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

Optimal. Leaf size=144 \[ \frac {5 a^3 \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} \sqrt {b c-a d}}+\frac {5 a^2 x \sqrt {a+b x^2}}{16 c^3 \left (c+d x^2\right )}+\frac {5 a x \left (a+b x^2\right )^{3/2}}{24 c^2 \left (c+d x^2\right )^2}+\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3} \]

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Rubi [A]  time = 0.07, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {378, 377, 208} \begin {gather*} \frac {5 a^2 x \sqrt {a+b x^2}}{16 c^3 \left (c+d x^2\right )}+\frac {5 a^3 \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} \sqrt {b c-a d}}+\frac {5 a x \left (a+b x^2\right )^{3/2}}{24 c^2 \left (c+d x^2\right )^2}+\frac {x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 378

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] - Dist[(c*q)/(a*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.81, size = 201, normalized size = 1.40 \begin {gather*} \frac {x \sqrt {a+b x^2} \left (\frac {\sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \left (a^2 \left (33 c^2+40 c d x^2+15 d^2 x^4\right )+2 a b c x^2 \left (13 c+5 d x^2\right )+8 b^2 c^2 x^4\right )}{\left (c+d x^2\right )^2 \sqrt {\frac {d x^2}{c}+1}}+\frac {15 a^2 \sin ^{-1}\left (\frac {\sqrt {x^2 \left (\frac {d}{c}-\frac {b}{a}\right )}}{\sqrt {\frac {d x^2}{c}+1}}\right )}{\sqrt {\frac {x^2 (a d-b c)}{a c}}}\right )}{48 c^4 \sqrt {\frac {b x^2}{a}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*Sqrt[a + b*x^2]*((Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*(8*b^2*c^2*x^4 + 2*a*b*c*x^2*(13*c + 5*d*x^2) + a^2
*(33*c^2 + 40*c*d*x^2 + 15*d^2*x^4)))/((c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]) + (15*a^2*ArcSin[Sqrt[(-(b/a) + d/c)
*x^2]/Sqrt[1 + (d*x^2)/c]])/Sqrt[((-(b*c) + a*d)*x^2)/(a*c)]))/(48*c^4*Sqrt[1 + (b*x^2)/a])

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IntegrateAlgebraic [B]  time = 20.71, size = 1180, normalized size = 8.19 \begin {gather*} \frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} d x^2}{\sqrt {c} \sqrt {b c-a d}}-\frac {d \sqrt {b x^2+a} x}{\sqrt {c} \sqrt {b c-a d}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {b c-a d}}\right ) a^3}{16 c^{7/2} \sqrt {b c-a d}}+\frac {\left (768 c^3 d^2 x^{10}+768 c^4 d x^8+256 c^5 x^6\right ) b^{11/2}+\sqrt {b x^2+a} \left (-768 c^3 d^2 x^9-768 c^4 d x^7-256 c^5 x^5\right ) b^5+\left (2112 a c^3 d^2 x^8+1472 a c^4 d x^6+384 a c^5 x^4\right ) b^{9/2}+\sqrt {b x^2+a} \left (-1728 a c^3 d^2 x^7-1088 a c^4 d x^5-256 a c^5 x^3\right ) b^4+\left (2352 a^2 c^3 d^2 x^6+912 a^2 c^4 d x^4+144 a^2 c^5 x^2\right ) b^{7/2}+\sqrt {b x^2+a} \left (-48 a^2 x c^5-464 a^2 d x^3 c^4-1584 a^2 d^2 x^5 c^3\right ) b^3+\left (-240 a^3 d^5 x^{10}-600 a^3 c d^4 x^8-440 a^3 c^2 d^3 x^6+1284 a^3 c^3 d^2 x^4+204 a^3 c^4 d x^2+8 a^3 c^5\right ) b^{5/2}+\sqrt {b x^2+a} \left (240 a^3 d^5 x^9+600 a^3 c d^4 x^7+440 a^3 c^2 d^3 x^5-660 a^3 c^3 d^2 x^3-60 a^3 c^4 d x\right ) b^2+\left (-300 a^4 d^5 x^8-760 a^4 c d^4 x^6-570 a^4 c^2 d^3 x^4+300 a^4 c^3 d^2 x^2+10 a^4 c^4 d\right ) b^{3/2}+\sqrt {b x^2+a} \left (180 a^4 d^5 x^7+460 a^4 c d^4 x^5+350 a^4 c^2 d^3 x^3-90 a^4 c^3 d^2 x\right ) b+\left (-75 a^5 d^5 x^6-195 a^5 c d^4 x^4-153 a^5 c^2 d^3 x^2+15 a^5 c^3 d^2\right ) \sqrt {b}+\sqrt {b x^2+a} \left (15 d^5 x^5 a^5+40 c d^4 x^3 a^5+33 c^2 d^3 x a^5\right )}{1536 b^3 c^3 d^3 \left (d x^2+c\right )^3 x^6-1536 b^{5/2} c^3 d^3 \sqrt {b x^2+a} \left (d x^2+c\right )^3 x^5+2304 a b^2 c^3 d^3 \left (d x^2+c\right )^3 x^4-1536 a b^{3/2} c^3 d^3 \sqrt {b x^2+a} \left (d x^2+c\right )^3 x^3+864 a^2 b c^3 d^3 \left (d x^2+c\right )^3 x^2-288 a^2 \sqrt {b} c^3 d^3 \sqrt {b x^2+a} \left (d x^2+c\right )^3 x+48 a^3 c^3 d^3 \left (d x^2+c\right )^3}-\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {b} d x^2}{\sqrt {c} \sqrt {a d-b c}}-\frac {d \sqrt {b x^2+a} x}{\sqrt {c} \sqrt {a d-b c}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {a d-b c}}\right )}{\sqrt {c} d^3 \sqrt {a d-b c}}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {b} d x^2}{\sqrt {c} \sqrt {b c-a d}}-\frac {d \sqrt {b x^2+a} x}{\sqrt {c} \sqrt {b c-a d}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {b c-a d}}\right )}{\sqrt {c} d^3 \sqrt {b c-a d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*Sqrt[a + b*x^2]*(-48*a^2*c^5*x - 464*a^2*c^4*d*x^3 - 1584*a^2*c^3*d^2*x^5) + Sqrt[a + b*x^2]*(33*a^5*c^2*
d^3*x + 40*a^5*c*d^4*x^3 + 15*a^5*d^5*x^5) + b^(7/2)*(144*a^2*c^5*x^2 + 912*a^2*c^4*d*x^4 + 2352*a^2*c^3*d^2*x
^6) + Sqrt[b]*(15*a^5*c^3*d^2 - 153*a^5*c^2*d^3*x^2 - 195*a^5*c*d^4*x^4 - 75*a^5*d^5*x^6) + b^4*Sqrt[a + b*x^2
]*(-256*a*c^5*x^3 - 1088*a*c^4*d*x^5 - 1728*a*c^3*d^2*x^7) + b*Sqrt[a + b*x^2]*(-90*a^4*c^3*d^2*x + 350*a^4*c^
2*d^3*x^3 + 460*a^4*c*d^4*x^5 + 180*a^4*d^5*x^7) + b^(9/2)*(384*a*c^5*x^4 + 1472*a*c^4*d*x^6 + 2112*a*c^3*d^2*
x^8) + b^(3/2)*(10*a^4*c^4*d + 300*a^4*c^3*d^2*x^2 - 570*a^4*c^2*d^3*x^4 - 760*a^4*c*d^4*x^6 - 300*a^4*d^5*x^8
) + b^5*Sqrt[a + b*x^2]*(-256*c^5*x^5 - 768*c^4*d*x^7 - 768*c^3*d^2*x^9) + b^2*Sqrt[a + b*x^2]*(-60*a^3*c^4*d*
x - 660*a^3*c^3*d^2*x^3 + 440*a^3*c^2*d^3*x^5 + 600*a^3*c*d^4*x^7 + 240*a^3*d^5*x^9) + b^(11/2)*(256*c^5*x^6 +
768*c^4*d*x^8 + 768*c^3*d^2*x^10) + b^(5/2)*(8*a^3*c^5 + 204*a^3*c^4*d*x^2 + 1284*a^3*c^3*d^2*x^4 - 440*a^3*c
^2*d^3*x^6 - 600*a^3*c*d^4*x^8 - 240*a^3*d^5*x^10))/(48*a^3*c^3*d^3*(c + d*x^2)^3 + 864*a^2*b*c^3*d^3*x^2*(c +
d*x^2)^3 + 2304*a*b^2*c^3*d^3*x^4*(c + d*x^2)^3 + 1536*b^3*c^3*d^3*x^6*(c + d*x^2)^3 - 288*a^2*Sqrt[b]*c^3*d^
3*x*Sqrt[a + b*x^2]*(c + d*x^2)^3 - 1536*a*b^(3/2)*c^3*d^3*x^3*Sqrt[a + b*x^2]*(c + d*x^2)^3 - 1536*b^(5/2)*c^
3*d^3*x^5*Sqrt[a + b*x^2]*(c + d*x^2)^3) - (b^3*ArcTan[(Sqrt[b]*Sqrt[c])/Sqrt[-(b*c) + a*d] + (Sqrt[b]*d*x^2)/
(Sqrt[c]*Sqrt[-(b*c) + a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[-(b*c) + a*d])])/(Sqrt[c]*d^3*Sqrt[-(b*c) +
a*d]) + (5*a^3*ArcTanh[(Sqrt[b]*Sqrt[c])/Sqrt[b*c - a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[b*c - a*d]) - (d*x*S
qrt[a + b*x^2])/(Sqrt[c]*Sqrt[b*c - a*d])])/(16*c^(7/2)*Sqrt[b*c - a*d]) - (b^3*ArcTanh[(Sqrt[b]*Sqrt[c])/Sqrt
[b*c - a*d] + (Sqrt[b]*d*x^2)/(Sqrt[c]*Sqrt[b*c - a*d]) - (d*x*Sqrt[a + b*x^2])/(Sqrt[c]*Sqrt[b*c - a*d])])/(S
qrt[c]*d^3*Sqrt[b*c - a*d])

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fricas [B]  time = 1.51, size = 706, normalized size = 4.90 \begin {gather*} \left [\frac {15 \, {\left (a^{3} d^{3} x^{6} + 3 \, a^{3} c d^{2} x^{4} + 3 \, a^{3} c^{2} d x^{2} + a^{3} c^{3}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (8 \, b^{3} c^{4} + 2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{5} + 2 \, {\left (13 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d - 20 \, a^{3} c^{2} d^{2}\right )} x^{3} + 33 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x\right )} \sqrt {b x^{2} + a}}{192 \, {\left (b c^{8} - a c^{7} d + {\left (b c^{5} d^{3} - a c^{4} d^{4}\right )} x^{6} + 3 \, {\left (b c^{6} d^{2} - a c^{5} d^{3}\right )} x^{4} + 3 \, {\left (b c^{7} d - a c^{6} d^{2}\right )} x^{2}\right )}}, -\frac {15 \, {\left (a^{3} d^{3} x^{6} + 3 \, a^{3} c d^{2} x^{4} + 3 \, a^{3} c^{2} d x^{2} + a^{3} c^{3}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (8 \, b^{3} c^{4} + 2 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 15 \, a^{3} c d^{3}\right )} x^{5} + 2 \, {\left (13 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d - 20 \, a^{3} c^{2} d^{2}\right )} x^{3} + 33 \, {\left (a^{2} b c^{4} - a^{3} c^{3} d\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, {\left (b c^{8} - a c^{7} d + {\left (b c^{5} d^{3} - a c^{4} d^{4}\right )} x^{6} + 3 \, {\left (b c^{6} d^{2} - a c^{5} d^{3}\right )} x^{4} + 3 \, {\left (b c^{7} d - a c^{6} d^{2}\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/192*(15*(a^3*d^3*x^6 + 3*a^3*c*d^2*x^4 + 3*a^3*c^2*d*x^2 + a^3*c^3)*sqrt(b*c^2 - a*c*d)*log(((8*b^2*c^2 - 8
*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*((2*b*c - a*d)*x^3 + a*c*x)*sqrt(b*c^2 -
a*c*d)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*((8*b^3*c^4 + 2*a*b^2*c^3*d + 5*a^2*b*c^2*d^2 - 15*a
^3*c*d^3)*x^5 + 2*(13*a*b^2*c^4 + 7*a^2*b*c^3*d - 20*a^3*c^2*d^2)*x^3 + 33*(a^2*b*c^4 - a^3*c^3*d)*x)*sqrt(b*x
^2 + a))/(b*c^8 - a*c^7*d + (b*c^5*d^3 - a*c^4*d^4)*x^6 + 3*(b*c^6*d^2 - a*c^5*d^3)*x^4 + 3*(b*c^7*d - a*c^6*d
^2)*x^2), -1/96*(15*(a^3*d^3*x^6 + 3*a^3*c*d^2*x^4 + 3*a^3*c^2*d*x^2 + a^3*c^3)*sqrt(-b*c^2 + a*c*d)*arctan(1/
2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + (a*b*c^2 - a^2*c*d
)*x)) - 2*((8*b^3*c^4 + 2*a*b^2*c^3*d + 5*a^2*b*c^2*d^2 - 15*a^3*c*d^3)*x^5 + 2*(13*a*b^2*c^4 + 7*a^2*b*c^3*d
- 20*a^3*c^2*d^2)*x^3 + 33*(a^2*b*c^4 - a^3*c^3*d)*x)*sqrt(b*x^2 + a))/(b*c^8 - a*c^7*d + (b*c^5*d^3 - a*c^4*d
^4)*x^6 + 3*(b*c^6*d^2 - a*c^5*d^3)*x^4 + 3*(b*c^7*d - a*c^6*d^2)*x^2)]

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giac [B]  time = 2.86, size = 846, normalized size = 5.88 \begin {gather*} -\frac {5 \, a^{3} \sqrt {b} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{16 \, \sqrt {-b^{2} c^{2} + a b c d} c^{3}} + \frac {48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} b^{\frac {7}{2}} c^{3} d^{2} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} \sqrt {b} d^{5} + 192 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {9}{2}} c^{4} d + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {7}{2}} c^{3} d^{2} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {3}{2}} c d^{4} + 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} \sqrt {b} d^{5} + 256 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {11}{2}} c^{5} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {9}{2}} c^{4} d + 288 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {7}{2}} c^{3} d^{2} - 440 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} c^{2} d^{3} + 440 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {3}{2}} c d^{4} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} \sqrt {b} d^{5} + 192 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {9}{2}} c^{4} d + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {7}{2}} c^{3} d^{2} + 360 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} c^{2} d^{3} - 420 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {3}{2}} c d^{4} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} \sqrt {b} d^{5} + 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {7}{2}} c^{3} d^{2} + 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} c^{2} d^{3} + 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {3}{2}} c d^{4} - 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} \sqrt {b} d^{5} + 8 \, a^{6} b^{\frac {5}{2}} c^{2} d^{3} + 10 \, a^{7} b^{\frac {3}{2}} c d^{4} + 15 \, a^{8} \sqrt {b} d^{5}}{24 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{3} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16*a^3*sqrt(b)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b^2*c^2 + a*b*c*d))/(sqrt
(-b^2*c^2 + a*b*c*d)*c^3) + 1/24*(48*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*c^3*d^2 - 15*(sqrt(b)*x - sqrt(b
*x^2 + a))^10*a^3*sqrt(b)*d^5 + 192*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(9/2)*c^4*d + 48*(sqrt(b)*x - sqrt(b*x^2
+ a))^8*a*b^(7/2)*c^3*d^2 - 150*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(3/2)*c*d^4 + 75*(sqrt(b)*x - sqrt(b*x^
2 + a))^8*a^4*sqrt(b)*d^5 + 256*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(11/2)*c^5 - 64*(sqrt(b)*x - sqrt(b*x^2 + a)
)^6*a*b^(9/2)*c^4*d + 288*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(7/2)*c^3*d^2 - 440*(sqrt(b)*x - sqrt(b*x^2 +
a))^6*a^3*b^(5/2)*c^2*d^3 + 440*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^(3/2)*c*d^4 - 150*(sqrt(b)*x - sqrt(b*x^
2 + a))^6*a^5*sqrt(b)*d^5 + 192*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(9/2)*c^4*d + 48*(sqrt(b)*x - sqrt(b*x^2
+ a))^4*a^3*b^(7/2)*c^3*d^2 + 360*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(5/2)*c^2*d^3 - 420*(sqrt(b)*x - sqrt
(b*x^2 + a))^4*a^5*b^(3/2)*c*d^4 + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^6*sqrt(b)*d^5 + 48*(sqrt(b)*x - sqrt(
b*x^2 + a))^2*a^4*b^(7/2)*c^3*d^2 + 72*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^(5/2)*c^2*d^3 + 120*(sqrt(b)*x -
sqrt(b*x^2 + a))^2*a^6*b^(3/2)*c*d^4 - 75*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*sqrt(b)*d^5 + 8*a^6*b^(5/2)*c^2*
d^3 + 10*a^7*b^(3/2)*c*d^4 + 15*a^8*sqrt(b)*d^5)/(((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x
^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)^3*c^3*d^3)

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maple [B]  time = 0.04, size = 21220, normalized size = 147.36 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{5/2}}{{\left (d\,x^2+c\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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