3.696 \(\int \frac{(a+b x)^{5/2}}{x^3 (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=220 \[ -\frac{(a+b x)^{5/2} (3 b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}+\frac{5 (a+b x)^{3/2} (3 b c-7 a d) (b c-a d)}{12 a c^3 (c+d x)^{3/2}}+\frac{5 \sqrt{a+b x} (3 b c-7 a d) (b c-a d)}{4 c^4 \sqrt{c+d x}}-\frac{5 \sqrt{a} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 c^{9/2}}-\frac{(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}} \]

[Out]

(5*(3*b*c - 7*a*d)*(b*c - a*d)*(a + b*x)^(3/2))/(12*a*c^3*(c + d*x)^(3/2)) - ((3*b*c - 7*a*d)*(a + b*x)^(5/2))
/(4*a*c^2*x*(c + d*x)^(3/2)) - (a + b*x)^(7/2)/(2*a*c*x^2*(c + d*x)^(3/2)) + (5*(3*b*c - 7*a*d)*(b*c - a*d)*Sq
rt[a + b*x])/(4*c^4*Sqrt[c + d*x]) - (5*Sqrt[a]*(3*b*c - 7*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(4*c^(9/2))

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Rubi [A]  time = 0.106627, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \[ -\frac{(a+b x)^{5/2} (3 b c-7 a d)}{4 a c^2 x (c+d x)^{3/2}}+\frac{5 (a+b x)^{3/2} (3 b c-7 a d) (b c-a d)}{12 a c^3 (c+d x)^{3/2}}+\frac{5 \sqrt{a+b x} (3 b c-7 a d) (b c-a d)}{4 c^4 \sqrt{c+d x}}-\frac{5 \sqrt{a} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 c^{9/2}}-\frac{(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/2)/(x^3*(c + d*x)^(5/2)),x]

[Out]

(5*(3*b*c - 7*a*d)*(b*c - a*d)*(a + b*x)^(3/2))/(12*a*c^3*(c + d*x)^(3/2)) - ((3*b*c - 7*a*d)*(a + b*x)^(5/2))
/(4*a*c^2*x*(c + d*x)^(3/2)) - (a + b*x)^(7/2)/(2*a*c*x^2*(c + d*x)^(3/2)) + (5*(3*b*c - 7*a*d)*(b*c - a*d)*Sq
rt[a + b*x])/(4*c^4*Sqrt[c + d*x]) - (5*Sqrt[a]*(3*b*c - 7*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(4*c^(9/2))

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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)^{5/2}}{x^3 (c+d x)^{5/2}} \, dx &=-\frac{(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}-\frac{\left (-\frac{3 b c}{2}+\frac{7 a d}{2}\right ) \int \frac{(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx}{2 a c}\\ &=-\frac{(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac{(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac{(5 (3 b c-7 a d) (b c-a d)) \int \frac{(a+b x)^{3/2}}{x (c+d x)^{5/2}} \, dx}{8 a c^2}\\ &=\frac{5 (3 b c-7 a d) (b c-a d) (a+b x)^{3/2}}{12 a c^3 (c+d x)^{3/2}}-\frac{(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac{(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac{(5 (3 b c-7 a d) (b c-a d)) \int \frac{\sqrt{a+b x}}{x (c+d x)^{3/2}} \, dx}{8 c^3}\\ &=\frac{5 (3 b c-7 a d) (b c-a d) (a+b x)^{3/2}}{12 a c^3 (c+d x)^{3/2}}-\frac{(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac{(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac{5 (3 b c-7 a d) (b c-a d) \sqrt{a+b x}}{4 c^4 \sqrt{c+d x}}+\frac{(5 a (3 b c-7 a d) (b c-a d)) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 c^4}\\ &=\frac{5 (3 b c-7 a d) (b c-a d) (a+b x)^{3/2}}{12 a c^3 (c+d x)^{3/2}}-\frac{(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac{(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac{5 (3 b c-7 a d) (b c-a d) \sqrt{a+b x}}{4 c^4 \sqrt{c+d x}}+\frac{(5 a (3 b c-7 a d) (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{4 c^4}\\ &=\frac{5 (3 b c-7 a d) (b c-a d) (a+b x)^{3/2}}{12 a c^3 (c+d x)^{3/2}}-\frac{(3 b c-7 a d) (a+b x)^{5/2}}{4 a c^2 x (c+d x)^{3/2}}-\frac{(a+b x)^{7/2}}{2 a c x^2 (c+d x)^{3/2}}+\frac{5 (3 b c-7 a d) (b c-a d) \sqrt{a+b x}}{4 c^4 \sqrt{c+d x}}-\frac{5 \sqrt{a} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.265011, size = 159, normalized size = 0.72 \[ \frac{-\frac{1}{2} x (3 b c-7 a d) \left (3 c^{5/2} (a+b x)^{5/2}-5 x (b c-a d) \left (\sqrt{c} \sqrt{a+b x} (4 a c+3 a d x+b c x)-3 a^{3/2} (c+d x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )\right )-3 c^{7/2} (a+b x)^{7/2}}{6 a c^{9/2} x^2 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/2)/(x^3*(c + d*x)^(5/2)),x]

