3.1.89 \(\int \frac {(c+d x^2)^4}{(a+b x^2)^{5/2}} \, dx\) [89]

Optimal. Leaf size=255 \[ -\frac {d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 b^4}-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}} \]

[Out]

1/3*(-a*d+b*c)*x*(d*x^2+c)^3/a/b/(b*x^2+a)^(3/2)+1/8*d^2*(35*a^2*d^2-80*a*b*c*d+48*b^2*c^2)*arctanh(x*b^(1/2)/
(b*x^2+a)^(1/2))/b^(9/2)+1/3*(-a*d+b*c)*(7*a*d+2*b*c)*x*(d*x^2+c)^2/a^2/b^2/(b*x^2+a)^(1/2)-1/24*d*(105*a^3*d^
3-170*a^2*b*c*d^2+40*a*b^2*c^2*d+16*b^3*c^3)*x*(b*x^2+a)^(1/2)/a^2/b^4-1/12*d*(-35*a^2*d^2+24*a*b*c*d+8*b^2*c^
2)*x*(d*x^2+c)*(b*x^2+a)^(1/2)/a^2/b^3

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Rubi [A]
time = 0.17, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {424, 540, 542, 396, 223, 212} \begin {gather*} \frac {x \left (c+d x^2\right )^2 (b c-a d) (7 a d+2 b c)}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {d^2 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{12 a^2 b^3}-\frac {d x \sqrt {a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 b^4}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(d*(16*b^3*c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(a^2*b^4) - (d*(8*b^
2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(12*a^2*b^3) + ((b*c - a*d)*(2*b*c + 7*a*d)*x*
(c + d*x^2)^2)/(3*a^2*b^2*Sqrt[a + b*x^2]) + ((b*c - a*d)*x*(c + d*x^2)^3)/(3*a*b*(a + b*x^2)^(3/2)) + (d^2*(4
8*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2))

Rule 212

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 396

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

Rule 424

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 540

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

Rule 542

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

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\int \frac {\left (c+d x^2\right )^2 \left (c (2 b c+a d)-d (4 b c-7 a d) x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a b}\\ &=\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {\left (c+d x^2\right ) \left (a c d (4 b c-7 a d)+d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx}{3 a^2 b^2}\\ &=-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {a c d \left (8 b^2 c^2-52 a b c d+35 a^2 d^2\right )+d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x^2}{\sqrt {a+b x^2}} \, dx}{12 a^2 b^3}\\ &=-\frac {d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 b^4}-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^4}\\ &=-\frac {d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 b^4}-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^4}\\ &=-\frac {d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 b^4}-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.40, size = 202, normalized size = 0.79 \begin {gather*} \frac {x \left (-105 a^5 d^4+16 b^5 c^4 x^2+20 a^4 b d^3 \left (12 c-7 d x^2\right )+8 a b^4 c^3 \left (3 c+4 d x^2\right )+a^3 b^2 d^2 \left (-144 c^2+320 c d x^2-21 d^2 x^4\right )+6 a^2 b^3 d^2 x^2 \left (-32 c^2+8 c d x^2+d^2 x^4\right )\right )}{24 a^2 b^4 \left (a+b x^2\right )^{3/2}}-\frac {d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-105*a^5*d^4 + 16*b^5*c^4*x^2 + 20*a^4*b*d^3*(12*c - 7*d*x^2) + 8*a*b^4*c^3*(3*c + 4*d*x^2) + a^3*b^2*d^2*
(-144*c^2 + 320*c*d*x^2 - 21*d^2*x^4) + 6*a^2*b^3*d^2*x^2*(-32*c^2 + 8*c*d*x^2 + d^2*x^4)))/(24*a^2*b^4*(a + b
*x^2)^(3/2)) - (d^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(8*b^(9/2))

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Maple [A]
time = 0.14, size = 360, normalized size = 1.41

method	result	size
default	\(d^{4} \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )+4 c \,d^{3} \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+6 c^{2} d^{2} \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+4 c^{3} d \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )+c^{4} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )\)	\(360\)
risch	\(\text {Expression too large to display}\)	\(1486\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^4*(1/4*x^7/b/(b*x^2+a)^(3/2)-7/4*a/b*(1/2*x^5/b/(b*x^2+a)^(3/2)-5/2*a/b*(-1/3*x^3/b/(b*x^2+a)^(3/2)+1/b*(-x/
b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))))+4*c*d^3*(1/2*x^5/b/(b*x^2+a)^(3/2)-5/2*a/b*(-1/3
*x^3/b/(b*x^2+a)^(3/2)+1/b*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2)))))+6*c^2*d^2*(-1/3*x^
3/b/(b*x^2+a)^(3/2)+1/b*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))+4*c^3*d*(-1/2*x/b/(b*x
^2+a)^(3/2)+1/2*a/b*(1/3*x/a/(b*x^2+a)^(3/2)+2/3*x/a^2/(b*x^2+a)^(1/2)))+c^4*(1/3*x/a/(b*x^2+a)^(3/2)+2/3*x/a^
2/(b*x^2+a)^(1/2))

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Maxima [A]
time = 0.29, size = 392, normalized size = 1.54 \begin {gather*} \frac {d^{4} x^{7}}{4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, c d^{3} x^{5}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {7 \, a d^{4} x^{5}}{8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - 2 \, c^{2} d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {10 \, a c d^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b} - \frac {35 \, a^{2} d^{4} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{24 \, b^{2}} + \frac {2 \, c^{4} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c^{4} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {4 \, c^{3} d x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {4 \, c^{3} d x}{3 \, \sqrt {b x^{2} + a} a b} - \frac {2 \, c^{2} d^{2} x}{\sqrt {b x^{2} + a} b^{2}} + \frac {10 \, a c d^{3} x}{3 \, \sqrt {b x^{2} + a} b^{3}} - \frac {35 \, a^{2} d^{4} x}{24 \, \sqrt {b x^{2} + a} b^{4}} + \frac {6 \, c^{2} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} - \frac {10 \, a c d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {7}{2}}} + \frac {35 \, a^{2} d^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*d^4*x^7/((b*x^2 + a)^(3/2)*b) + 2*c*d^3*x^5/((b*x^2 + a)^(3/2)*b) - 7/8*a*d^4*x^5/((b*x^2 + a)^(3/2)*b^2)
- 2*c^2*d^2*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2)) + 10/3*a*c*d^3*x*(3*x^2/((b*x^2 + a)
^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b - 35/24*a^2*d^4*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(
3/2)*b^2))/b^2 + 2/3*c^4*x/(sqrt(b*x^2 + a)*a^2) + 1/3*c^4*x/((b*x^2 + a)^(3/2)*a) - 4/3*c^3*d*x/((b*x^2 + a)^
(3/2)*b) + 4/3*c^3*d*x/(sqrt(b*x^2 + a)*a*b) - 2*c^2*d^2*x/(sqrt(b*x^2 + a)*b^2) + 10/3*a*c*d^3*x/(sqrt(b*x^2
+ a)*b^3) - 35/24*a^2*d^4*x/(sqrt(b*x^2 + a)*b^4) + 6*c^2*d^2*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 10*a*c*d^3*arcs
inh(b*x/sqrt(a*b))/b^(7/2) + 35/8*a^2*d^4*arcsinh(b*x/sqrt(a*b))/b^(9/2)

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Fricas [A]
time = 0.65, size = 684, normalized size = 2.68 \begin {gather*} \left [\frac {3 \, {\left (48 \, a^{4} b^{2} c^{2} d^{2} - 80 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (48 \, a^{2} b^{4} c^{2} d^{2} - 80 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (48 \, a^{3} b^{3} c^{2} d^{2} - 80 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (6 \, a^{2} b^{4} d^{4} x^{7} + 3 \, {\left (16 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 4 \, {\left (4 \, b^{6} c^{4} + 8 \, a b^{5} c^{3} d - 48 \, a^{2} b^{4} c^{2} d^{2} + 80 \, a^{3} b^{3} c d^{3} - 35 \, a^{4} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (8 \, a b^{5} c^{4} - 48 \, a^{3} b^{3} c^{2} d^{2} + 80 \, a^{4} b^{2} c d^{3} - 35 \, a^{5} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}, -\frac {3 \, {\left (48 \, a^{4} b^{2} c^{2} d^{2} - 80 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (48 \, a^{2} b^{4} c^{2} d^{2} - 80 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (48 \, a^{3} b^{3} c^{2} d^{2} - 80 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (6 \, a^{2} b^{4} d^{4} x^{7} + 3 \, {\left (16 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 4 \, {\left (4 \, b^{6} c^{4} + 8 \, a b^{5} c^{3} d - 48 \, a^{2} b^{4} c^{2} d^{2} + 80 \, a^{3} b^{3} c d^{3} - 35 \, a^{4} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (8 \, a b^{5} c^{4} - 48 \, a^{3} b^{3} c^{2} d^{2} + 80 \, a^{4} b^{2} c d^{3} - 35 \, a^{5} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{24 \, {\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(3*(48*a^4*b^2*c^2*d^2 - 80*a^5*b*c*d^3 + 35*a^6*d^4 + (48*a^2*b^4*c^2*d^2 - 80*a^3*b^3*c*d^3 + 35*a^4*b
^2*d^4)*x^4 + 2*(48*a^3*b^3*c^2*d^2 - 80*a^4*b^2*c*d^3 + 35*a^5*b*d^4)*x^2)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^
2 + a)*sqrt(b)*x - a) + 2*(6*a^2*b^4*d^4*x^7 + 3*(16*a^2*b^4*c*d^3 - 7*a^3*b^3*d^4)*x^5 + 4*(4*b^6*c^4 + 8*a*b
^5*c^3*d - 48*a^2*b^4*c^2*d^2 + 80*a^3*b^3*c*d^3 - 35*a^4*b^2*d^4)*x^3 + 3*(8*a*b^5*c^4 - 48*a^3*b^3*c^2*d^2 +
80*a^4*b^2*c*d^3 - 35*a^5*b*d^4)*x)*sqrt(b*x^2 + a))/(a^2*b^7*x^4 + 2*a^3*b^6*x^2 + a^4*b^5), -1/24*(3*(48*a^
4*b^2*c^2*d^2 - 80*a^5*b*c*d^3 + 35*a^6*d^4 + (48*a^2*b^4*c^2*d^2 - 80*a^3*b^3*c*d^3 + 35*a^4*b^2*d^4)*x^4 + 2
*(48*a^3*b^3*c^2*d^2 - 80*a^4*b^2*c*d^3 + 35*a^5*b*d^4)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (6*
a^2*b^4*d^4*x^7 + 3*(16*a^2*b^4*c*d^3 - 7*a^3*b^3*d^4)*x^5 + 4*(4*b^6*c^4 + 8*a*b^5*c^3*d - 48*a^2*b^4*c^2*d^2
+ 80*a^3*b^3*c*d^3 - 35*a^4*b^2*d^4)*x^3 + 3*(8*a*b^5*c^4 - 48*a^3*b^3*c^2*d^2 + 80*a^4*b^2*c*d^3 - 35*a^5*b*
d^4)*x)*sqrt(b*x^2 + a))/(a^2*b^7*x^4 + 2*a^3*b^6*x^2 + a^4*b^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.72, size = 237, normalized size = 0.93 \begin {gather*} \frac {{\left ({\left (3 \, {\left (\frac {2 \, d^{4} x^{2}}{b} + \frac {16 \, a^{2} b^{6} c d^{3} - 7 \, a^{3} b^{5} d^{4}}{a^{2} b^{7}}\right )} x^{2} + \frac {4 \, {\left (4 \, b^{8} c^{4} + 8 \, a b^{7} c^{3} d - 48 \, a^{2} b^{6} c^{2} d^{2} + 80 \, a^{3} b^{5} c d^{3} - 35 \, a^{4} b^{4} d^{4}\right )}}{a^{2} b^{7}}\right )} x^{2} + \frac {3 \, {\left (8 \, a b^{7} c^{4} - 48 \, a^{3} b^{5} c^{2} d^{2} + 80 \, a^{4} b^{4} c d^{3} - 35 \, a^{5} b^{3} d^{4}\right )}}{a^{2} b^{7}}\right )} x}{24 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (48 \, b^{2} c^{2} d^{2} - 80 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((3*(2*d^4*x^2/b + (16*a^2*b^6*c*d^3 - 7*a^3*b^5*d^4)/(a^2*b^7))*x^2 + 4*(4*b^8*c^4 + 8*a*b^7*c^3*d - 48*
a^2*b^6*c^2*d^2 + 80*a^3*b^5*c*d^3 - 35*a^4*b^4*d^4)/(a^2*b^7))*x^2 + 3*(8*a*b^7*c^4 - 48*a^3*b^5*c^2*d^2 + 80
*a^4*b^4*c*d^3 - 35*a^5*b^3*d^4)/(a^2*b^7))*x/(b*x^2 + a)^(3/2) - 1/8*(48*b^2*c^2*d^2 - 80*a*b*c*d^3 + 35*a^2*
d^4)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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