3.640 \(\int \frac{(a+b x)^{3/2}}{x^4 (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=240 \[ -\frac{d \sqrt{a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{24 a c^4 \sqrt{c+d x}}+\frac{(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{9/2}}-\frac{7 \sqrt{a+b x} (b c-a d)}{12 c^2 x^2 \sqrt{c+d x}}-\frac{\sqrt{a+b x} (3 b c-35 a d) (b c-a d)}{24 a c^3 x \sqrt{c+d x}}-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}} \]

[Out]

-(d*(3*b^2*c^2 - 100*a*b*c*d + 105*a^2*d^2)*Sqrt[a + b*x])/(24*a*c^4*Sqrt[c + d*x]) - (a*Sqrt[a + b*x])/(3*c*x
^3*Sqrt[c + d*x]) - (7*(b*c - a*d)*Sqrt[a + b*x])/(12*c^2*x^2*Sqrt[c + d*x]) - ((3*b*c - 35*a*d)*(b*c - a*d)*S
qrt[a + b*x])/(24*a*c^3*x*Sqrt[c + d*x]) + ((b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(3/2)*c^(9/2))

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Rubi [A]  time = 0.218044, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {98, 151, 152, 12, 93, 208} \[ -\frac{d \sqrt{a+b x} \left (105 a^2 d^2-100 a b c d+3 b^2 c^2\right )}{24 a c^4 \sqrt{c+d x}}+\frac{(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{9/2}}-\frac{7 \sqrt{a+b x} (b c-a d)}{12 c^2 x^2 \sqrt{c+d x}}-\frac{\sqrt{a+b x} (3 b c-35 a d) (b c-a d)}{24 a c^3 x \sqrt{c+d x}}-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(3/2)/(x^4*(c + d*x)^(3/2)),x]

[Out]

-(d*(3*b^2*c^2 - 100*a*b*c*d + 105*a^2*d^2)*Sqrt[a + b*x])/(24*a*c^4*Sqrt[c + d*x]) - (a*Sqrt[a + b*x])/(3*c*x
^3*Sqrt[c + d*x]) - (7*(b*c - a*d)*Sqrt[a + b*x])/(12*c^2*x^2*Sqrt[c + d*x]) - ((3*b*c - 35*a*d)*(b*c - a*d)*S
qrt[a + b*x])/(24*a*c^3*x*Sqrt[c + d*x]) + ((b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*ArcTanh[(Sqrt[c]*S
qrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(3/2)*c^(9/2))

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 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 152

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] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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{(a+b x)^{3/2}}{x^4 (c+d x)^{3/2}} \, dx &=-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}}-\frac{\int \frac{-\frac{7}{2} a (b c-a d)-3 b (b c-a d) x}{x^3 \sqrt{a+b x} (c+d x)^{3/2}} \, dx}{3 c}\\ &=-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}}-\frac{7 (b c-a d) \sqrt{a+b x}}{12 c^2 x^2 \sqrt{c+d x}}+\frac{\int \frac{\frac{1}{4} a (3 b c-35 a d) (b c-a d)-7 a b d (b c-a d) x}{x^2 \sqrt{a+b x} (c+d x)^{3/2}} \, dx}{6 a c^2}\\ &=-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}}-\frac{7 (b c-a d) \sqrt{a+b x}}{12 c^2 x^2 \sqrt{c+d x}}-\frac{(3 b c-35 a d) (b c-a d) \sqrt{a+b x}}{24 a c^3 x \sqrt{c+d x}}-\frac{\int \frac{\frac{3}{8} a (b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )+\frac{1}{4} a b d (3 b c-35 a d) (b c-a d) x}{x \sqrt{a+b x} (c+d x)^{3/2}} \, dx}{6 a^2 c^3}\\ &=-\frac{d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt{a+b x}}{24 a c^4 \sqrt{c+d x}}-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}}-\frac{7 (b c-a d) \sqrt{a+b x}}{12 c^2 x^2 \sqrt{c+d x}}-\frac{(3 b c-35 a d) (b c-a d) \sqrt{a+b x}}{24 a c^3 x \sqrt{c+d x}}+\frac{\int -\frac{3 a (b c-a d)^2 \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )}{16 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 a^2 c^4 (b c-a d)}\\ &=-\frac{d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt{a+b x}}{24 a c^4 \sqrt{c+d x}}-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}}-\frac{7 (b c-a d) \sqrt{a+b x}}{12 c^2 x^2 \sqrt{c+d x}}-\frac{(3 b c-35 a d) (b c-a d) \sqrt{a+b x}}{24 a c^3 x \sqrt{c+d x}}-\frac{\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 a c^4}\\ &=-\frac{d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt{a+b x}}{24 a c^4 \sqrt{c+d x}}-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}}-\frac{7 (b c-a d) \sqrt{a+b x}}{12 c^2 x^2 \sqrt{c+d x}}-\frac{(3 b c-35 a d) (b c-a d) \sqrt{a+b x}}{24 a c^3 x \sqrt{c+d x}}-\frac{\left ((b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{8 a c^4}\\ &=-\frac{d \left (3 b^2 c^2-100 a b c d+105 a^2 d^2\right ) \sqrt{a+b x}}{24 a c^4 \sqrt{c+d x}}-\frac{a \sqrt{a+b x}}{3 c x^3 \sqrt{c+d x}}-\frac{7 (b c-a d) \sqrt{a+b x}}{12 c^2 x^2 \sqrt{c+d x}}-\frac{(3 b c-35 a d) (b c-a d) \sqrt{a+b x}}{24 a c^3 x \sqrt{c+d x}}+\frac{(b c-a d) \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.147545, size = 188, normalized size = 0.78 \[ \frac{\left (-45 a^2 b c d^2+35 a^3 d^3+9 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} c^{9/2}}-\frac{\sqrt{a+b x} \left (a^2 \left (-14 c^2 d x+8 c^3+35 c d^2 x^2+105 d^3 x^3\right )+2 a b c x \left (7 c^2-19 c d x-50 d^2 x^2\right )+3 b^2 c^2 x^2 (c+d x)\right )}{24 a c^4 x^3 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(3/2)/(x^4*(c + d*x)^(3/2)),x]