[Out]

(-3*c^(7/2)*(a + b*x)^(7/2) - ((3*b*c - 7*a*d)*x*(3*c^(5/2)*(a + b*x)^(5/2) - 5*(b*c - a*d)*x*(Sqrt[c]*Sqrt[a
+ b*x]*(4*a*c + b*c*x + 3*a*d*x) - 3*a^(3/2)*(c + d*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c +
d*x])])))/2)/(6*a*c^(9/2)*x^2*(c + d*x)^(3/2))

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Maple [B]  time = 0.028, size = 758, normalized size = 3.5 \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)^(5/2)/x^3/(d*x+c)^(5/2),x)

[Out]

-1/24*(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^4-150*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+45*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+210*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^3-300*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+90
*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+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^2*a^3*c^2*d^2-150*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^2*a^2*b*c^3*d+45*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^2*a*b^2*c^4-
210*x^3*a^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+230*x^3*a*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-32*x
^3*b^2*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-280*x^2*a^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+316*x^2
*a*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-48*x^2*b^2*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)-42*x*a^2*c^2
*d*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+54*x*a*b*c^3*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)+12*a^2*c^3*((b*x+a)*(d
*x+c))^(1/2)*(a*c)^(1/2))/c^4/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/x^2/(d*x+c)^(3/2)

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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)^(5/2)/x^3/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 23.6324, size = 1422, normalized size = 6.46 \begin{align*} \left [\frac{15 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (3 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{4} - 10 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt{\frac{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^{2} +{\left (b c^{2} + a c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a}{c}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \,{\left (6 \, a^{2} c^{3} -{\left (16 \, b^{2} c^{2} d - 115 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} - 2 \,{\left (12 \, b^{2} c^{3} - 79 \, a b c^{2} d + 70 \, a^{2} c d^{2}\right )} x^{2} + 3 \,{\left (9 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (c^{4} d^{2} x^{4} + 2 \, c^{5} d x^{3} + c^{6} x^{2}\right )}}, \frac{15 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (3 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} x^{3} +{\left (3 \, b^{2} c^{4} - 10 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt{-\frac{a}{c}} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{a}{c}}}{2 \,{\left (a b d x^{2} + a^{2} c +{\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \,{\left (6 \, a^{2} c^{3} -{\left (16 \, b^{2} c^{2} d - 115 \, a b c d^{2} + 105 \, a^{2} d^{3}\right )} x^{3} - 2 \,{\left (12 \, b^{2} c^{3} - 79 \, a b c^{2} d + 70 \, a^{2} c d^{2}\right )} x^{2} + 3 \,{\left (9 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (c^{4} d^{2} x^{4} + 2 \, c^{5} d x^{3} + c^{6} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)/x^3/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(15*((3*b^2*c^2*d^2 - 10*a*b*c*d^3 + 7*a^2*d^4)*x^4 + 2*(3*b^2*c^3*d - 10*a*b*c^2*d^2 + 7*a^2*c*d^3)*x^3
+ (3*b^2*c^4 - 10*a*b*c^3*d + 7*a^2*c^2*d^2)*x^2)*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^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) -
4*(6*a^2*c^3 - (16*b^2*c^2*d - 115*a*b*c*d^2 + 105*a^2*d^3)*x^3 - 2*(12*b^2*c^3 - 79*a*b*c^2*d + 70*a^2*c*d^2)
*x^2 + 3*(9*a*b*c^3 - 7*a^2*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^4*d^2*x^4 + 2*c^5*d*x^3 + c^6*x^2), 1/24
*(15*((3*b^2*c^2*d^2 - 10*a*b*c*d^3 + 7*a^2*d^4)*x^4 + 2*(3*b^2*c^3*d - 10*a*b*c^2*d^2 + 7*a^2*c*d^3)*x^3 + (3
*b^2*c^4 - 10*a*b*c^3*d + 7*a^2*c^2*d^2)*x^2)*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt
(d*x + c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - 2*(6*a^2*c^3 - (16*b^2*c^2*d - 115*a*b*c*d^2 +
105*a^2*d^3)*x^3 - 2*(12*b^2*c^3 - 79*a*b*c^2*d + 70*a^2*c*d^2)*x^2 + 3*(9*a*b*c^3 - 7*a^2*c^2*d)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(c^4*d^2*x^4 + 2*c^5*d*x^3 + c^6*x^2)]

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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)**(5/2)/x**3/(d*x+c)**(5/2),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)^(5/2)/x^3/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError