b^{2} c^{2} e \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}\, a^{3}}+\frac {d^{2} f \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{4 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}\, b}+\frac {\left (d x +c \right )^{\frac {3}{2}} d^{2} f}{4 \left (b d x +a d \right )^{2} \left (a d -b c \right )}+\frac {5 \sqrt {d x +c}\, d^{2} e}{4 \left (b d x +a d \right )^{2} a}-\frac {\sqrt {d x +c}\, b c d e}{\left (b d x +a d \right )^{2} a^{2}}-\frac {\sqrt {d x +c}\, d^{2} f}{4 \left (b d x +a d \right )^{2} b}-\frac {2 \sqrt {c}\, e \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a)^3,x)

[Out]

1/4*d^2/(b*d*x+a*d)^2/(a*d-b*c)*(d*x+c)^(3/2)*f+3/4*d^2/a/(b*d*x+a*d)^2/(a*d-b*c)*(d*x+c)^(3/2)*b*e-d/a^2/(b*d
*x+a*d)^2/(a*d-b*c)*(d*x+c)^(3/2)*b^2*c*e-1/4*d^2/(b*d*x+a*d)^2/b*(d*x+c)^(1/2)*f+5/4*d^2/a/(b*d*x+a*d)^2*(d*x
+c)^(1/2)*e-d/a^2/(b*d*x+a*d)^2*b*(d*x+c)^(1/2)*c*e+1/4*d^2/(a*d-b*c)/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/
2)/((a*d-b*c)*b)^(1/2)*b)*f+3/4*d^2/a/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b
)*e-3*d/a^2/(a*d-b*c)*b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*c*e+2/a^3/(a*d-b*c)*b^
2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*c^2*e-2*e*arctanh((d*x+c)^(1/2)/c^(1/2))*c^(
1/2)/a^3

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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((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a)^3,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(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c positive or negative?

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mupad [B]  time = 4.54, size = 4852, normalized size = 23.33

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)*(c + d*x)^(1/2))/(x*(a + b*x)^3),x)

[Out]

(c^(1/2)*e*atan(((c^(1/2)*e*(((c + d*x)^(1/2)*(a^6*d^6*f^2 + 9*a^4*b^2*d^6*e^2 + 128*b^6*c^4*d^2*e^2 + 6*a^5*b
*d^6*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 320*a*b^5*c^3*d^3*e^2 - 72*a^3*b^3*c*d^5*e^2 + 16*a^3*b^3*c^2*d^4*e*f - 2
4*a^4*b^2*c*d^5*e*f))/(8*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d)) + (c^(1/2)*e*((5*a^8*b^3*c*d^5*e - a^9*b^2
*c*d^5*f + 4*a^6*b^5*c^3*d^3*e - 9*a^7*b^4*c^2*d^4*e + a^8*b^3*c^2*d^4*f)/(a^8*b*d^2 + a^6*b^3*c^2 - 2*a^7*b^2
*c*d) + (c^(1/2)*e*(c + d*x)^(1/2)*(64*a^9*b^3*d^5 - 256*a^8*b^4*c*d^4 - 128*a^6*b^6*c^3*d^2 + 320*a^7*b^5*c^2
*d^3))/(8*a^3*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d))))/a^3)*1i)/a^3 + (c^(1/2)*e*(((c + d*x)^(1/2)*(a^6*d^
6*f^2 + 9*a^4*b^2*d^6*e^2 + 128*b^6*c^4*d^2*e^2 + 6*a^5*b*d^6*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 320*a*b^5*c^3*d^
3*e^2 - 72*a^3*b^3*c*d^5*e^2 + 16*a^3*b^3*c^2*d^4*e*f - 24*a^4*b^2*c*d^5*e*f))/(8*(a^6*b*d^2 + a^4*b^3*c^2 - 2
*a^5*b^2*c*d)) - (c^(1/2)*e*((5*a^8*b^3*c*d^5*e - a^9*b^2*c*d^5*f + 4*a^6*b^5*c^3*d^3*e - 9*a^7*b^4*c^2*d^4*e
+ a^8*b^3*c^2*d^4*f)/(a^8*b*d^2 + a^6*b^3*c^2 - 2*a^7*b^2*c*d) - (c^(1/2)*e*(c + d*x)^(1/2)*(64*a^9*b^3*d^5 -
256*a^8*b^4*c*d^4 - 128*a^6*b^6*c^3*d^2 + 320*a^7*b^5*c^2*d^3))/(8*a^3*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*
d))))/a^3)*1i)/a^3)/(((a^5*c*d^6*e*f^2)/4 - 12*a^2*b^3*c^2*d^5*e^3 - 8*b^5*c^4*d^3*e^3 + 18*a*b^4*c^3*d^4*e^3
+ (9*a^3*b^2*c*d^6*e^3)/4 + 2*a^2*b^3*c^3*d^4*e^2*f - 4*a^3*b^2*c^2*d^5*e^2*f + (3*a^4*b*c*d^6*e^2*f)/2)/(a^8*
b*d^2 + a^6*b^3*c^2 - 2*a^7*b^2*c*d) + (c^(1/2)*e*(((c + d*x)^(1/2)*(a^6*d^6*f^2 + 9*a^4*b^2*d^6*e^2 + 128*b^6
*c^4*d^2*e^2 + 6*a^5*b*d^6*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 320*a*b^5*c^3*d^3*e^2 - 72*a^3*b^3*c*d^5*e^2 + 16*a
^3*b^3*c^2*d^4*e*f - 24*a^4*b^2*c*d^5*e*f))/(8*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d)) + (c^(1/2)*e*((5*a^8
*b^3*c*d^5*e - a^9*b^2*c*d^5*f + 4*a^6*b^5*c^3*d^3*e - 9*a^7*b^4*c^2*d^4*e + a^8*b^3*c^2*d^4*f)/(a^8*b*d^2 + a
^6*b^3*c^2 - 2*a^7*b^2*c*d) + (c^(1/2)*e*(c + d*x)^(1/2)*(64*a^9*b^3*d^5 - 256*a^8*b^4*c*d^4 - 128*a^6*b^6*c^3
*d^2 + 320*a^7*b^5*c^2*d^3))/(8*a^3*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d))))/a^3))/a^3 - (c^(1/2)*e*(((c +
d*x)^(1/2)*(a^6*d^6*f^2 + 9*a^4*b^2*d^6*e^2 + 128*b^6*c^4*d^2*e^2 + 6*a^5*b*d^6*e*f + 256*a^2*b^4*c^2*d^4*e^2
- 320*a*b^5*c^3*d^3*e^2 - 72*a^3*b^3*c*d^5*e^2 + 16*a^3*b^3*c^2*d^4*e*f - 24*a^4*b^2*c*d^5*e*f))/(8*(a^6*b*d^
2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d)) - (c^(1/2)*e*((5*a^8*b^3*c*d^5*e - a^9*b^2*c*d^5*f + 4*a^6*b^5*c^3*d^3*e - 9
*a^7*b^4*c^2*d^4*e + a^8*b^3*c^2*d^4*f)/(a^8*b*d^2 + a^6*b^3*c^2 - 2*a^7*b^2*c*d) - (c^(1/2)*e*(c + d*x)^(1/2)
*(64*a^9*b^3*d^5 - 256*a^8*b^4*c*d^4 - 128*a^6*b^6*c^3*d^2 + 320*a^7*b^5*c^2*d^3))/(8*a^3*(a^6*b*d^2 + a^4*b^3
*c^2 - 2*a^5*b^2*c*d))))/a^3))/a^3))*2i)/a^3 - (((c + d*x)^(1/2)*(a^2*d^2*f - 5*a*b*d^2*e + 4*b^2*c*d*e))/(4*a
^2*b) - ((c + d*x)^(3/2)*(a^2*d^2*f + 3*a*b*d^2*e - 4*b^2*c*d*e))/(4*a^2*(a*d - b*c)))/(b^2*(c + d*x)^2 - (2*b
^2*c - 2*a*b*d)*(c + d*x) + a^2*d^2 + b^2*c^2 - 2*a*b*c*d) + (atan((((-b^3*(a*d - b*c)^3)^(1/2)*(((c + d*x)^(1
/2)*(a^6*d^6*f^2 + 9*a^4*b^2*d^6*e^2 + 128*b^6*c^4*d^2*e^2 + 6*a^5*b*d^6*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 320*a
*b^5*c^3*d^3*e^2 - 72*a^3*b^3*c*d^5*e^2 + 16*a^3*b^3*c^2*d^4*e*f - 24*a^4*b^2*c*d^5*e*f))/(8*(a^6*b*d^2 + a^4*
b^3*c^2 - 2*a^5*b^2*c*d)) - ((-b^3*(a*d - b*c)^3)^(1/2)*((5*a^8*b^3*c*d^5*e - a^9*b^2*c*d^5*f + 4*a^6*b^5*c^3*
d^3*e - 9*a^7*b^4*c^2*d^4*e + a^8*b^3*c^2*d^4*f)/(a^8*b*d^2 + a^6*b^3*c^2 - 2*a^7*b^2*c*d) - ((-b^3*(a*d - b*c
)^3)^(1/2)*(c + d*x)^(1/2)*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b^2*c*d*e)*(64*a^9*b^3*d^5 - 256*a^
8*b^4*c*d^4 - 128*a^6*b^6*c^3*d^2 + 320*a^7*b^5*c^2*d^3))/(64*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d)*(a^3*b
^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)))*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b^
2*c*d*e))/(8*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)))*(8*b^3*c^2*e + a^3*d^2*f + 3*a^
2*b*d^2*e - 12*a*b^2*c*d*e)*1i)/(8*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)) + ((-b^3*(
a*d - b*c)^3)^(1/2)*(((c + d*x)^(1/2)*(a^6*d^6*f^2 + 9*a^4*b^2*d^6*e^2 + 128*b^6*c^4*d^2*e^2 + 6*a^5*b*d^6*e*f
+ 256*a^2*b^4*c^2*d^4*e^2 - 320*a*b^5*c^3*d^3*e^2 - 72*a^3*b^3*c*d^5*e^2 + 16*a^3*b^3*c^2*d^4*e*f - 24*a^4*b^
2*c*d^5*e*f))/(8*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d)) + ((-b^3*(a*d - b*c)^3)^(1/2)*((5*a^8*b^3*c*d^5*e
- a^9*b^2*c*d^5*f + 4*a^6*b^5*c^3*d^3*e - 9*a^7*b^4*c^2*d^4*e + a^8*b^3*c^2*d^4*f)/(a^8*b*d^2 + a^6*b^3*c^2 -
2*a^7*b^2*c*d) + ((-b^3*(a*d - b*c)^3)^(1/2)*(c + d*x)^(1/2)*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b
^2*c*d*e)*(64*a^9*b^3*d^5 - 256*a^8*b^4*c*d^4 - 128*a^6*b^6*c^3*d^2 + 320*a^7*b^5*c^2*d^3))/(64*(a^6*b*d^2 + a
^4*b^3*c^2 - 2*a^5*b^2*c*d)*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)))*(8*b^3*c^2*e + a
^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b^2*c*d*e))/(8*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)
))*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b^2*c*d*e)*1i)/(8*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^
2*d + 3*a^5*b^4*c*d^2)))/(((a^5*c*d^6*e*f^2)/4 - 12*a^2*b^3*c^2*d^5*e^3 - 8*b^5*c^4*d^3*e^3 + 18*a*b^4*c^3*d^4
