3.672 \(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=457 \[ -\frac {16 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {16 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (33 a e^2+32 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {8 c \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{63 e^5}-\frac {20 c \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}} \]

[Out]

-2*(c*x^2+a)^(5/2)/e/(e*x+d)^(1/2)-20/63*c*(-7*e*x+8*d)*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/e^3-8/63*c*(d*(33*a*e^2+
32*c*d^2)-3*e*(7*a*e^2+8*c*d^2)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/e^5-16/63*(21*a^2*e^4+57*a*c*d^2*e^2+32*c^2*d
^4)*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2
)*c^(1/2)*(e*x+d)^(1/2)*(c*x^2/a+1)^(1/2)/e^6/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)
+16/63*d*(a*e^2+c*d^2)*(33*a*e^2+32*c*d^2)*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+
d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*c^(1/2)*(c*x^2/a+1)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^
(1/2)/e^6/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]  time = 0.46, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {733, 815, 844, 719, 424, 419} \[ -\frac {16 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {8 c \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{63 e^5}+\frac {16 \sqrt {-a} \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (33 a e^2+32 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {20 c \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*c*Sqrt[d + e*x]*(d*(32*c*d^2 + 33*a*e^2) - 3*e*(8*c*d^2 + 7*a*e^2)*x)*Sqrt[a + c*x^2])/(63*e^5) - (20*c*(8
*d - 7*e*x)*Sqrt[d + e*x]*(a + c*x^2)^(3/2))/(63*e^3) - (2*(a + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) - (16*Sqrt[-a]
*Sqrt[c]*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1
- (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(63*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt
[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (16*Sqrt[-a]*Sqrt[c]*d*(c*d^2 + a*e^2)*(32*c*d^2 + 33*a*e^2)*Sqrt[(Sqr
t[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/
Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(63*e^6*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 719

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

Rule 733

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

Rule 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx &=-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {(10 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{e}\\ &=-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {40 \int \frac {\left (-\frac {1}{2} a c d e+\frac {1}{2} c \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{\sqrt {d+e x}} \, dx}{21 e^3}\\ &=-\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {32 \int \frac {-a c^2 d e \left (2 c d^2+3 a e^2\right )+\frac {1}{4} c^2 \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{63 c e^5}\\ &=-\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {\left (8 c d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{63 e^6}+\frac {\left (8 c \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{63 e^6}\\ &=-\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {\left (16 a \sqrt {c} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{63 \sqrt {-a} e^6 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (16 a \sqrt {c} d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{63 \sqrt {-a} e^6 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {16 \sqrt {-a} \sqrt {c} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} \sqrt {c} d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 3.86, size = 684, normalized size = 1.50 \[ \frac {\sqrt {d+e x} \left (-\frac {2 \left (a+c x^2\right ) \left (63 a^2 e^4+2 a c e^2 \left (106 d^2+29 d e x-14 e^2 x^2\right )+c^2 \left (128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right )\right )}{e^5 (d+e x)}+\frac {16 \left (e^2 \left (a+c x^2\right ) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )-\sqrt {a} \sqrt {c} e (d+e x)^{3/2} \left (12 i a^{3/2} \sqrt {c} d e^3+21 a^2 e^4+8 i \sqrt {a} c^{3/2} d^3 e+57 a c d^2 e^2+32 c^2 d^4\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {c} (d+e x)^{3/2} \left (57 a^{3/2} c d^2 e^3+21 a^{5/2} e^5-21 i a^2 \sqrt {c} d e^4-57 i a c^{3/2} d^3 e^2+32 \sqrt {a} c^2 d^4 e-32 i c^{5/2} d^5\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^7 (d+e x) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{63 \sqrt {a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*((-2*(a + c*x^2)*(63*a^2*e^4 + 2*a*c*e^2*(106*d^2 + 29*d*e*x - 14*e^2*x^2) + c^2*(128*d^4 + 32*
d^3*e*x - 16*d^2*e^2*x^2 + 10*d*e^3*x^3 - 7*e^4*x^4)))/(e^5*(d + e*x)) + (16*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt
[c]]*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4)*(a + c*x^2) + Sqrt[c]*((-32*I)*c^(5/2)*d^5 + 32*Sqrt[a]*c^2*d^
4*e - (57*I)*a*c^(3/2)*d^3*e^2 + 57*a^(3/2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*a^(5/2)*e^5)*Sqrt[(e*((I
*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[
I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)
] - Sqrt[a]*Sqrt[c]*e*(32*c^2*d^4 + (8*I)*Sqrt[a]*c^(3/2)*d^3*e + 57*a*c*d^2*e^2 + (12*I)*a^(3/2)*Sqrt[c]*d*e^
3 + 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]
*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e
)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(e^7*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(63*Sqrt[a + c*x^2])

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fricas [F]  time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)*sqrt(e*x + d)/(e^2*x^2 + 2*d*e*x + d^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 + a)^(5/2)/(e*x + d)^(3/2), x)

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maple [B]  time = 0.12, size = 1736, normalized size = 3.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/63*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(-7*c^3*e^6*x^6+10*c^3*d*e^5*x^5+168*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1
/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)*Elli
pticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^3*e^6+62
4*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^2*
c*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-
a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+712*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*
c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c^2*d^4*e^2*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^
(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+256*EllipticE((-(e*x+d
)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*c^3*d^6*(-(e*x+d)/(-c*d+
(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(
1/2)*e)*e)^(1/2)-168*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/
2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c
*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^3*e^6-360*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)
,(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^2*c*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*(
(-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-264*(-(e*
x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c
*d+(-a*c)^(1/2)*e)*e)^(1/2)*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-
a*c)^(1/2)*e))^(1/2))*(-a*c)^(1/2)*a^2*d*e^5-192*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-
a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c^2*d^4*e^2*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c
)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-520*EllipticF((-(e*x
+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c*d^3*e^3*(-a*c)^(1/
2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1
/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-256*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2
)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*c^2*d^5*e*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c
)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-35*a*c^2*e^6*x^4-16*
c^3*d^2*e^4*x^4+68*a*c^2*d*e^5*x^3+32*c^3*d^3*e^3*x^3+35*a^2*c*e^6*x^2+196*a*c^2*d^2*e^4*x^2+128*c^3*d^4*e^2*x
^2+58*a^2*c*d*e^5*x+32*a*c^2*d^3*e^3*x+63*e^6*a^3+212*a^2*c*d^2*e^4+128*a*c^2*d^4*e^2)/e^7/(c*e*x^3+c*d*x^2+a*
e*x+a*d)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x^2 + a)^(5/2)/(e*x + d)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**(3/2), x)

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