[Out]

-(Sqrt[a + b*x]*(3*b^2*c^2*x^2*(c + d*x) + 2*a*b*c*x*(7*c^2 - 19*c*d*x - 50*d^2*x^2) + a^2*(8*c^3 - 14*c^2*d*x
+ 35*c*d^2*x^2 + 105*d^3*x^3)))/(24*a*c^4*x^3*Sqrt[c + d*x]) + ((b^3*c^3 + 9*a*b^2*c^2*d - 45*a^2*b*c*d^2 + 3
5*a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(3/2)*c^(9/2))

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Maple [B]  time = 0.027, size = 707, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)/x^4/(d*x+c)^(3/2),x)

[Out]

1/48*(b*x+a)^(1/2)*(105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^3*d^4-135*ln((a*
d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a^2*b*c*d^3+27*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^4*a*b^2*c^2*d^2+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c
)/x)*x^4*b^3*c^3*d+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^3*c*d^3-135*ln((a
*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a^2*b*c^2*d^2+27*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*
((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*x^3*a*b^2*c^3*d+3*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*
c)/x)*x^3*b^3*c^4-210*x^3*a^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+200*x^3*a*b*c*d^2*((b*x+a)*(d*x+c))^(1/2
)*(a*c)^(1/2)-6*x^3*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-70*x^2*a^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*
c)^(1/2)+76*x^2*a*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-6*x^2*b^2*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2
)+28*x*a^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-28*x*a*b*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-16*a^2*c
^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2))/c^4/a/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/x^3/(d*x+c)^(1/2)

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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((b*x+a)^(3/2)/x^4/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 23.9515, size = 1374, normalized size = 5.72 \begin{align*} \left [\frac{3 \,{\left ({\left (b^{3} c^{3} d + 9 \, a b^{2} c^{2} d^{2} - 45 \, a^{2} b c d^{3} + 35 \, a^{3} d^{4}\right )} x^{4} +{\left (b^{3} c^{4} + 9 \, a b^{2} c^{3} d - 45 \, a^{2} b c^{2} d^{2} + 35 \, a^{3} c d^{3}\right )} x^{3}\right )} \sqrt{a c} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \,{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \,{\left (8 \, a^{3} c^{4} +{\left (3 \, a b^{2} c^{3} d - 100 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3}\right )} x^{3} +{\left (3 \, a b^{2} c^{4} - 38 \, a^{2} b c^{3} d + 35 \, a^{3} c^{2} d^{2}\right )} x^{2} + 14 \,{\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \,{\left (a^{2} c^{5} d x^{4} + a^{2} c^{6} x^{3}\right )}}, -\frac{3 \,{\left ({\left (b^{3} c^{3} d + 9 \, a b^{2} c^{2} d^{2} - 45 \, a^{2} b c d^{3} + 35 \, a^{3} d^{4}\right )} x^{4} +{\left (b^{3} c^{4} + 9 \, a b^{2} c^{3} d - 45 \, a^{2} b c^{2} d^{2} + 35 \, a^{3} c d^{3}\right )} x^{3}\right )} \sqrt{-a c} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (a b c d x^{2} + a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \,{\left (8 \, a^{3} c^{4} +{\left (3 \, a b^{2} c^{3} d - 100 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3}\right )} x^{3} +{\left (3 \, a b^{2} c^{4} - 38 \, a^{2} b c^{3} d + 35 \, a^{3} c^{2} d^{2}\right )} x^{2} + 14 \,{\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (a^{2} c^{5} d x^{4} + a^{2} c^{6} x^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)/x^4/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

[1/96*(3*((b^3*c^3*d + 9*a*b^2*c^2*d^2 - 45*a^2*b*c*d^3 + 35*a^3*d^4)*x^4 + (b^3*c^4 + 9*a*b^2*c^3*d - 45*a^2*
b*c^2*d^2 + 35*a^3*c*d^3)*x^3)*sqrt(a*c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b*
c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(8*a^3*c^4 + (3*a*b^2*c^
3*d - 100*a^2*b*c^2*d^2 + 105*a^3*c*d^3)*x^3 + (3*a*b^2*c^4 - 38*a^2*b*c^3*d + 35*a^3*c^2*d^2)*x^2 + 14*(a^2*b
*c^4 - a^3*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^5*d*x^4 + a^2*c^6*x^3), -1/48*(3*((b^3*c^3*d + 9*a*b^
2*c^2*d^2 - 45*a^2*b*c*d^3 + 35*a^3*d^4)*x^4 + (b^3*c^4 + 9*a*b^2*c^3*d - 45*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*x^3
)*sqrt(-a*c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2
+ (a*b*c^2 + a^2*c*d)*x)) + 2*(8*a^3*c^4 + (3*a*b^2*c^3*d - 100*a^2*b*c^2*d^2 + 105*a^3*c*d^3)*x^3 + (3*a*b^2*
c^4 - 38*a^2*b*c^3*d + 35*a^3*c^2*d^2)*x^2 + 14*(a^2*b*c^4 - a^3*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c
^5*d*x^4 + a^2*c^6*x^3)]

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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((b*x+a)**(3/2)/x**4/(d*x+c)**(3/2),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((b*x+a)^(3/2)/x^4/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError