\(\int \frac {(A+B x) (a+b x^2)^{3/2}}{(d x)^{3/2}} \, dx\) [1357]

Optimal result

Integrand size = 24, antiderivative size = 334 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{3/2}} \, dx=\frac {24 a A \sqrt {b} \sqrt {d x} \sqrt {a+b x^2}}{5 d^2 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {4 \sqrt {d x} (5 a B+21 A b x) \sqrt {a+b x^2}}{35 d^2}-\frac {2 (7 A-B x) \left (a+b x^2\right )^{3/2}}{7 d \sqrt {d x}}-\frac {24 a^{5/4} A \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{2}\right )}{5 d^{3/2} \sqrt {a+b x^2}}+\frac {4 a^{5/4} \left (21 A \sqrt {b}+5 \sqrt {a} B\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{2}\right )}{35 \sqrt [4]{b} d^{3/2} \sqrt {a+b x^2}} \] Output:

24/5*a*A*b^(1/2)*(d*x)^(1/2)*(b*x^2+a)^(1/2)/d^2/(a^(1/2)+b^(1/2)*x)+4/35* 
(d*x)^(1/2)*(21*A*b*x+5*B*a)*(b*x^2+a)^(1/2)/d^2-2/7*(-B*x+7*A)*(b*x^2+a)^ 
(3/2)/d/(d*x)^(1/2)-24/5*a^(5/4)*A*b^(1/4)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/ 
(a^(1/2)+b^(1/2)*x)^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*(d*x)^(1/2)/a^ 
(1/4)/d^(1/2))),1/2*2^(1/2))/d^(3/2)/(b*x^2+a)^(1/2)+4/35*a^(5/4)*(21*A*b^ 
(1/2)+5*a^(1/2)*B)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^( 
1/2)*InverseJacobiAM(2*arctan(b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2)),1/2*2^( 
1/2))/b^(1/4)/d^(3/2)/(b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.24 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{3/2}} \, dx=\frac {2 a x \sqrt {a+b x^2} \left (-A \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{4},\frac {3}{4},-\frac {b x^2}{a}\right )+B x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{(d x)^{3/2} \sqrt {1+\frac {b x^2}{a}}} \] Input:

Integrate[((A + B*x)*(a + b*x^2)^(3/2))/(d*x)^(3/2),x]
 

Output:

(2*a*x*Sqrt[a + b*x^2]*(-(A*Hypergeometric2F1[-3/2, -1/4, 3/4, -((b*x^2)/a 
)]) + B*x*Hypergeometric2F1[-3/2, 1/4, 5/4, -((b*x^2)/a)]))/((d*x)^(3/2)*S 
qrt[1 + (b*x^2)/a])
 

Rubi [A] (warning: unable to verify)

Time = 0.80 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {547, 27, 548, 27, 556, 555, 1512, 27, 761, 1510}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} (A+B x)}{(d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 547

\(\displaystyle -\frac {12 \int -\frac {(a B+7 A b x) \sqrt {b x^2+a}}{2 \sqrt {d x}}dx}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 \int \frac {(a B+7 A b x) \sqrt {b x^2+a}}{\sqrt {d x}}dx}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

\(\Big \downarrow \) 548

\(\displaystyle \frac {6 \left (\frac {4}{15} a \int \frac {5 a B+21 A b x}{2 \sqrt {d x} \sqrt {b x^2+a}}dx+\frac {2 \sqrt {d x} \sqrt {a+b x^2} (5 a B+21 A b x)}{15 d}\right )}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 \left (\frac {2}{15} a \int \frac {5 a B+21 A b x}{\sqrt {d x} \sqrt {b x^2+a}}dx+\frac {2 \sqrt {d x} \sqrt {a+b x^2} (5 a B+21 A b x)}{15 d}\right )}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

\(\Big \downarrow \) 556

\(\displaystyle \frac {6 \left (\frac {2 a \sqrt {x} \int \frac {5 a B+21 A b x}{\sqrt {x} \sqrt {b x^2+a}}dx}{15 \sqrt {d x}}+\frac {2 \sqrt {d x} \sqrt {a+b x^2} (5 a B+21 A b x)}{15 d}\right )}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

\(\Big \downarrow \) 555

\(\displaystyle \frac {6 \left (\frac {4 a \sqrt {x} \int \frac {5 a B+21 A b x}{\sqrt {b x^2+a}}d\sqrt {x}}{15 \sqrt {d x}}+\frac {2 \sqrt {d x} \sqrt {a+b x^2} (5 a B+21 A b x)}{15 d}\right )}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

\(\Big \downarrow \) 1512

\(\displaystyle \frac {6 \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+21 A \sqrt {b}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-21 \sqrt {a} A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}\right )}{15 \sqrt {d x}}+\frac {2 \sqrt {d x} \sqrt {a+b x^2} (5 a B+21 A b x)}{15 d}\right )}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+21 A \sqrt {b}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-21 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{15 \sqrt {d x}}+\frac {2 \sqrt {d x} \sqrt {a+b x^2} (5 a B+21 A b x)}{15 d}\right )}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {6 \left (\frac {4 a \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (5 \sqrt {a} B+21 A \sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}-21 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{15 \sqrt {d x}}+\frac {2 \sqrt {d x} \sqrt {a+b x^2} (5 a B+21 A b x)}{15 d}\right )}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {6 \left (\frac {4 a \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (5 \sqrt {a} B+21 A \sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}-21 A \sqrt {b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{15 \sqrt {d x}}+\frac {2 \sqrt {d x} \sqrt {a+b x^2} (5 a B+21 A b x)}{15 d}\right )}{7 d}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A-B x)}{7 d \sqrt {d x}}\)

Input:

Int[((A + B*x)*(a + b*x^2)^(3/2))/(d*x)^(3/2),x]
 

Output:

(-2*(7*A - B*x)*(a + b*x^2)^(3/2))/(7*d*Sqrt[d*x]) + (6*((2*Sqrt[d*x]*(5*a 
*B + 21*A*b*x)*Sqrt[a + b*x^2])/(15*d) + (4*a*Sqrt[x]*(-21*A*Sqrt[b]*(-((S 
qrt[x]*Sqrt[a + b*x^2])/(Sqrt[a] + Sqrt[b]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[ 
b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4 
)*Sqrt[x])/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^2])) + (a^(1/4)*(21*A*Sqr 
t[b] + 5*Sqrt[a]*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt 
[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*b^(1/4)* 
Sqrt[a + b*x^2])))/(15*Sqrt[d*x])))/(7*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 547

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

rule 548

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[(e*x)^(m + 1)*(c*(m + 2*p + 2) + d*(m + 2*p + 1)*x)*((a + b*x^ 
2)^p/(e*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*a*(p/((m + 2*p + 1)*(m + 
 2*p + 2)))   Int[(e*x)^m*(a + b*x^2)^(p - 1)*(c*(m + 2*p + 2) + d*(m + 2*p 
 + 1)*x), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[ 
p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 555

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> 
Simp[2   Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]], x] /; Free 
Q[{a, c, f, g}, x]
 

rule 556

Int[((c_) + (d_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symb 
ol] :> Simp[Sqrt[x]/Sqrt[e*x]   Int[(c + d*x)/(Sqrt[x]*Sqrt[a + b*x^2]), x] 
, x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

rule 1512

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q 
 Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c 
, d, e}, x] && PosQ[c/a]
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {\frac {24 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b}{5}-\frac {12 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b}{5}+\frac {4 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-a b}\, a^{2}}{7}+\frac {2 B \,b^{3} x^{5}}{7}+\frac {2 A \,b^{3} x^{4}}{5}+\frac {8 B a \,b^{2} x^{3}}{7}-\frac {8 A a \,b^{2} x^{2}}{5}+\frac {6 B \,a^{2} b x}{7}-2 A \,a^{2} b}{\sqrt {b \,x^{2}+a}\, b d \sqrt {d x}}\)	\(340\)
risch	\(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-5 B b \,x^{3}-7 A b \,x^{2}-15 B a x +35 A a \right )}{35 d \sqrt {d x}}+\frac {4 a \left (\frac {5 B a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b d \,x^{3}+a d x}}+\frac {21 A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b d \,x^{3}+a d x}}\right ) \sqrt {d \left (b \,x^{2}+a \right ) x}}{35 d \sqrt {d x}\, \sqrt {b \,x^{2}+a}}\)	\(348\)
elliptic	\(\frac {\sqrt {d \left (b \,x^{2}+a \right ) x}\, \left (-\frac {2 \left (b d \,x^{2}+a d \right ) a A}{d^{2} \sqrt {x \left (b d \,x^{2}+a d \right )}}+\frac {2 B b \,x^{2} \sqrt {b d \,x^{3}+a d x}}{7 d^{2}}+\frac {2 A b x \sqrt {b d \,x^{3}+a d x}}{5 d^{2}}+\frac {6 a B \sqrt {b d \,x^{3}+a d x}}{7 d^{2}}+\frac {4 a^{2} B \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{7 d b \sqrt {b d \,x^{3}+a d x}}+\frac {12 a A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 d \sqrt {b d \,x^{3}+a d x}}\right )}{\sqrt {d x}\, \sqrt {b \,x^{2}+a}}\)	\(403\)

Input:

int((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2/35*(84*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(- 
a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)*EllipticE(((b*x+(-a*b)^(1/2) 
)/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^2*b-42*A*((b*x+(-a*b)^(1/2))/ 
(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1 
/2)*b*x)^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/ 
2))*2^(1/2)*a^2*b+10*B*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a* 
b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)*EllipticF(((b*x+ 
(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*(-a*b)^(1/2)*a^2+5* 
B*b^3*x^5+7*A*b^3*x^4+20*B*a*b^2*x^3-28*A*a*b^2*x^2+15*B*a^2*b*x-35*A*a^2* 
b)/(b*x^2+a)^(1/2)/b/d/(d*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.32 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{3/2}} \, dx=\frac {2 \, {\left (20 \, \sqrt {b d} B a^{2} x {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - 84 \, \sqrt {b d} A a b x {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (5 \, B b^{2} x^{3} + 7 \, A b^{2} x^{2} + 15 \, B a b x - 35 \, A a b\right )} \sqrt {b x^{2} + a} \sqrt {d x}\right )}}{35 \, b d^{2} x} \] Input:

integrate((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(3/2),x, algorithm="fricas")
 

Output:

2/35*(20*sqrt(b*d)*B*a^2*x*weierstrassPInverse(-4*a/b, 0, x) - 84*sqrt(b*d 
)*A*a*b*x*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) + 
(5*B*b^2*x^3 + 7*A*b^2*x^2 + 15*B*a*b*x - 35*A*a*b)*sqrt(b*x^2 + a)*sqrt(d 
*x))/(b*d^2*x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.70 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.60 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{3/2}} \, dx=\frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 d^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {A \sqrt {a} b x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 d^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B a^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 d^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt {a} b x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 d^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} \] Input:

integrate((B*x+A)*(b*x**2+a)**(3/2)/(d*x)**(3/2),x)
 

Output:

A*a**(3/2)*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), b*x**2*exp_polar(I*pi)/ 
a)/(2*d**(3/2)*sqrt(x)*gamma(3/4)) + A*sqrt(a)*b*x**(3/2)*gamma(3/4)*hyper 
((-1/2, 3/4), (7/4,), b*x**2*exp_polar(I*pi)/a)/(2*d**(3/2)*gamma(7/4)) + 
B*a**(3/2)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x**2*exp_polar( 
I*pi)/a)/(2*d**(3/2)*gamma(5/4)) + B*sqrt(a)*b*x**(5/2)*gamma(5/4)*hyper(( 
-1/2, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/a)/(2*d**(3/2)*gamma(9/4))
 

Maxima [F]

\[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)*(B*x + A)/(d*x)^(3/2), x)
 

Giac [F]

\[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(3/2)*(B*x + A)/(d*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(A + B*x))/(d*x)^(3/2),x)
 

Output:

int(((a + b*x^2)^(3/2)*(A + B*x))/(d*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{3/2}} \, dx=\frac {2 \sqrt {d}\, \left (49 \sqrt {b \,x^{2}+a}\, a^{2}+7 \sqrt {b \,x^{2}+a}\, a b \,x^{2}+15 \sqrt {b \,x^{2}+a}\, a b x +5 \sqrt {b \,x^{2}+a}\, b^{2} x^{3}+42 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{4}+a \,x^{2}}d x \right ) a^{3}+10 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{3}+a x}d x \right ) a^{2} b \right )}{35 \sqrt {x}\, d^{2}} \] Input:

int((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(3/2),x)
 

Output:

(2*sqrt(d)*(49*sqrt(a + b*x**2)*a**2 + 7*sqrt(a + b*x**2)*a*b*x**2 + 15*sq 
rt(a + b*x**2)*a*b*x + 5*sqrt(a + b*x**2)*b**2*x**3 + 42*sqrt(x)*int((sqrt 
(x)*sqrt(a + b*x**2))/(a*x**2 + b*x**4),x)*a**3 + 10*sqrt(x)*int((sqrt(x)* 
sqrt(a + b*x**2))/(a*x + b*x**3),x)*a**2*b))/(35*sqrt(x)*d**2)