3.774 \(\int \frac{(c+d x)^{5/2}}{x^4 (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=230 \[ \frac{(c+d x)^{5/2} (7 b c-a d)}{12 a^2 c x^2 \sqrt{a+b x}}-\frac{5 (c+d x)^{3/2} (b c-a d) (7 b c-a d)}{24 a^3 c x \sqrt{a+b x}}-\frac{5 \sqrt{c+d x} (b c-a d)^2 (7 b c-a d)}{8 a^4 c \sqrt{a+b x}}+\frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} \sqrt{c}}-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}} \]

[Out]

(-5*(b*c - a*d)^2*(7*b*c - a*d)*Sqrt[c + d*x])/(8*a^4*c*Sqrt[a + b*x]) - (5*(b*c - a*d)*(7*b*c - a*d)*(c + d*x
)^(3/2))/(24*a^3*c*x*Sqrt[a + b*x]) + ((7*b*c - a*d)*(c + d*x)^(5/2))/(12*a^2*c*x^2*Sqrt[a + b*x]) - (c + d*x)
^(7/2)/(3*a*c*x^3*Sqrt[a + b*x]) + (5*(b*c - a*d)^2*(7*b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqr
t[c + d*x])])/(8*a^(9/2)*Sqrt[c])

________________________________________________________________________________________

Rubi [A]  time = 0.113387, antiderivative size = 230, 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} \[ \frac{(c+d x)^{5/2} (7 b c-a d)}{12 a^2 c x^2 \sqrt{a+b x}}-\frac{5 (c+d x)^{3/2} (b c-a d) (7 b c-a d)}{24 a^3 c x \sqrt{a+b x}}-\frac{5 \sqrt{c+d x} (b c-a d)^2 (7 b c-a d)}{8 a^4 c \sqrt{a+b x}}+\frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} \sqrt{c}}-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2)/(x^4*(a + b*x)^(3/2)),x]

[Out]

(-5*(b*c - a*d)^2*(7*b*c - a*d)*Sqrt[c + d*x])/(8*a^4*c*Sqrt[a + b*x]) - (5*(b*c - a*d)*(7*b*c - a*d)*(c + d*x
)^(3/2))/(24*a^3*c*x*Sqrt[a + b*x]) + ((7*b*c - a*d)*(c + d*x)^(5/2))/(12*a^2*c*x^2*Sqrt[a + b*x]) - (c + d*x)
^(7/2)/(3*a*c*x^3*Sqrt[a + b*x]) + (5*(b*c - a*d)^2*(7*b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqr
t[c + d*x])])/(8*a^(9/2)*Sqrt[c])

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{(c+d x)^{5/2}}{x^4 (a+b x)^{3/2}} \, dx &=-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}}-\frac{\left (\frac{7 b c}{2}-\frac{a d}{2}\right ) \int \frac{(c+d x)^{5/2}}{x^3 (a+b x)^{3/2}} \, dx}{3 a c}\\ &=\frac{(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt{a+b x}}-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}}+\frac{(5 (b c-a d) (7 b c-a d)) \int \frac{(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx}{24 a^2 c}\\ &=-\frac{5 (b c-a d) (7 b c-a d) (c+d x)^{3/2}}{24 a^3 c x \sqrt{a+b x}}+\frac{(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt{a+b x}}-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}}-\frac{\left (5 (b c-a d)^2 (7 b c-a d)\right ) \int \frac{\sqrt{c+d x}}{x (a+b x)^{3/2}} \, dx}{16 a^3 c}\\ &=-\frac{5 (b c-a d)^2 (7 b c-a d) \sqrt{c+d x}}{8 a^4 c \sqrt{a+b x}}-\frac{5 (b c-a d) (7 b c-a d) (c+d x)^{3/2}}{24 a^3 c x \sqrt{a+b x}}+\frac{(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt{a+b x}}-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}}-\frac{\left (5 (b c-a d)^2 (7 b c-a d)\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 a^4}\\ &=-\frac{5 (b c-a d)^2 (7 b c-a d) \sqrt{c+d x}}{8 a^4 c \sqrt{a+b x}}-\frac{5 (b c-a d) (7 b c-a d) (c+d x)^{3/2}}{24 a^3 c x \sqrt{a+b x}}+\frac{(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt{a+b x}}-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}}-\frac{\left (5 (b c-a d)^2 (7 b c-a d)\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^4}\\ &=-\frac{5 (b c-a d)^2 (7 b c-a d) \sqrt{c+d x}}{8 a^4 c \sqrt{a+b x}}-\frac{5 (b c-a d) (7 b c-a d) (c+d x)^{3/2}}{24 a^3 c x \sqrt{a+b x}}+\frac{(7 b c-a d) (c+d x)^{5/2}}{12 a^2 c x^2 \sqrt{a+b x}}-\frac{(c+d x)^{7/2}}{3 a c x^3 \sqrt{a+b x}}+\frac{5 (b c-a d)^2 (7 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{9/2} \sqrt{c}}\\ \end{align*}

