Optimal. Leaf size=220 \[ \frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{5 d^3 \sqrt{a+b x}}+\frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}-\frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.234948, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {47, 50, 63, 307, 224, 221, 1200, 1199, 424} \[ \frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}+\frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}-\frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 63
Rule 307
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{(c+d x)^{5/4}} \, dx &=-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}}+\frac{(6 b) \int \frac{\sqrt{a+b x}}{\sqrt [4]{c+d x}} \, dx}{d}\\ &=-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}}+\frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}-\frac{(12 b (b c-a d)) \int \frac{1}{\sqrt{a+b x} \sqrt [4]{c+d x}} \, dx}{5 d^2}\\ &=-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}}+\frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}-\frac{(48 b (b c-a d)) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^3}\\ &=-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}}+\frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}+\frac{\left (48 \sqrt{b} (b c-a d)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^3}-\frac{\left (48 \sqrt{b} (b c-a d)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^3}\\ &=-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}}+\frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}+\frac{\left (48 \sqrt{b} (b c-a d)^{3/2} \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^3 \sqrt{a+b x}}-\frac{\left (48 \sqrt{b} (b c-a d)^{3/2} \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^3 \sqrt{a+b x}}\\ &=-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}}+\frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}+\frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}-\frac{\left (48 \sqrt{b} (b c-a d)^{3/2} \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}}{\sqrt{1-\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{5 d^3 \sqrt{a+b x}}\\ &=-\frac{4 (a+b x)^{3/2}}{d \sqrt [4]{c+d x}}+\frac{24 b \sqrt{a+b x} (c+d x)^{3/4}}{5 d^2}-\frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}+\frac{48 \sqrt [4]{b} (b c-a d)^{7/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{5 d^3 \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0517911, size = 73, normalized size = 0.33 \[ \frac{2 (a+b x)^{5/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4} \, _2F_1\left (\frac{5}{4},\frac{5}{2};\frac{7}{2};\frac{d (a+b x)}{a d-b c}\right )}{5 b (c+d x)^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{3}{4}}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{3}{2}}}{\left (c + d x\right )^{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{{\left (d x + c\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]