*e^3 + (9*a^3*b^2*c*d^6*e^3)/4 + 2*a^2*b^3*c^3*d^4*e^2*f - 4*a^3*b^2*c^2*d^5*e^2*f + (3*a^4*b*c*d^6*e^2*f)/2)/
(a^8*b*d^2 + a^6*b^3*c^2 - 2*a^7*b^2*c*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*(((c + d*x)^(1/2)*(a^6*d^6*f^2 + 9*a^4
*b^2*d^6*e^2 + 128*b^6*c^4*d^2*e^2 + 6*a^5*b*d^6*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 320*a*b^5*c^3*d^3*e^2 - 72*a^
3*b^3*c*d^5*e^2 + 16*a^3*b^3*c^2*d^4*e*f - 24*a^4*b^2*c*d^5*e*f))/(8*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d)
) - ((-b^3*(a*d - b*c)^3)^(1/2)*((5*a^8*b^3*c*d^5*e - a^9*b^2*c*d^5*f + 4*a^6*b^5*c^3*d^3*e - 9*a^7*b^4*c^2*d^
4*e + a^8*b^3*c^2*d^4*f)/(a^8*b*d^2 + a^6*b^3*c^2 - 2*a^7*b^2*c*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*(c + d*x)^(1/
2)*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b^2*c*d*e)*(64*a^9*b^3*d^5 - 256*a^8*b^4*c*d^4 - 128*a^6*b^
6*c^3*d^2 + 320*a^7*b^5*c^2*d^3))/(64*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d)*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3
*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)))*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b^2*c*d*e))/(8*(a^3*b^6*c^
3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)))*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b^2*c*d
*e))/(8*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)) + ((-b^3*(a*d - b*c)^3)^(1/2)*(((c +
d*x)^(1/2)*(a^6*d^6*f^2 + 9*a^4*b^2*d^6*e^2 + 128*b^6*c^4*d^2*e^2 + 6*a^5*b*d^6*e*f + 256*a^2*b^4*c^2*d^4*e^2
- 320*a*b^5*c^3*d^3*e^2 - 72*a^3*b^3*c*d^5*e^2 + 16*a^3*b^3*c^2*d^4*e*f - 24*a^4*b^2*c*d^5*e*f))/(8*(a^6*b*d^2
+ a^4*b^3*c^2 - 2*a^5*b^2*c*d)) + ((-b^3*(a*d - b*c)^3)^(1/2)*((5*a^8*b^3*c*d^5*e - a^9*b^2*c*d^5*f + 4*a^6*b
^5*c^3*d^3*e - 9*a^7*b^4*c^2*d^4*e + a^8*b^3*c^2*d^4*f)/(a^8*b*d^2 + a^6*b^3*c^2 - 2*a^7*b^2*c*d) + ((-b^3*(a*
d - b*c)^3)^(1/2)*(c + d*x)^(1/2)*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b^2*c*d*e)*(64*a^9*b^3*d^5 -
256*a^8*b^4*c*d^4 - 128*a^6*b^6*c^3*d^2 + 320*a^7*b^5*c^2*d^3))/(64*(a^6*b*d^2 + a^4*b^3*c^2 - 2*a^5*b^2*c*d)
*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)))*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e -
12*a*b^2*c*d*e))/(8*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2)))*(8*b^3*c^2*e + a^3*d^2*f
+ 3*a^2*b*d^2*e - 12*a*b^2*c*d*e))/(8*(a^3*b^6*c^3 - a^6*b^3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2))))*(-b^
3*(a*d - b*c)^3)^(1/2)*(8*b^3*c^2*e + a^3*d^2*f + 3*a^2*b*d^2*e - 12*a*b^2*c*d*e)*1i)/(4*(a^3*b^6*c^3 - a^6*b^
3*d^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*(d*x+c)**(1/2)/x/(b*x+a)**3,x)

[Out]

Timed out

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