Mathematica [A]  time = 0.278389, size = 168, normalized size = 0.73 \[ \frac{\frac{1}{2} x (7 b c-a d) \left (2 a^{5/2} (c+d x)^{5/2}-5 x (b c-a d) \left (\sqrt{a} \sqrt{c+d x} (a (c-2 d x)+3 b c x)-3 \sqrt{c} x \sqrt{a+b x} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )\right )-4 a^{7/2} (c+d x)^{7/2}}{12 a^{9/2} c x^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(5/2)/(x^4*(a + b*x)^(3/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.026, size = 704, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(5/2)/x^4/(b*x+a)^(3/2),x)

[Out]

-1/48*(d*x+c)^(1/2)*(15*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*b*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^4*a^2*b^2*c*d^2+225*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^3*c^2*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^4*b^4*c^3+15*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^4*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^3*b*c*d^2+225*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^2*c^2*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*b^3*c^3+162*x^3*a^2*b*d^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-380*x^3*a*b^2*c*d*(a*c)^(1/2)*((b
*x+a)*(d*x+c))^(1/2)+210*x^3*b^3*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+66*x^2*a^3*d^2*(a*c)^(1/2)*((b*x+a)*(
d*x+c))^(1/2)-136*x^2*a^2*b*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+70*x^2*a*b^2*c^2*(a*c)^(1/2)*((b*x+a)*(d*x
+c))^(1/2)+52*x*a^3*c*d*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-28*x*a^2*b*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)
+16*a^3*c^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/a^4/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/x^3/(b*x+a)^(1/2)

________________________________________________________________________________________

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((d*x+c)^(5/2)/x^4/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 19.0879, size = 1378, normalized size = 5.99 \begin{align*} \left [-\frac{15 \,{\left ({\left (7 \, b^{4} c^{3} - 15 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} +{\left (7 \, a b^{3} c^{3} - 15 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} 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^{4} c^{3} +{\left (105 \, a b^{3} c^{3} - 190 \, a^{2} b^{2} c^{2} d + 81 \, a^{3} b c d^{2}\right )} x^{3} +{\left (35 \, a^{2} b^{2} c^{3} - 68 \, a^{3} b c^{2} d + 33 \, a^{4} c d^{2}\right )} x^{2} - 2 \,{\left (7 \, a^{3} b c^{3} - 13 \, a^{4} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \,{\left (a^{5} b c x^{4} + a^{6} c x^{3}\right )}}, -\frac{15 \,{\left ({\left (7 \, b^{4} c^{3} - 15 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} +{\left (7 \, a b^{3} c^{3} - 15 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} 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^{4} c^{3} +{\left (105 \, a b^{3} c^{3} - 190 \, a^{2} b^{2} c^{2} d + 81 \, a^{3} b c d^{2}\right )} x^{3} +{\left (35 \, a^{2} b^{2} c^{3} - 68 \, a^{3} b c^{2} d + 33 \, a^{4} c d^{2}\right )} x^{2} - 2 \,{\left (7 \, a^{3} b c^{3} - 13 \, a^{4} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (a^{5} b c x^{4} + a^{6} c x^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)/x^4/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(15*((7*b^4*c^3 - 15*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 - a^3*b*d^3)*x^4 + (7*a*b^3*c^3 - 15*a^2*b^2*c^2*d +
9*a^3*b*c*d^2 - a^4*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^4*c^3 + (105*a*b^3*
c^3 - 190*a^2*b^2*c^2*d + 81*a^3*b*c*d^2)*x^3 + (35*a^2*b^2*c^3 - 68*a^3*b*c^2*d + 33*a^4*c*d^2)*x^2 - 2*(7*a^
3*b*c^3 - 13*a^4*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*b*c*x^4 + a^6*c*x^3), -1/48*(15*((7*b^4*c^3 - 15*
a*b^3*c^2*d + 9*a^2*b^2*c*d^2 - a^3*b*d^3)*x^4 + (7*a*b^3*c^3 - 15*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 - a^4*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^4*c^3 + (105*a*b^3*c^3 - 190*a^2*b^2*c^2*d + 81*a^3*b*c*d^2)*x^3 + (35*a^2
*b^2*c^3 - 68*a^3*b*c^2*d + 33*a^4*c*d^2)*x^2 - 2*(7*a^3*b*c^3 - 13*a^4*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))
/(a^5*b*c*x^4 + a^6*c*x^3)]

________________________________________________________________________________________

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((d*x+c)**(5/2)/x**4/(b*x+a)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

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((d*x+c)^(5/2)/x^4/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError