3.673 \(\int \frac{(a+b x)^{5/2} (c+d x)^{5/2}}{x^7} \, dx\)
Optimal. Leaf size=280 \[ \frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^4}{768 a^2 c^3 x^2}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}{512 a^3 c^3 x}+\frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{7/2} c^{7/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^3}{192 a c^3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2} (b c-a d)^2}{32 c^3 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2} (b c-a d)}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6} \]
[Out]
(-5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*a^3*c^3*x) + (5*(b*c - a*d)^4*Sqrt[a + b*x]*(c + d*x)^(3/2
))/(768*a^2*c^3*x^2) - ((b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(5/2))/(192*a*c^3*x^3) - ((b*c - a*d)^2*Sqrt[a +
b*x]*(c + d*x)^(7/2))/(32*c^3*x^4) - ((b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(12*c^2*x^5) - ((a + b*x)^
(5/2)*(c + d*x)^(7/2))/(6*c*x^6) + (5*(b*c - a*d)^6*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/
(512*a^(7/2)*c^(7/2))
________________________________________________________________________________________
Rubi [A] time = 0.180726, antiderivative size = 280, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used =
{94, 93, 208} \[ \frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d)^4}{768 a^2 c^3 x^2}-\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^5}{512 a^3 c^3 x}+\frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{7/2} c^{7/2}}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (b c-a d)^3}{192 a c^3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2} (b c-a d)^2}{32 c^3 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{7/2} (b c-a d)}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x]
[Out]
(-5*(b*c - a*d)^5*Sqrt[a + b*x]*Sqrt[c + d*x])/(512*a^3*c^3*x) + (5*(b*c - a*d)^4*Sqrt[a + b*x]*(c + d*x)^(3/2
))/(768*a^2*c^3*x^2) - ((b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(5/2))/(192*a*c^3*x^3) - ((b*c - a*d)^2*Sqrt[a +
b*x]*(c + d*x)^(7/2))/(32*c^3*x^4) - ((b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(7/2))/(12*c^2*x^5) - ((a + b*x)^
(5/2)*(c + d*x)^(7/2))/(6*c*x^6) + (5*(b*c - a*d)^6*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/
(512*a^(7/2)*c^(7/2))
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} (c+d x)^{5/2}}{x^7} \, dx &=-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac{(5 (b c-a d)) \int \frac{(a+b x)^{3/2} (c+d x)^{5/2}}{x^6} \, dx}{12 c}\\ &=-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac{(b c-a d)^2 \int \frac{\sqrt{a+b x} (c+d x)^{5/2}}{x^5} \, dx}{8 c^2}\\ &=-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac{(b c-a d)^3 \int \frac{(c+d x)^{5/2}}{x^4 \sqrt{a+b x}} \, dx}{64 c^3}\\ &=-\frac{(b c-a d)^3 \sqrt{a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}-\frac{\left (5 (b c-a d)^4\right ) \int \frac{(c+d x)^{3/2}}{x^3 \sqrt{a+b x}} \, dx}{384 a c^3}\\ &=\frac{5 (b c-a d)^4 \sqrt{a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac{(b c-a d)^3 \sqrt{a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac{\left (5 (b c-a d)^5\right ) \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx}{512 a^2 c^3}\\ &=-\frac{5 (b c-a d)^5 \sqrt{a+b x} \sqrt{c+d x}}{512 a^3 c^3 x}+\frac{5 (b c-a d)^4 \sqrt{a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac{(b c-a d)^3 \sqrt{a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}-\frac{\left (5 (b c-a d)^6\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{1024 a^3 c^3}\\ &=-\frac{5 (b c-a d)^5 \sqrt{a+b x} \sqrt{c+d x}}{512 a^3 c^3 x}+\frac{5 (b c-a d)^4 \sqrt{a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac{(b c-a d)^3 \sqrt{a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}-\frac{\left (5 (b c-a d)^6\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{512 a^3 c^3}\\ &=-\frac{5 (b c-a d)^5 \sqrt{a+b x} \sqrt{c+d x}}{512 a^3 c^3 x}+\frac{5 (b c-a d)^4 \sqrt{a+b x} (c+d x)^{3/2}}{768 a^2 c^3 x^2}-\frac{(b c-a d)^3 \sqrt{a+b x} (c+d x)^{5/2}}{192 a c^3 x^3}-\frac{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{7/2}}{32 c^3 x^4}-\frac{(b c-a d) (a+b x)^{3/2} (c+d x)^{7/2}}{12 c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6}+\frac{5 (b c-a d)^6 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{7/2} c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.798463, size = 268, normalized size = 0.96 \[ -\frac{(b c-a d) \left (128 a^{7/2} c^{3/2} (a+b x)^{3/2} (c+d x)^{7/2}+x (b c-a d) \left (x (b c-a d) \left (8 a^{5/2} \sqrt{c} \sqrt{a+b x} (c+d x)^{5/2}-5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )\right )+48 a^{7/2} \sqrt{c} \sqrt{a+b x} (c+d x)^{7/2}\right )\right )}{1536 a^{7/2} c^{7/2} x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 c x^6} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^7,x]
