3.555 \(\int \frac{(a+c x^2)^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=203 \[ -\frac{5 a^2 c^2 \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{5 a^3 c^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{7/2}}-\frac{5 a c \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac{\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

[Out]

(-5*a^2*c^2*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (5*a*c*(a*e - c*d*x)*(a + c*x^
2)^(3/2))/(24*(c*d^2 + a*e^2)^2*(d + e*x)^4) - ((a*e - c*d*x)*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)*(d + e*x)^
6) - (5*a^3*c^3*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(7/2))

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Rubi [A]  time = 0.105414, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {721, 725, 206} \[ -\frac{5 a^2 c^2 \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{5 a^3 c^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{7/2}}-\frac{5 a c \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac{\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

(-5*a^2*c^2*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (5*a*c*(a*e - c*d*x)*(a + c*x^
2)^(3/2))/(24*(c*d^2 + a*e^2)^2*(d + e*x)^4) - ((a*e - c*d*x)*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)*(d + e*x)^
6) - (5*a^3*c^3*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(7/2))

Rule 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx &=-\frac{(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}+\frac{(5 a c) \int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{6 \left (c d^2+a e^2\right )}\\ &=-\frac{5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac{(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}+\frac{\left (5 a^2 c^2\right ) \int \frac{\sqrt{a+c x^2}}{(d+e x)^3} \, dx}{8 \left (c d^2+a e^2\right )^2}\\ &=-\frac{5 a^2 c^2 (a e-c d x) \sqrt{a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac{(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}+\frac{\left (5 a^3 c^3\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{16 \left (c d^2+a e^2\right )^3}\\ &=-\frac{5 a^2 c^2 (a e-c d x) \sqrt{a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac{(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac{\left (5 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{c d^2+a e^2-x^2} \, dx,x,\frac{a e-c d x}{\sqrt{a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^3}\\ &=-\frac{5 a^2 c^2 (a e-c d x) \sqrt{a+c x^2}}{16 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{5 a c (a e-c d x) \left (a+c x^2\right )^{3/2}}{24 \left (c d^2+a e^2\right )^2 (d+e x)^4}-\frac{(a e-c d x) \left (a+c x^2\right )^{5/2}}{6 \left (c d^2+a e^2\right ) (d+e x)^6}-\frac{5 a^3 c^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{16 \left (c d^2+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.536996, size = 305, normalized size = 1.5 \[ \frac{1}{48} \left (\frac{\sqrt{a+c x^2} \left (-a^3 c^2 e \left (122 d^2 e^2 x^2+54 d^3 e x+33 d^4+54 d e^3 x^3+33 e^4 x^4\right )+a^2 c^3 d x \left (122 d^2 e^2 x^2+54 d^3 e x+33 d^4+54 d e^3 x^3+33 e^4 x^4\right )-2 a^4 c e^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )-8 a^5 e^5+2 a c^4 d^3 x^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )+8 c^5 d^5 x^5\right )}{(d+e x)^6 \left (a e^2+c d^2\right )^3}-\frac{15 a^3 c^3 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{7/2}}+\frac{15 a^3 c^3 \log (d+e x)}{\left (a e^2+c d^2\right )^{7/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

((Sqrt[a + c*x^2]*(-8*a^5*e^5 + 8*c^5*d^5*x^5 - 2*a^4*c*e^3*(13*d^2 + 6*d*e*x + 13*e^2*x^2) + 2*a*c^4*d^3*x^3*
(13*d^2 + 6*d*e*x + 13*e^2*x^2) - a^3*c^2*e*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4
) + a^2*c^3*d*x*(33*d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4)))/((c*d^2 + a*e^2)^3*(d +
e*x)^6) + (15*a^3*c^3*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2) - (15*a^3*c^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*
Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))/48

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Maple [B]  time = 0.214, size = 7616, normalized size = 37.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^7,x)

[Out]

result too large to display

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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((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**7,x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**7, x)

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Giac [B]  time = 2.34626, size = 2558, normalized size = 12.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