[Out]
-((a + b*x)^(5/2)*(c + d*x)^(7/2))/(6*c*x^6) - ((b*c - a*d)*(128*a^(7/2)*c^(3/2)*(a + b*x)^(3/2)*(c + d*x)^(7/
2) + (b*c - a*d)*x*(48*a^(7/2)*Sqrt[c]*Sqrt[a + b*x]*(c + d*x)^(7/2) + (b*c - a*d)*x*(8*a^(5/2)*Sqrt[c]*Sqrt[a
+ b*x]*(c + d*x)^(5/2) - 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*a*
d*x) + 3*(b*c - a*d)^2*x^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])))))/(1536*a^(7/2)*c^(7/2)
*x^5)
________________________________________________________________________________________
Maple [B] time = 0.021, size = 1271, normalized size = 4.5 \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)*(d*x+c)^(5/2)/x^7,x)
[Out]
1/3072*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*b^5*c^5+20*(a*
c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^5*c*d^4+20*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a*b^
4*c^5-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^5*c^2*d^3-1280*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c
)^(1/2)*x*a^4*b*c^5-90*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^5*b*c*d^5
+225*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^4*b^2*c^2*d^4-300*ln((a*d*x
+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^3*b^3*c^3*d^3+225*ln((a*d*x+b*c*x+2*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^2*b^4*c^4*d^2-90*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2
+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a*b^5*c^5*d-16*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^2*b^3*c
^5-864*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^3*b^2*c^5-864*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*x^2*a^5*c^3*d^2-30*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^5*d^5-1280*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*x*a^5*c^4*d-512*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*a^5*c^5*(a*c)^(1/2)+15*ln((a*d*x+b*c*x+2*(a
*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*a^6*d^6+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x^6*b^6*c^6+170*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^4*b*c*d^4-396*(
a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a^3*b^2*c^2*d^3-396*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)
*x^5*a^2*b^3*c^3*d^2+170*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^5*a*b^4*c^4*d-112*(a*c)^(1/2)*(b*d*x^2+
a*d*x+b*c*x+a*c)^(1/2)*x^4*a^4*b*c^2*d^3-2376*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^3*b^2*c^3*d^2-
112*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^4*a^2*b^3*c^4*d-2544*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(
1/2)*x^3*a^4*b*c^3*d^2-2544*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^3*a^3*b^2*c^4*d-3392*(a*c)^(1/2)*(b*
d*x^2+a*d*x+b*c*x+a*c)^(1/2)*x^2*a^4*b*c^4*d)/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^6/(a*c)^(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((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [A] time = 145.712, size = 1968, normalized size = 7.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x, algorithm="fricas")
[Out]
[1/6144*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*
c*d^5 + a^6*d^6)*sqrt(a*c)*x^6*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*(256*a^6*c^6 + (15*a*b^5*c^6 - 85*
a^2*b^4*c^5*d + 198*a^3*b^3*c^4*d^2 + 198*a^4*b^2*c^3*d^3 - 85*a^5*b*c^2*d^4 + 15*a^6*c*d^5)*x^5 - 2*(5*a^2*b^
4*c^6 - 28*a^3*b^3*c^5*d - 594*a^4*b^2*c^4*d^2 - 28*a^5*b*c^3*d^3 + 5*a^6*c^2*d^4)*x^4 + 8*(a^3*b^3*c^6 + 159*
a^4*b^2*c^5*d + 159*a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^3 + 16*(27*a^4*b^2*c^6 + 106*a^5*b*c^5*d + 27*a^6*c^4*d^2)*
x^2 + 640*(a^5*b*c^6 + a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^6), -1/3072*(15*(b^6*c^6 - 6*a*b^
5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(-a*c)*x
^6*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*(256*a^6*c^6 + (15*a*b^5*c^6 - 85*a^2*b^4*c^5*d + 198*a^3*b^3*c^4*d^2 + 198*a^4*b^2*c^3*d^
3 - 85*a^5*b*c^2*d^4 + 15*a^6*c*d^5)*x^5 - 2*(5*a^2*b^4*c^6 - 28*a^3*b^3*c^5*d - 594*a^4*b^2*c^4*d^2 - 28*a^5*
b*c^3*d^3 + 5*a^6*c^2*d^4)*x^4 + 8*(a^3*b^3*c^6 + 159*a^4*b^2*c^5*d + 159*a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^3 + 1
6*(27*a^4*b^2*c^6 + 106*a^5*b*c^5*d + 27*a^6*c^4*d^2)*x^2 + 640*(a^5*b*c^6 + a^6*c^5*d)*x)*sqrt(b*x + a)*sqrt(
d*x + c))/(a^4*c^4*x^6)]
________________________________________________________________________________________
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)*(d*x+c)**(5/2)/x**7,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((b*x+a)^(5/2)*(d*x+c)^(5/2)/x^7,x, algorithm="giac")
[Out]
Exception raised: TypeError