5/8*a^3*c^3*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/((c^3*d^6 + 3*a*c^2*d^
4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(-c*d^2 - a*e^2)) + 1/24*(768*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^8*d^10*
e + 256*(sqrt(c)*x - sqrt(c*x^2 + a))^6*c^(17/2)*d^11 + 960*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(15/2)*d^9*e^2 +
640*(sqrt(c)*x - sqrt(c*x^2 + a))^9*c^7*d^8*e^3 - 768*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^8*d^10*e + 240*(sqr
t(c)*x - sqrt(c*x^2 + a))^10*c^(13/2)*d^7*e^4 - 1088*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(15/2)*d^9*e^2 + 48*(
sqrt(c)*x - sqrt(c*x^2 + a))^11*c^6*d^6*e^5 + 576*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^7*d^8*e^3 + 2160*(sqrt(c
)*x - sqrt(c*x^2 + a))^8*a*c^(13/2)*d^7*e^4 + 960*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^2*c^(15/2)*d^9*e^2 + 1840*
(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^6*d^6*e^5 - 576*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c^7*d^8*e^3 + 720*(sqr
t(c)*x - sqrt(c*x^2 + a))^10*a*c^(11/2)*d^5*e^6 - 3744*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*c^(13/2)*d^7*e^4 +
144*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a*c^5*d^4*e^7 - 2592*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^6*d^6*e^5 - 64
0*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c^7*d^8*e^3 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(11/2)*d^5*e^6 +
2160*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*c^(13/2)*d^7*e^4 + 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^5*d^4*
e^7 + 2592*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^3*c^6*d^6*e^5 + 720*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^2*c^(9/2)*
d^3*e^8 - 3320*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^3*c^(11/2)*d^5*e^6 + 240*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*
c^(13/2)*d^7*e^4 + 144*(sqrt(c)*x - sqrt(c*x^2 + a))^11*a^2*c^4*d^2*e^9 - 5640*(sqrt(c)*x - sqrt(c*x^2 + a))^7
*a^3*c^5*d^4*e^7 - 1840*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^4*c^6*d^6*e^5 - 2910*(sqrt(c)*x - sqrt(c*x^2 + a))^8
*a^3*c^(9/2)*d^3*e^8 + 1080*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^4*c^(11/2)*d^5*e^6 - 340*(sqrt(c)*x - sqrt(c*x^2
+ a))^9*a^3*c^4*d^2*e^9 + 7080*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^4*c^5*d^4*e^7 - 48*(sqrt(c)*x - sqrt(c*x^2 +
a))*a^5*c^6*d^6*e^5 + 75*(sqrt(c)*x - sqrt(c*x^2 + a))^10*a^3*c^(7/2)*d*e^10 + 5680*(sqrt(c)*x - sqrt(c*x^2 +
a))^6*a^4*c^(9/2)*d^3*e^8 + 792*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^5*c^(11/2)*d^5*e^6 + 33*(sqrt(c)*x - sqrt(c
*x^2 + a))^11*a^3*c^3*e^11 + 1800*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^4*c^4*d^2*e^9 - 2040*(sqrt(c)*x - sqrt(c*x
^2 + a))^3*a^5*c^5*d^4*e^7 + 45*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^4*c^(7/2)*d*e^10 - 4620*(sqrt(c)*x - sqrt(c*
x^2 + a))^4*a^5*c^(9/2)*d^3*e^8 + 8*a^6*c^(11/2)*d^5*e^6 + 5*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^4*c^3*e^11 - 21
60*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^5*c^4*d^2*e^9 - 168*(sqrt(c)*x - sqrt(c*x^2 + a))*a^6*c^5*d^4*e^7 - 330*(
sqrt(c)*x - sqrt(c*x^2 + a))^6*a^5*c^(7/2)*d*e^10 + 1104*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^6*c^(9/2)*d^3*e^8 +
90*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^5*c^3*e^11 + 1640*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^6*c^4*d^2*e^9 + 450*
(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^6*c^(7/2)*d*e^10 + 26*a^7*c^(9/2)*d^3*e^8 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))
^5*a^6*c^3*e^11 - 252*(sqrt(c)*x - sqrt(c*x^2 + a))*a^7*c^4*d^2*e^9 - 273*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^7*
c^(7/2)*d*e^10 + 5*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^7*c^3*e^11 + 33*a^8*c^(7/2)*d*e^10 + 33*(sqrt(c)*x - sqrt
(c*x^2 + a))*a^8*c^3*e^11)/((c^3*d^6*e^6 + 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 + a^3*e^12)*((sqrt(c)*x - sqrt(c
*x^2 + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + a))*sqrt(c)*d - a*e)^6)